Theorem itg2addnclem 35522

Description: An alternate expression for the ∫₂ integral that includes an arbitrarily small but strictly positive "buffer zone" wherever the simple function is nonzero. (Contributed by Brendan Leahy, 10-Oct-2017.) (Revised by Brendan Leahy, 10-Mar-2018.)

Hypothesis

Ref	Expression
itg2addnclem.1	⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}

Assertion

Ref	Expression
itg2addnclem	⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < ))

Distinct variable group:   𝑥,𝑦,𝑧,𝑔,𝐹

Allowed substitution hints:   𝐿(𝑥,𝑦,𝑧,𝑔)

Proof of Theorem itg2addnclem

Dummy variables 𝑠 𝑢 𝑓 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2734	. . 3 ⊢ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} = {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}
2	1	itg2val 24598	. 2 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
3		itg2addnclem.1	. . . 4 ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}
4	3	supeq1i 9052	. . 3 ⊢ sup(𝐿, ℝ*, < ) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )
5		xrltso 12714	. . . . 5 ⊢ < Or ℝ*
6	5	a1i 11	. . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → < Or ℝ*)
7		simprr 773	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 = (∫1‘𝑓))
8		itg1cl 24554	. . . . . . . . . 10 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘𝑓) ∈ ℝ)
9	8	rexrd 10866	. . . . . . . . 9 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘𝑓) ∈ ℝ*)
10	9	adantr 484	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ*)
11	7, 10	eqeltrd 2834	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 ∈ ℝ*)
12	11	rexlimiva 3193	. . . . . 6 ⊢ (∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → 𝑥 ∈ ℝ*)
13	12	abssi 3973	. . . . 5 ⊢ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ*
14		supxrcl 12888	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ* → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ∈ ℝ*)
15	13, 14	mp1i 13	. . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ∈ ℝ*)
16		fveq1 6705	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑧) = (𝑓‘𝑧))
17	16	eqeq1d 2736	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((𝑔‘𝑧) = 0 ↔ (𝑓‘𝑧) = 0))
18	16	oveq1d 7217	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((𝑔‘𝑧) + 𝑦) = ((𝑓‘𝑧) + 𝑦))
19	17, 18	ifbieq2d 4455	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦)) = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)))
20	19	mpteq2dv 5140	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))))
21	20	breq1d 5053	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘r ≤ 𝐹))
22	21	rexbidv 3209	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘r ≤ 𝐹))
23		fveq2 6706	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (∫1‘𝑔) = (∫1‘𝑓))
24	23	eqeq2d 2745	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (𝑥 = (∫1‘𝑔) ↔ 𝑥 = (∫1‘𝑓)))
25	22, 24	anbi12d 634	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))))
26	25	cbvrexvw 3352	. . . . . . . . 9 ⊢ (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑓 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)))
27		breq2 5047	. . . . . . . . . . . . . . . . 17 ⊢ (0 = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) → ((𝑓‘𝑧) ≤ 0 ↔ (𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))))
28		breq2 5047	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓‘𝑧) + 𝑦) = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) → ((𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑦) ↔ (𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))))
29		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑧) = 0 → (𝑓‘𝑧) = 0)
30		0le0 11914	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ≤ 0
31	29, 30	eqbrtrdi 5082	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑧) = 0 → (𝑓‘𝑧) ≤ 0)
32	31	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) → (𝑓‘𝑧) ≤ 0)
33		rpge0 12582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
34	33	ad2antlr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 0 ≤ 𝑦)
35		i1ff 24545	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ dom ∫1 → 𝑓:ℝ⟶ℝ)
36	35	ffvelrnda 6893	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
37	36	adantlr 715	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
38		rpre 12577	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
39	38	ad2antlr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
40	37, 39	addge01d 11403	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (0 ≤ 𝑦 ↔ (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑦)))
41	34, 40	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑦))
42	41	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓‘𝑧) = 0) → (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑦))
43	27, 28, 32, 42	ifbothda 4467	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)))
44	43	adantlll 718	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)))
45	35	ad2antlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝑓:ℝ⟶ℝ)
46	45	ffvelrnda 6893	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
47	46	rexrd 10866	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ*)
48		0re 10818	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
49	38	ad2antlr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
50	46, 49	readdcld 10845	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 𝑦) ∈ ℝ)
51		ifcl 4474	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ ((𝑓‘𝑧) + 𝑦) ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∈ ℝ)
52	48, 50, 51	sylancr 590	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∈ ℝ)
53	52	rexrd 10866	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∈ ℝ*)
54		iccssxr 13001	. . . . . . . . . . . . . . . . . . 19 ⊢ (0[,]+∞) ⊆ ℝ*
55		fss 6551	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐹:ℝ⟶ℝ*)
56	54, 55	mpan2 691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹:ℝ⟶ℝ*)
57	56	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝐹:ℝ⟶ℝ*)
58	57	ffvelrnda 6893	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ ℝ*)
59		xrletr 12731	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑧) ∈ ℝ* ∧ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∈ ℝ* ∧ (𝐹‘𝑧) ∈ ℝ*) → (((𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∧ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
60	47, 53, 58, 59	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∧ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
61	44, 60	mpand 695	. . . . . . . . . . . . . 14 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧) → (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
62	61	ralimdva 3093	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (∀𝑧 ∈ ℝ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧) → ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
63		reex 10803	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
64	63	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ℝ ∈ V)
65		eqidd 2735	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))))
66		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹:ℝ⟶(0[,]+∞))
67	66	feqmptd 6769	. . . . . . . . . . . . . . 15 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹‘𝑧)))
68	67	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹‘𝑧)))
69	64, 52, 58, 65, 68	ofrfval2 7478	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧)))
70	35	feqmptd 6769	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓‘𝑧)))
71	70	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓‘𝑧)))
72	64, 46, 58, 71, 68	ofrfval2 7478	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (𝑓 ∘r ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
73	62, 69, 72	3imtr4d 297	. . . . . . . . . . . 12 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘r ≤ 𝐹 → 𝑓 ∘r ≤ 𝐹))
74	73	rexlimdva 3196	. . . . . . . . . . 11 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘r ≤ 𝐹 → 𝑓 ∘r ≤ 𝐹))
75	74	anim1d 614	. . . . . . . . . 10 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → (𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))))
76	75	reximdva 3186	. . . . . . . . 9 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∃𝑓 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))))
77	26, 76	syl5bi 245	. . . . . . . 8 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) → ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))))
78	77	ss2abdv 3967	. . . . . . 7 ⊢ (𝐹:ℝ⟶(0[,]+∞) → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⊆ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))})
79	78	sseld 3890	. . . . . 6 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} → 𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}))
80		simp3r 1204	. . . . . . . . . . 11 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 = (∫1‘𝑓))
81	9	3ad2ant2 1136	. . . . . . . . . . 11 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ*)
