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Theorem itg2addnclem 32631
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((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑔))}
Assertion
Ref Expression
itg2addnclem (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = 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.
StepHypRef Expression
1 eqid 2610 . . 3 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))} = {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}
21itg2val 23301 . 2 (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
3 itg2addnclem.1 . . . 4 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑔))}
43supeq1i 8236 . . 3 sup(𝐿, ℝ*, < ) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑔))}, ℝ*, < )
5 xrltso 11850 . . . . 5 < Or ℝ*
65a1i 11 . . . 4 (𝐹:ℝ⟶(0[,]+∞) → < Or ℝ*)
7 simprr 792 . . . . . . . 8 ((𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑥 = (∫1𝑓))) → 𝑥 = (∫1𝑓))
8 itg1cl 23258 . . . . . . . . . 10 (𝑓 ∈ dom ∫1 → (∫1𝑓) ∈ ℝ)
98rexrd 9968 . . . . . . . . 9 (𝑓 ∈ dom ∫1 → (∫1𝑓) ∈ ℝ*)
109adantr 480 . . . . . . . 8 ((𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑥 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ*)
117, 10eqeltrd 2688 . . . . . . 7 ((𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑥 = (∫1𝑓))) → 𝑥 ∈ ℝ*)
1211rexlimiva 3010 . . . . . 6 (∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓)) → 𝑥 ∈ ℝ*)
1312abssi 3640 . . . . 5 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*
14 supxrcl 12017 . . . . 5 ({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ* → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ*)
1513, 14mp1i 13 . . . 4 (𝐹:ℝ⟶(0[,]+∞) → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ*)
16 fveq1 6102 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (𝑔𝑧) = (𝑓𝑧))
1716eqeq1d 2612 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((𝑔𝑧) = 0 ↔ (𝑓𝑧) = 0))
1816oveq1d 6564 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((𝑔𝑧) + 𝑦) = ((𝑓𝑧) + 𝑦))
1917, 18ifbieq2d 4061 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦)) = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)))
2019mpteq2dv 4673 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))))
2120breq1d 4593 . . . . . . . . . . . 12 (𝑔 = 𝑓 → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ↔ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘𝑟𝐹))
2221rexbidv 3034 . . . . . . . . . . 11 (𝑔 = 𝑓 → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘𝑟𝐹))
23 fveq2 6103 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (∫1𝑔) = (∫1𝑓))
2423eqeq2d 2620 . . . . . . . . . . 11 (𝑔 = 𝑓 → (𝑥 = (∫1𝑔) ↔ 𝑥 = (∫1𝑓)))
2522, 24anbi12d 743 . . . . . . . . . 10 (𝑔 = 𝑓 → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑓))))
2625cbvrexv 3148 . . . . . . . . 9 (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑔)) ↔ ∃𝑓 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑓)))
27 breq2 4587 . . . . . . . . . . . . . . . . 17 (0 = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) → ((𝑓𝑧) ≤ 0 ↔ (𝑓𝑧) ≤ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))))
28 breq2 4587 . . . . . . . . . . . . . . . . 17 (((𝑓𝑧) + 𝑦) = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) → ((𝑓𝑧) ≤ ((𝑓𝑧) + 𝑦) ↔ (𝑓𝑧) ≤ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))))
29 id 22 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑧) = 0 → (𝑓𝑧) = 0)
30 0le0 10987 . . . . . . . . . . . . . . . . . . 19 0 ≤ 0
3129, 30syl6eqbr 4622 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑧) = 0 → (𝑓𝑧) ≤ 0)
3231adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → (𝑓𝑧) ≤ 0)
33 rpge0 11721 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
3433ad2antlr 759 . . . . . . . . . . . . . . . . . . 19 (((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 0 ≤ 𝑦)
35 i1ff 23249 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 ∈ dom ∫1𝑓:ℝ⟶ℝ)
3635ffvelrnda 6267 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ dom ∫1𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
3736adantlr 747 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
38 rpre 11715 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
3938ad2antlr 759 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