82	80, 81	eqeltrd 2834	. . . . . . . . . 10 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 ∈ ℝ*)
83	82	rexlimdv3a 3198	. . . . . . . . 9 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → 𝑥 ∈ ℝ*))
84	83	abssdv 3972	. . . . . . . 8 ⊢ (𝐹:ℝ⟶(0[,]+∞) → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ*)
85		xrsupss 12882	. . . . . . . 8 ⊢ ({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ* → ∃𝑎 ∈ ℝ* (∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ* (𝑏 < 𝑎 → ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠)))
86	84, 85	syl 17	. . . . . . 7 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∃𝑎 ∈ ℝ* (∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ* (𝑏 < 𝑎 → ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠)))
87	6, 86	supub 9064	. . . . . 6 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) < 𝑏))
88	79, 87	syld 47	. . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) < 𝑏))
89	88	imp 410	. . . 4 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}) → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) < 𝑏)
90		supxrlub 12898	. . . . . . . 8 ⊢ (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ* ∧ 𝑏 ∈ ℝ*) → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠))
91	13, 90	mpan 690	. . . . . . 7 ⊢ (𝑏 ∈ ℝ* → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠))
92	91	adantl 485	. . . . . 6 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠))
93		simprrr 782	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → 𝑠 = (∫1‘𝑓))
94	93	breq2d 5055	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑏 < 𝑠 ↔ 𝑏 < (∫1‘𝑓)))
95		simplll 775	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → 𝐹:ℝ⟶(0[,]+∞))
96		i1f0 24556	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × {0}) ∈ dom ∫1
97		2rp 12574	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ+
98	97	ne0ii 4242	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ+ ≠ ∅
99		ffvelrn 6891	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ (0[,]+∞))
100		elxrge0 13028	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑧) ∈ (0[,]+∞) ↔ ((𝐹‘𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑧)))
101	99, 100	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑧)))
102	101	simprd 499	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹‘𝑧))
103	102	ralrimiva 3098	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∀𝑧 ∈ ℝ 0 ≤ (𝐹‘𝑧))
104	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ℝ ∈ V)
105		c0ex 10810	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ V
106	105	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → 0 ∈ V)
107		eqidd 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
108	104, 106, 99, 107, 67	ofrfval2 7478	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ((𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐹‘𝑧)))
109	103, 108	mpbird 260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
110	109	ralrimivw 3099	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∀𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
111		r19.2z 4396	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℝ+ ≠ ∅ ∧ ∀𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
112	98, 110, 111	sylancr 590	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
113		fveq2 6706	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = (ℝ × {0}) → (∫1‘𝑔) = (∫1‘(ℝ × {0})))
114		itg10 24557	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∫1‘(ℝ × {0})) = 0
115	113, 114	eqtr2di 2791	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = (ℝ × {0}) → 0 = (∫1‘𝑔))
116	115	biantrud 535	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = (ℝ × {0}) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑔))))
117		fveq1 6705	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔 = (ℝ × {0}) → (𝑔‘𝑧) = ((ℝ × {0})‘𝑧))
118	105	fvconst2 7008	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
119	117, 118	sylan9eq 2794	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = 0)
120		iftrue 4435	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔‘𝑧) = 0 → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦)) = 0)
121	119, 120	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦)) = 0)
122	121	mpteq2dva 5139	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ 0))
123	122	breq1d 5053	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹))
124	123	rexbidv 3209	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = (ℝ × {0}) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹))
125	116, 124	bitr3d 284	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = (ℝ × {0}) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑔)) ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹))
126	125	rspcev 3530	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑔)))
127	96, 112, 126	sylancr 590	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑔)))
128		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = -∞ → 𝑏 = -∞)
129		mnflt 12698	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ ℝ → -∞ < 0)
130	48, 129	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = -∞ → -∞ < 0)
131	128, 130	eqbrtrd 5065	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = -∞ → 𝑏 < 0)
132		eqeq1 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 0 → (𝑎 = (∫1‘𝑔) ↔ 0 = (∫1‘𝑔)))
133	132	anbi2d 632	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 0 → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑔))))
134	133	rexbidv 3209	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 0 → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑔))))
135		breq2 5047	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 0 → (𝑏 < 𝑎 ↔ 𝑏 < 0))
136	134, 135	anbi12d 634	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 0 → ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎) ↔ (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑔)) ∧ 𝑏 < 0)))
137	105, 136	spcev 3514	. . . . . . . . . . . . . . . . . 18 ⊢ ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑔)) ∧ 𝑏 < 0) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
138	127, 131, 137	syl2an 599	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 = -∞) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
139	95, 138	sylan 583	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 = -∞) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
140		simp-4r 784	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ∈ ℝ*)
141	8	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ)
142	141	ad3antlr 731	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → (∫1‘𝑓) ∈ ℝ)
143		simpllr 776	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → 𝑏 ∈ ℝ*)
144		ngtmnft 12739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 ∈ ℝ* → (𝑏 = -∞ ↔ ¬ -∞ < 𝑏))
145	144	biimprd 251	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 ∈ ℝ* → (¬ -∞ < 𝑏 → 𝑏 = -∞))
146	145	necon1ad 2952	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℝ* → (𝑏 ≠ -∞ → -∞ < 𝑏))
147	146	imp 410	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℝ* ∧ 𝑏 ≠ -∞) → -∞ < 𝑏)
148	143, 147	sylan 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → -∞ < 𝑏)
149		simpr 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → 𝑏 ∈ ℝ*)
150	9	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ*)
151	149, 150	anim12i 616	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑏 ∈ ℝ* ∧ (∫1‘𝑓) ∈ ℝ*))
152		xrltle 12722	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℝ* ∧ (∫1‘𝑓) ∈ ℝ*) → (𝑏 < (∫1‘𝑓) → 𝑏 ≤ (∫1‘𝑓)))
153	152	imp 410	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑏 ∈ ℝ* ∧ (∫1‘𝑓) ∈ ℝ*) ∧ 𝑏 < (∫1‘𝑓)) → 𝑏 ≤ (∫1‘𝑓))
154	151, 153	sylan 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → 𝑏 ≤ (∫1‘𝑓))
155	154	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ≤ (∫1‘𝑓))
156		xrre 12742	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑏 ∈ ℝ* ∧ (∫1‘𝑓) ∈ ℝ) ∧ (-∞ < 𝑏 ∧ 𝑏 ≤ (∫1‘𝑓))) → 𝑏 ∈ ℝ)
157	140, 142, 148, 155, 156	syl22anc 839	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ∈ ℝ)
158	127	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑔)))
159		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑏 < (∫1‘𝑓))
160		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑓 ∈ dom ∫1)
161		simpl 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑓 ∈ dom ∫1)
162		cnvimass 5938	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ dom 𝑓
163	162, 35	fssdm 6554	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 ∈ dom ∫1 → (◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
164	163	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
165		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0)
166		fdm 6543	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓:ℝ⟶ℝ → dom 𝑓 = ℝ)
167	166	eqcomd 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓:ℝ⟶ℝ → ℝ = dom 𝑓)
168		ffun 6537	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑓:ℝ⟶ℝ → Fun 𝑓)
169		difpreima 6874	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (Fun 𝑓 → (◡𝑓 “ (ran 𝑓 ∖ {0})) = ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})))
170	168, 169	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓:ℝ⟶ℝ → (◡𝑓 “ (ran 𝑓 ∖ {0})) = ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})))
171		cnvimarndm 5939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (◡𝑓 “ ran 𝑓) = dom 𝑓
172	171	difeq1i 4023	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})) = (dom 𝑓 ∖ (◡𝑓 “ {0}))
173	170, 172	eqtrdi 2790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓:ℝ⟶ℝ → (◡𝑓 “ (ran 𝑓 ∖ {0})) = (dom 𝑓 ∖ (◡𝑓 “ {0})))
174	167, 173	difeq12d 4028	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓:ℝ⟶ℝ → (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) = (dom 𝑓 ∖ (dom 𝑓 ∖ (◡𝑓 “ {0}))))
175		cnvimass 5938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (◡𝑓 “ {0}) ⊆ dom 𝑓
176		dfss4 4163	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((◡𝑓 “ {0}) ⊆ dom 𝑓 ↔ (dom 𝑓 ∖ (dom 𝑓 ∖ (◡𝑓 “ {0}))) = (◡𝑓 “ {0}))
177	175, 176	mpbi 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (dom 𝑓 ∖ (dom 𝑓 ∖ (◡𝑓 “ {0}))) = (◡𝑓 “ {0})
178	174, 177	eqtrdi 2790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓:ℝ⟶ℝ → (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) = (◡𝑓 “ {0}))
179	178	eleq2d 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ 𝑧 ∈ (◡𝑓 “ {0})))
180		ffn 6534	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓:ℝ⟶ℝ → 𝑓 Fn ℝ)
181		fniniseg 6869	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 Fn ℝ → (𝑧 ∈ (◡𝑓 “ {0}) ↔ (𝑧 ∈ ℝ ∧ (𝑓‘𝑧) = 0)))
182		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑧 ∈ ℝ ∧ (𝑓‘𝑧) = 0) → (𝑓‘𝑧) = 0)
183	181, 182	syl6bi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 Fn ℝ → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
184	180, 183	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
185	179, 184	sylbid 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓‘𝑧) = 0))
186	35, 185	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 ∈ dom ∫1 → (𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓‘𝑧) = 0))
187	186	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (𝑓‘𝑧) = 0)
188	187	adantlr 715	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) ∧ 𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (𝑓‘𝑧) = 0)
189	161, 164, 165, 188	itg10a 24580	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (∫1‘𝑓) = 0)
190	160, 189	sylan 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (∫1‘𝑓) = 0)
191	159, 190	breqtrd 5069	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑏 < 0)
192	158, 191, 137	syl2anc 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
193		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → 𝑓 ∈ dom ∫1)
194		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
195	193, 194	anim12i 616	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → (𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ))
196	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ℝ ∈ V)
197		fvex 6719	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓‘𝑢) ∈ V
198	197	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑓‘𝑢) ∈ V)
199		ovex 7235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V
200	199, 105	ifex 4479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V
201	200	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V)
202	35	feqmptd 6769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 = (𝑢 ∈ ℝ ↦ (𝑓‘𝑢)))
203	202	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 = (𝑢 ∈ ℝ ↦ (𝑓‘𝑢)))
204		eqidd 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
205	196, 198, 201, 203, 204	offval2 7477	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘f − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑢 ∈ ℝ ↦ ((𝑓‘𝑢) − if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))))
206		ovif2 7298	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓‘𝑢) − if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑢) − 0))
207	171, 166	syl5eq 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓:ℝ⟶ℝ → (◡𝑓 “ ran 𝑓) = ℝ)
208	207	difeq1d 4026	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓:ℝ⟶ℝ → ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})) = (ℝ ∖ (◡𝑓 “ {0})))
209	170, 208	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑓:ℝ⟶ℝ → (◡𝑓 “ (ran 𝑓 ∖ {0})) = (ℝ ∖ (◡𝑓 “ {0})))
210	209	eleq2d 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑓:ℝ⟶ℝ → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0}))))
211	35, 210	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑓 ∈ dom ∫1 → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0}))))
212	211	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0}))))
213		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
214	213	biantrurd 536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (◡𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ ¬ 𝑢 ∈ (◡𝑓 “ {0}))))
215		eldif 3867	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0})) ↔ (𝑢 ∈ ℝ ∧ ¬ 𝑢 ∈ (◡𝑓 “ {0})))
216	214, 215	bitr4di 292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (◡𝑓 “ {0}) ↔ 𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0}))))
217	212, 216	bitr4d 285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ 𝑢 ∈ (◡𝑓 “ {0})))
218	217	con2bid 358	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (◡𝑓 “ {0}) ↔ ¬ 𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))))
219		fniniseg 6869	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑓 Fn ℝ → (𝑢 ∈ (◡𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0)))
220	35, 180, 219	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑓 ∈ dom ∫1 → (𝑢 ∈ (◡𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0)))
221	220	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (◡𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0)))
222	218, 221	bitr3d 284	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ (𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0)))
223		oveq1 7209	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑓‘𝑢) = 0 → ((𝑓‘𝑢) − 0) = (0 − 0))
224		0m0e0 11933	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (0 − 0) = 0
225	223, 224	eqtrdi 2790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑓‘𝑢) = 0 → ((𝑓‘𝑢) − 0) = 0)
226	225	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0) → ((𝑓‘𝑢) − 0) = 0)
227	222, 226	syl6bi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓‘𝑢) − 0) = 0))
228	227	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) ∧ ¬ 𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑢) − 0) = 0)
229	228	ifeq2da 4461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑢) − 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
230	206, 229	syl5eq 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → ((𝑓‘𝑢) − if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
231	230	mpteq2dva 5139	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ ((𝑓‘𝑢) − if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
232	205, 231	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘f − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
233		simpll 767	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 ∈ dom ∫1)
234	199	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V)
235		1ex 10812	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 1 ∈ V
236	235, 105	ifex 4479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V
237	236	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V)
238		fconstmpt 5600	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑢 ∈ ℝ ↦ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
239	238	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑢 ∈ ℝ ↦ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
240		eqidd 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))
241	196, 234, 237, 239, 240	offval2 7477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘f · (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑢 ∈ ℝ ↦ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))))
242		ovif2 7298	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0))
243		resubcl 11125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((∫1‘𝑓) ∈ ℝ ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
244	8, 243	sylan 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
245	244	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
246		2re 11887	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ 2 ∈ ℝ
247		i1fima 24547	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓 ∈ dom ∫1 → (◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol)
248		mblvol 24399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