4037, 39addge01d 10494 . . . . . . . . . . . . . . . . . . 19 (((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (0 ≤ 𝑦 ↔ (𝑓𝑧) ≤ ((𝑓𝑧) + 𝑦)))
4134, 40mpbid 221 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ≤ ((𝑓𝑧) + 𝑦))
4241adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓𝑧) = 0) → (𝑓𝑧) ≤ ((𝑓𝑧) + 𝑦))
4327, 28, 32, 42ifbothda 4073 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ≤ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)))
4443adantlll 750 . . . . . . . . . . . . . . 15 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ≤ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)))
4535ad2antlr 759 . . . . . . . . . . . . . . . . . 18 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝑓:ℝ⟶ℝ)
4645ffvelrnda 6267 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
4746rexrd 9968 . . . . . . . . . . . . . . . 16 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ*)
48 0re 9919 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
4938ad2antlr 759 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
5046, 49readdcld 9948 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + 𝑦) ∈ ℝ)
51 ifcl 4080 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ ∧ ((𝑓𝑧) + 𝑦) ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ∈ ℝ)
5248, 50, 51sylancr 694 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ∈ ℝ)
5352rexrd 9968 . . . . . . . . . . . . . . . 16 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ∈ ℝ*)
54 iccssxr 12127 . . . . . . . . . . . . . . . . . . 19 (0[,]+∞) ⊆ ℝ*
55 fss 5969 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℝ⟶(0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐹:ℝ⟶ℝ*)
5654, 55mpan2 703 . . . . . . . . . . . . . . . . . 18 (𝐹:ℝ⟶(0[,]+∞) → 𝐹:ℝ⟶ℝ*)
5756ad2antrr 758 . . . . . . . . . . . . . . . . 17 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝐹:ℝ⟶ℝ*)
5857ffvelrnda 6267 . . . . . . . . . . . . . . . 16 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ∈ ℝ*)
59 xrletr 11865 . . . . . . . . . . . . . . . 16 (((𝑓𝑧) ∈ ℝ* ∧ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ∈ ℝ* ∧ (𝐹𝑧) ∈ ℝ*) → (((𝑓𝑧) ≤ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ∧ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧)))
6047, 53, 58, 59syl3anc 1318 . . . . . . . . . . . . . . 15 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) ≤ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ∧ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧)))
6144, 60mpand 707 . . . . . . . . . . . . . 14 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ≤ (𝐹𝑧) → (𝑓𝑧) ≤ (𝐹𝑧)))
6261ralimdva 2945 . . . . . . . . . . . . 13 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (∀𝑧 ∈ ℝ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ≤ (𝐹𝑧) → ∀𝑧 ∈ ℝ (𝑓𝑧) ≤ (𝐹𝑧)))
63 reex 9906 . . . . . . . . . . . . . . 15 ℝ ∈ V
6463a1i 11 . . . . . . . . . . . . . 14 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ℝ ∈ V)
65 eqidd 2611 . . . . . . . . . . . . . 14 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))))
66 id 22 . . . . . . . . . . . . . . . 16 (𝐹:ℝ⟶(0[,]+∞) → 𝐹:ℝ⟶(0[,]+∞))
6766feqmptd 6159 . . . . . . . . . . . . . . 15 (𝐹:ℝ⟶(0[,]+∞) → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹𝑧)))
6867ad2antrr 758 . . . . . . . . . . . . . 14 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹𝑧)))
6964, 52, 58, 65, 68ofrfval2 6813 . . . . . . . . . . . . 13 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘𝑟𝐹 ↔ ∀𝑧 ∈ ℝ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ≤ (𝐹𝑧)))
7035feqmptd 6159 . . . . . . . . . . . . . . 15 (𝑓 ∈ dom ∫1𝑓 = (𝑧 ∈ ℝ ↦ (𝑓𝑧)))
7170ad2antlr 759 . . . . . . . . . . . . . 14 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓𝑧)))
7264, 46, 58, 71, 68ofrfval2 6813 . . . . . . . . . . . . 13 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (𝑓𝑟𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓𝑧) ≤ (𝐹𝑧)))
7362, 69, 723imtr4d 282 . . . . . . . . . . . 12 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘𝑟𝐹𝑓𝑟𝐹))
7473rexlimdva 3013 . . . . . . . . . . 11 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘𝑟𝐹𝑓𝑟𝐹))
7574anim1d 586 . . . . . . . . . 10 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑓)) → (𝑓𝑟𝐹𝑥 = (∫1𝑓))))