249	247, 248	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑓 ∈ dom ∫1 → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
250		neldifsn 4695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ¬ 0 ∈ (ran 𝑓 ∖ {0})
251		i1fima2 24548	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑓 ∈ dom ∫1 ∧ ¬ 0 ∈ (ran 𝑓 ∖ {0})) → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
252	250, 251	mpan2 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑓 ∈ dom ∫1 → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
253	249, 252	eqeltrrd 2835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑓 ∈ dom ∫1 → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
254		remulcl 10797	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((2 ∈ ℝ ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
255	246, 253, 254	sylancr 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑓 ∈ dom ∫1 → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
256	255	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
257		2cnd 11891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 2 ∈ ℂ)
258	253	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
259	258	recnd 10844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
260		2ne0 11917	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ 2 ≠ 0
261	260	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 2 ≠ 0)
262		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)
263	257, 259, 261, 262	mulne0d 11467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ≠ 0)
264	245, 256, 263	redivcld 11643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
265	264	recnd 10844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
266	265	mulid1d 10833	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1) = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
267	265	mul01d 11014	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0) = 0)
268	266, 267	ifeq12d 4450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
269	242, 268	syl5eq 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
270	269	mpteq2dv 5140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
271	241, 270	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘f · (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
272		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))
273	272	i1f1 24559	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol ∧ (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
274	247, 252, 273	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ dom ∫1 → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
275	274	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
276	275, 264	i1fmulc 24573	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘f · (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) ∈ dom ∫1)
277	271, 276	eqeltrrd 2835	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1)
278		i1fsub 24578	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1) → (𝑓 ∘f − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
279	233, 277, 278	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘f − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
280	232, 279	eqeltrrd 2835	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
281		iftrue 4435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
282		iftrue 4435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
283	282	breq2d 5055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
284	283, 282	ifbieq1d 4453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
285		iftrue 4435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
286	284, 285	sylan9eqr 2796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
287	281, 286	eqtr4d 2777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
288		iffalse 4438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
289		ianor 982	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∨ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))))
290	283	ifbid 4452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
291		iffalse 4438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
292	290, 291	sylan9eqr 2796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
293	292	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0))
294		iffalse 4438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
295		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 0 = 0
296		eqeq1 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) → (if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 ↔ if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0))
297		eqeq1 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (0 = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) → (0 = 0 ↔ if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0))
298	296, 297	ifboth 4468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 ∧ 0 = 0) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
299	294, 295, 298	sylancl 589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
300	293, 299	pm2.61d1 183	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
301	300, 299	jaoi 857	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∨ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
302	289, 301	sylbi 220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
303	288, 302	eqtr4d 2777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
304	287, 303	pm2.61i 185	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0)
305		eleq1w 2816	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))))
306		fveq2 6706	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
307	306	oveq1d 7217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑢 = 𝑧 → ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
308	305, 307	ifbieq1d 4453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑢 = 𝑧 → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
309		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
310		ovex 7235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ V
311	310, 105	ifex 4479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V
312	308, 309, 311	fvmpt 6807	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 ∈ ℝ → ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
313	312	breq2d 5055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ ℝ → (0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) ↔ 0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
314	313, 312	ifbieq1d 4453	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ ℝ → if(0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
315	304, 314	eqtr4id 2793	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ ℝ → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), 0))
316	315	mpteq2ia 5135	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑧 ∈ ℝ ↦ if(0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), 0))
317	316	i1fpos 24576	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
318	280, 317	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
319	195, 318	sylan 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
320	195, 264	sylan 583	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
321	8	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (∫1‘𝑓) ∈ ℝ)
322	321, 194, 243	syl2an 599	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
323	322	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
324	255	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
325	324	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
326		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑏 < (∫1‘𝑓))
327		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑏 ∈ ℝ)
328	141	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → (∫1‘𝑓) ∈ ℝ)
329	327, 328	posdifd 11402	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → (𝑏 < (∫1‘𝑓) ↔ 0 < ((∫1‘𝑓) − 𝑏)))
330	326, 329	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → 0 < ((∫1‘𝑓) − 𝑏))
331	330	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < ((∫1‘𝑓) − 𝑏))
332	253	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
333	332	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
334		mblss 24400	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol → (◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
335		ovolge0 24350	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ → 0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
336	247, 334, 335	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ dom ∫1 → 0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
337		ltlen 10916	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((0 ∈ ℝ ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ (0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)))
338	48, 253, 337	sylancr 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → (0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ (0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)))
339	338	biimprd 251	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ dom ∫1 → ((0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
340	336, 339	mpand 695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ dom ∫1 → ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0 → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
341	340	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0 → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
342	341	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
343	342	adantlr 715	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
344		2pos 11916	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 < 2
345		mulgt0 10893	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((2 ∈ ℝ ∧ 0 < 2) ∧ ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ ∧ 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → 0 < (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
346	246, 344, 345	mpanl12 702	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ ∧ 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) → 0 < (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
347	333, 343, 346	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
348	323, 325, 331, 347	divgt0d 11750	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
349	320, 348	elrpd 12608	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ+)
350		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → 𝑓 ∘r ≤ 𝐹)
351	350	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 ∘r ≤ 𝐹)
352		ffn 6534	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹 Fn ℝ)
353	35, 180	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 Fn ℝ)
354	353	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → 𝑓 Fn ℝ)
355		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → 𝑓 Fn ℝ)
356		simpl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → 𝐹 Fn ℝ)
357	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → ℝ ∈ V)
358		inidm 4123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (ℝ ∩ ℝ) = ℝ
359		eqidd 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) = (𝑓‘𝑧))
360		eqidd 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
361	355, 356, 357, 357, 358, 359, 360	ofrfval 7467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → (𝑓 ∘r ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
362	352, 354, 361	syl2an 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑓 ∘r ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
363	362	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘r ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
364		simpl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → 𝑓 ∈ dom ∫1)
365	364	anim2i 620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1))
366	365, 194	anim12i 616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ))
367		breq1 5046	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0 = if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → (0 ≤ (𝐹‘𝑧) ↔ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
368		breq1 5046	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧) ↔ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
369		simplll 775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝐹:ℝ⟶(0[,]+∞))
370	369	ffvelrnda 6893	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ (0[,]+∞))
371	370, 100	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑧)))
372	371	simprd 499	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹‘𝑧))
373	372	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) ∧ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) → 0 ≤ (𝐹‘𝑧))
374		oveq1 7209	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
375	374	breq1d 5053	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → ((((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧) ↔ (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧)))
376		oveq1 7209	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (0 = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → (0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
377	376	breq1d 5053	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0 = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → ((0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧) ↔ (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧)))
378	35	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓:ℝ⟶ℝ)
379	378	ffvelrnda 6893	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
380	379	recnd 10844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℂ)
381	244	recnd 10844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) − 𝑏) ∈ ℂ)
382	381	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − 𝑏) ∈ ℂ)
383	255	recnd 10844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓 ∈ dom ∫1 → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℂ)
384	383	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℂ)
385	382, 384, 263	divcld 11591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
386	385	adantlll 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
387	386	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
388	380, 387	npcand 11176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (𝑓‘𝑧))
389	388	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (𝑓‘𝑧))
390		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧))
391	389, 390	eqbrtrd 5065	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
392	391	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) ∧ ¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) ∧ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
393	288	pm2.24d 154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 → (0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧)))
394	393	impcom 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 ∧ ¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
395	394	adantll 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) ∧ ¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) ∧ ¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
396	375, 377, 392, 395	ifbothda 4467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) ∧ ¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) → (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
397	367, 368, 373, 396	ifbothda 4467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧))
398	397	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) ≤ (𝐹‘𝑧) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
399	366, 398	sylanl1 680	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) ≤ (𝐹‘𝑧) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
400	399	ralimdva 3093	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧) → ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
401	363, 400	sylbid 243	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘r ≤ 𝐹 → ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
402	351, 401	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧))
403		ovex 7235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ V
404	105, 403	ifex 4479	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ∈ V
405	404	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ∈ V)
406		eqidd 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) = (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))))
407	104, 405, 99, 406, 67	ofrfval2 7478	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ((𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘r ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
408	407	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘r ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
409	402, 408	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘r ≤ 𝐹)
410		oveq2 7210	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦) = (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
411	410	ifeq2d 4449	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦)) = if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
412	411	mpteq2dv 5140	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))))
413	412	breq1d 5053	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → ((𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘r ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘r ≤ 𝐹))
414	413	rspcev 3530	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ+ ∧ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘r ≤ 𝐹) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘r ≤ 𝐹)
415	349, 409, 414	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘r ≤ 𝐹)
416		fveq2 6706	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = 𝑔 → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))
417	416	eqcoms 2742	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))
418	417	biantrud 535	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))))
419		nfmpt1 5142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ Ⅎ𝑧(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
420	419	nfeq2 2917	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑧 𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
421		fveq1 6705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (𝑔‘𝑧) = ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧))
422	310, 105	ifex 4479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V
423		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
424	423	fvmpt2 6818	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑧 ∈ ℝ ∧ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V) → ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
425	422, 424	mpan2 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ ℝ → ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
426	421, 425	sylan9eq 2794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
427	426	eqeq1d 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) = 0 ↔ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0))
428	426	oveq1d 7217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) + 𝑦) = (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))
429	427, 428	ifbieq2d 4455	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦)) = if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦)))
430	420, 429	mpteq2da 5138	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))))
431	430	breq1d 5053	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘r ≤ 𝐹))
432	431	rexbidv 3209	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘r ≤ 𝐹))
433	418, 432	bitr3d 284	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)) ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘r ≤ 𝐹))
434	433	rspcev 3530	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 ∧ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘r ≤ 𝐹) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)))
435	319, 415, 434	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)))
436		simplrr 778	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 ∈ ℝ)
437	199	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V)
438	235, 105	ifex 4479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V
439	438	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V)
440		fconstmpt 5600	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑧 ∈ ℝ ↦ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
441	440	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑧 ∈ ℝ ↦ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
442		eqidd 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))
443	196, 437, 439, 441, 442	offval2 7477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘f · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑧 ∈ ℝ ↦ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))))
444		ovif2 7298	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0))
445	266, 267	ifeq12d 4450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0)) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
446	444, 445	syl5eq 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
447	446	mpteq2dv 5140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
448	443, 447	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘f · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
449		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))
450	449	i1f1 24559	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol ∧ (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
451	247, 252, 450	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ dom ∫1 → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
452	451	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
453	452, 264	i1fmulc 24573	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘f · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) ∈ dom ∫1)
454	448, 453	eqeltrrd 2835	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1)
455		i1fsub 24578	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1) → (𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
456	233, 454, 455	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
457		itg1cl 24554	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1 → (∫1‘(𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ∈ ℝ)
458	456, 457	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ∈ ℝ)
459	458	adantlrl 720	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ∈ ℝ)
460	318	adantlrl 720	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
461		itg1cl 24554	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) ∈ ℝ)
462	460, 461	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) ∈ ℝ)
463		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘𝑓))
464		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
465	8	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (∫1‘𝑓) ∈ ℝ)
466	97	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → 2 ∈ ℝ+)
467	464, 465, 466	ltdiv1d 12656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (𝑏 < (∫1‘𝑓) ↔ (𝑏 / 2) < ((∫1‘𝑓) / 2)))
468		recn 10802	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑏 ∈ ℝ → 𝑏 ∈ ℂ)
469	468	2halvesd 12059	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑏 ∈ ℝ → ((𝑏 / 2) + (𝑏 / 2)) = 𝑏)
470	469	oveq1d 7217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑏 ∈ ℝ → (((𝑏 / 2) + (𝑏 / 2)) − (𝑏 / 2)) = (𝑏 − (𝑏 / 2)))
471	468	halfcld 12058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑏 ∈ ℝ → (𝑏 / 2) ∈ ℂ)
472	471, 471	pncand 11173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑏 ∈ ℝ → (((𝑏 / 2) + (𝑏 / 2)) − (𝑏 / 2)) = (𝑏 / 2))
473	470, 472	eqtr3d 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 ∈ ℝ → (𝑏 − (𝑏 / 2)) = (𝑏 / 2))
474	473	breq1d 5053	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑏 ∈ ℝ → ((𝑏 − (𝑏 / 2)) < ((∫1‘𝑓) / 2) ↔ (𝑏 / 2) < ((∫1‘𝑓) / 2)))
475	474	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((𝑏 − (𝑏 / 2)) < ((∫1‘𝑓) / 2) ↔ (𝑏 / 2) < ((∫1‘𝑓) / 2)))
476		rehalfcl 12039	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 ∈ ℝ → (𝑏 / 2) ∈ ℝ)
477	476	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (𝑏 / 2) ∈ ℝ)
478	8	rehalfcld 12060	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → ((∫1‘𝑓) / 2) ∈ ℝ)
479	478	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) / 2) ∈ ℝ)
480	464, 477, 479	ltsubaddd 11411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((𝑏 − (𝑏 / 2)) < ((∫1‘𝑓) / 2) ↔ 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2))))
481	467, 475, 480	3bitr2d 310	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (𝑏 < (∫1‘𝑓) ↔ 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2))))
482	481	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑏 < (∫1‘𝑓) ↔ 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2))))
483	482	adantlrl 720	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑏 < (∫1‘𝑓) ↔ 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2))))
484	463, 483	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2)))
485	452, 264	itg1mulc 24574	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘f · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))))
486	448	fveq2d 6710	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘f · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))))
487	449	itg11 24560	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol ∧ (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
488	247, 252, 487	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
489	488	oveq2d 7218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
490	489	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
491	252	recnd 10844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 ∈ dom ∫1 → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
492	491	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
493	265, 492	mulcomd 10837	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) = ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
494	249	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
495	494	oveq1d 7217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) = ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)))
496	259, 382	mulcomd 10837	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) = (((∫1‘𝑓) − 𝑏) · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
497	495, 496	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) = (((∫1‘𝑓) − 𝑏) · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
498	497	oveq1d 7217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) = ((((∫1‘𝑓) − 𝑏) · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
499	492, 382, 384, 263	divassd 11626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) = ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
500	382, 257, 259, 261, 262	divcan5rd 11618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) = (((∫1‘𝑓) − 𝑏) / 2))
501	498, 499, 500	3eqtr3d 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (((∫1‘𝑓) − 𝑏) / 2))
502	490, 493, 501	3eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = (((∫1‘𝑓) − 𝑏) / 2))
503	485, 486, 502	3eqtr3d 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (((∫1‘𝑓) − 𝑏) / 2))
504	503	oveq2d 7218	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = ((∫1‘𝑓) − (((∫1‘𝑓) − 𝑏) / 2)))
505		itg1sub 24579	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1) → (∫1‘(𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = ((∫1‘𝑓) − (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))))
506	233, 454, 505	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = ((∫1‘𝑓) − (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))))
507	8	recnd 10844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘𝑓) ∈ ℂ)
508	507	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘𝑓) ∈ ℂ)
509	468	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 ∈ ℂ)
510	508, 509, 257, 261	divsubdird 11630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / 2) = (((∫1‘𝑓) / 2) − (𝑏 / 2)))
511	510	oveq2d 7218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − (((∫1‘𝑓) − 𝑏) / 2)) = ((∫1‘𝑓) − (((∫1‘𝑓) / 2) − (𝑏 / 2))))
512	507	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (∫1‘𝑓) ∈ ℂ)
513	512	halfcld 12058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) / 2) ∈ ℂ)
514	471	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (𝑏 / 2) ∈ ℂ)
515	512, 513, 514	subsubd 11200	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) − (((∫1‘𝑓) / 2) − (𝑏 / 2))) = (((∫1‘𝑓) − ((∫1‘𝑓) / 2)) + (𝑏 / 2)))
516	515	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − (((∫1‘𝑓) / 2) − (𝑏 / 2))) = (((∫1‘𝑓) − ((∫1‘𝑓) / 2)) + (𝑏 / 2)))
517	507	2halvesd 12059	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 ∈ dom ∫1 → (((∫1‘𝑓) / 2) + ((∫1‘𝑓) / 2)) = (∫1‘𝑓))
518	517	oveq1d 7217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → ((((∫1‘𝑓) / 2) + ((∫1‘𝑓) / 2)) − ((∫1‘𝑓) / 2)) = ((∫1‘𝑓) − ((∫1‘𝑓) / 2)))
519	507	halfcld 12058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 ∈ dom ∫1 → ((∫1‘𝑓) / 2) ∈ ℂ)
520	519, 519	pncand 11173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → ((((∫1‘𝑓) / 2) + ((∫1‘𝑓) / 2)) − ((∫1‘𝑓) / 2)) = ((∫1‘𝑓) / 2))
521	518, 520	eqtr3d 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ dom ∫1 → ((∫1‘𝑓) − ((∫1‘𝑓) / 2)) = ((∫1‘𝑓) / 2))
522	521	oveq1d 7217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ dom ∫1 → (((∫1‘𝑓) − ((∫1‘𝑓) / 2)) + (𝑏 / 2)) = (((∫1‘𝑓) / 2) + (𝑏 / 2)))
523	522	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − ((∫1‘𝑓) / 2)) + (𝑏 / 2)) = (((∫1‘𝑓) / 2) + (𝑏 / 2)))
524	511, 516, 523	3eqtrrd 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) / 2) + (𝑏 / 2)) = ((∫1‘𝑓) − (((∫1‘𝑓) − 𝑏) / 2)))
525	504, 506, 524	3eqtr4d 2784	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = (((∫1‘𝑓) / 2) + (𝑏 / 2)))
526	525	adantlrl 720	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = (((∫1‘𝑓) / 2) + (𝑏 / 2)))
527	484, 526	breqtrrd 5071	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))))
528	456	adantlrl 720	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
529		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0))
530	529	adantlrl 720	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0))
531	233, 36	sylan 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
532	264	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
533	531, 532	resubcld 11243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ ℝ)
534	533	leidd 11381	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
535	534	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
536	285	breq2d 5055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
537	536	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
538	535, 537	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
539	533	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ ℝ)
540	48	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → 0 ∈ ℝ)
541	48	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → 0 ∈ ℝ)
542	533, 541	ltnled 10962	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) < 0 ↔ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
543	542	biimpar 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) < 0)
544	539, 540, 543	ltled 10963	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ 0)
545		iffalse 4438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
546	545	breq2d 5055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ 0))
547	546	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ 0))
548	544, 547	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
549	538, 548	pm2.61dan 813	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
550	530, 549	sylan 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
551	550	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
552		iftrue 4435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
553	552	oveq2d 7218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
554		iba 531	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ↔ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})))))
555	554	bicomd 226	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → ((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
556	555	ifbid 4452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
557	553, 556	breq12d 5056	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
558	557	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
559	551, 558	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
560	35	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓:ℝ⟶ℝ)
561	170	eleq2d 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑧 ∈ ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0}))))
562		eldif 3867	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑧 ∈ ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})) ↔ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0})))
563	561, 562	bitrdi 290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0}))))
564	563	notbid 321	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑓:ℝ⟶ℝ → (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0}))))
565	564	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0}))))
566		pm4.53 986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0})) ↔ (¬ 𝑧 ∈ (◡𝑓 “ ran 𝑓) ∨ 𝑧 ∈ (◡𝑓 “ {0})))
567	207	eleq2d 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ ran 𝑓) ↔ 𝑧 ∈ ℝ))
568	567	biimpar 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ (◡𝑓 “ ran 𝑓))
569	568	pm2.24d 154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (◡𝑓 “ ran 𝑓) → (𝑓‘𝑧) = 0))
570	181	simplbda 503	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑓 Fn ℝ ∧ 𝑧 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑧) = 0)
571	570	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓 Fn ℝ → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
572	180, 571	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
573	572	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
574	569, 573	jaod 859	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → ((¬ 𝑧 ∈ (◡𝑓 “ ran 𝑓) ∨ 𝑧 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑧) = 0))
575	566, 574	syl5bi 245	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑧) = 0))
576	565, 575	sylbid 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (𝑓‘𝑧) = 0))
577	576	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓‘𝑧) = 0)
578	560, 577	sylanl1 680	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓‘𝑧) = 0)
579	578	oveq1d 7217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − 0) = (0 − 0))
580	579, 224	eqtrdi 2790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − 0) = 0)
581	580, 30	eqbrtrdi 5082	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − 0) ≤ 0)
582		iffalse 4438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) = 0)
583	582	oveq2d 7218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = ((𝑓‘𝑧) − 0))
584	289, 288	sylbir 238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∨ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
585	584	olcs 876	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
586	583, 585	breq12d 5056	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − 0) ≤ 0))
587	586	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − 0) ≤ 0))
588	581, 587	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
589	559, 588	pm2.61dan 813	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
590	589	ralrimiva 3098	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∀𝑧 ∈ ℝ ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
591	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ℝ ∈ V)
592		ovex 7235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ V
593	592	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ V)
594	422	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V)
595		fvex 6719	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓‘𝑧) ∈ V
596	595	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ V)
597	199, 105	ifex 4479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V
598	597	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V)
599	70	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓‘𝑧)))
600		eqidd 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
601	591, 596, 598, 599, 600	offval2 7477	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑧 ∈ ℝ ↦ ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))))
602		eqidd 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
603	591, 593, 594, 601, 602	ofrfval2 7478	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∘r ≤ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ↔ ∀𝑧 ∈ ℝ ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
604	590, 603	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∘r ≤ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
605		itg1le 24583	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1 ∧ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 ∧ (𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∘r ≤ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (∫1‘(𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ≤ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
606	528, 460, 604, 605	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘f − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ≤ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
607	436, 459, 462, 527, 606	ltletrd 10975	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
608	607	adantllr 719	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
609	608	adantlll 718	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
610		fvex 6719	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) ∈ V
611		eqeq1 2738	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (𝑎 = (∫1‘𝑔) ↔ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)))
612	611	anbi2d 632	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))))
613	612	rexbidv 3209	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))))
614		breq2 5047	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (𝑏 < 𝑎 ↔ 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))))
615	613, 614	anbi12d 634	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎) ↔ (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)) ∧ 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))))
616	610, 615	spcev 3514	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)) ∧ 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
617	435, 609, 616	syl2anc 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
618	192, 617	pm2.61dane 3022	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
619	618	expr 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → (𝑏 ∈ ℝ → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
620	619	adantllr 719	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → (𝑏 ∈ ℝ → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
621	620	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → (𝑏 ∈ ℝ → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
622	157, 621	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
623	139, 622	pm2.61dane 3022	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
624	623	ex 416	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑏 < (∫1‘𝑓) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
625	94, 624	sylbid 243	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑏 < 𝑠 → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
626	625	imp 410	. . . . . . . . . . . 12 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < 𝑠) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
627	626	an32s 652	. . . . . . . . . . 11 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ 𝑏 < 𝑠) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
628	627	rexlimdvaa 3197	. . . . . . . . . 10 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ 𝑏 < 𝑠) → (∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
629	628	expimpd 457	. . . . . . . . 9 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → ((𝑏 < 𝑠 ∧ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
630	629	ancomsd 469	. . . . . . . 8 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → ((∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)) ∧ 𝑏 < 𝑠) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
631	630	exlimdv 1941	. . . . . . 7 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (∃𝑠(∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)) ∧ 𝑏 < 𝑠) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
632		eqeq1 2738	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → (𝑥 = (∫1‘𝑓) ↔ 𝑠 = (∫1‘𝑓)))
633	632	anbi2d 632	. . . . . . . . 9 ⊢ (𝑥 = 𝑠 → ((𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ (𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))))
634	633	rexbidv 3209	. . . . . . . 8 ⊢ (𝑥 = 𝑠 → (∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))))
635	634	rexab 3601	. . . . . . 7 ⊢ (∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠 ↔ ∃𝑠(∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)) ∧ 𝑏 < 𝑠))
636		eqeq1 2738	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑥 = (∫1‘𝑔) ↔ 𝑎 = (∫1‘𝑔)))
637	636	anbi2d 632	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔))))
638	637	rexbidv 3209	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔))))
639	638	rexab 3601	. . . . . . 7 ⊢ (∃𝑎 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑏 < 𝑎 ↔ ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
640	631, 635, 639	3imtr4g 299	. . . . . 6 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠 → ∃𝑎 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑏 < 𝑎))
641	92, 640	sylbid 243	. . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) → ∃𝑎 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑏 < 𝑎))
642	641	impr 458	. . . 4 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑏 ∈ ℝ* ∧ 𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))) → ∃𝑎 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑏 < 𝑎)
643	6, 15, 89, 642	eqsupd 9062	. . 3 ⊢ (𝐹:ℝ⟶(0[,]+∞) → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
644	4, 643	syl5eq 2786	. 2 ⊢ (𝐹:ℝ⟶(0[,]+∞) → sup(𝐿, ℝ*, < ) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
645	2, 644	eqtr4d 2777	1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543  ∃wex 1787   ∈ wcel 2110  {cab 2712   ≠ wne 2935  ∀wral 3054  ∃wrex 3055  Vcvv 3401   ∖ cdif 3854   ⊆ wss 3857  ∅c0 4227  ifcif 4429  {csn 4531   class class class wbr 5043   ↦ cmpt 5124   Or wor 5456   × cxp 5538  ^◡ccnv 5539  dom cdm 5540  ran crn 5541   “ cima 5543  Fun wfun 6363   Fn wfn 6364  ⟶wf 6365  ‘cfv 6369  (class class class)co 7202   ∘_f cof 7456   ∘_r cofr 7457  supcsup 9045  ℂcc 10710  ℝcr 10711  0cc0 10712  1c1 10713   + caddc 10715   · cmul 10717  +∞cpnf 10847  -∞cmnf 10848  ℝ^*cxr 10849   < clt 10850   ≤ cle 10851   − cmin 11045   / cdiv 11472  2c2 11868  ℝ⁺crp 12569  [,]cicc 12921  vol*covol 24331  volcvol 24332  ∫₁citg1 24484  ∫₂citg2 24485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-inf2 9245  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789  ax-pre-sup 10790  ax-addf 10791

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-disj 5009  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-se 5499  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-isom 6378  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-of 7458  df-ofr 7459  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-2o 8192  df-er 8380  df-map 8499  df-pm 8500  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-fi 9016  df-sup 9047  df-inf 9048  df-oi 9115  df-dju 9500  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-div 11473  df-nn 11814  df-2 11876  df-3 11877  df-n0 12074  df-z 12160  df-uz 12422  df-q 12528  df-rp 12570  df-xneg 12687  df-xadd 12688  df-xmul 12689  df-ioo 12922  df-ico 12924  df-icc 12925  df-fz 13079  df-fzo 13222  df-fl 13350  df-seq 13558  df-exp 13619  df-hash 13880  df-cj 14645  df-re 14646  df-im 14647  df-sqrt 14781  df-abs 14782  df-clim 15032  df-sum 15233  df-rest 16899  df-topgen 16920  df-psmet 20327  df-xmet 20328  df-met 20329  df-bl 20330  df-mopn 20331  df-top 21763  df-topon 21780  df-bases 21815  df-cmp 22256  df-ovol 24333  df-vol 24334  df-mbf 24488  df-itg1 24489  df-itg2 24490

This theorem is referenced by:  itg2addnc  35525