7675reximdva 3000 . . . . . . . . 9 (𝐹:ℝ⟶(0[,]+∞) → (∃𝑓 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑓)) → ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))))
7726, 76syl5bi 231 . . . . . . . 8 (𝐹:ℝ⟶(0[,]+∞) → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑔)) → ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))))
7877ss2abdv 3638 . . . . . . 7 (𝐹:ℝ⟶(0[,]+∞) → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑔))} ⊆ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))})
7978sseld 3567 . . . . . 6 (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑔))} → 𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}))
80 simp3r 1083 . . . . . . . . . . 11 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑥 = (∫1𝑓))) → 𝑥 = (∫1𝑓))
8193ad2ant2 1076 . . . . . . . . . . 11 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑥 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ*)
8280, 81eqeltrd 2688 . . . . . . . . . 10 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑥 = (∫1𝑓))) → 𝑥 ∈ ℝ*)
8382rexlimdv3a 3015 . . . . . . . . 9 (𝐹:ℝ⟶(0[,]+∞) → (∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓)) → 𝑥 ∈ ℝ*))
8483abssdv 3639 . . . . . . . 8 (𝐹:ℝ⟶(0[,]+∞) → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*)
85 xrsupss 12011 . . . . . . . 8 ({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ* → ∃𝑎 ∈ ℝ* (∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))} ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ* (𝑏 < 𝑎 → ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}𝑏 < 𝑠)))
8684, 85syl 17 . . . . . . 7 (𝐹:ℝ⟶(0[,]+∞) → ∃𝑎 ∈ ℝ* (∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))} ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ* (𝑏 < 𝑎 → ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}𝑏 < 𝑠)))
876, 86supub 8248 . . . . . 6 (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))} → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) < 𝑏))
8879, 87syld 46 . . . . 5 (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑔))} → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) < 𝑏))
8988imp 444 . . . 4 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑥 = (∫1𝑔))}) → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) < 𝑏)
90 supxrlub 12027 . . . . . . . 8 (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*𝑏 ∈ ℝ*) → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}𝑏 < 𝑠))
9113, 90mpan 702 . . . . . . 7 (𝑏 ∈ ℝ* → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}𝑏 < 𝑠))
9291adantl 481 . . . . . 6 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑥 = (∫1𝑓))}𝑏 < 𝑠))
93 simprrr 801 . . . . . . . . . . . . . . 15 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) → 𝑠 = (∫1𝑓))
9493breq2d 4595 . . . . . . . . . . . . . 14 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) → (𝑏 < 𝑠𝑏 < (∫1𝑓)))
95 simplll 794 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) → 𝐹:ℝ⟶(0[,]+∞))
96 i1f0 23260 . . . . . . . . . . . . . . . . . . 19 (ℝ × {0}) ∈ dom ∫1
97 2rp 11713 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ+
9897ne0ii 3882 . . . . . . . . . . . . . . . . . . . 20 + ≠ ∅
99 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ∈ (0[,]+∞))
100 elxrge0 12152 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹𝑧) ∈ (0[,]+∞) ↔ ((𝐹𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹𝑧)))
10199, 100sylib 207 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹𝑧)))
102101simprd 478 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹𝑧))
103102ralrimiva 2949 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:ℝ⟶(0[,]+∞) → ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧))
10463a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:ℝ⟶(0[,]+∞) → ℝ ∈ V)
105 c0ex 9913 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
106105a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → 0 ∈ V)
107 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
108104, 106, 99, 107, 67ofrfval2 6813 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:ℝ⟶(0[,]+∞) → ((𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧)))
109103, 108mpbird 246 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
110109ralrimivw 2950 . . . . . . . . . . . . . . . . . . . 20 (𝐹:ℝ⟶(0[,]+∞) → ∀𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
111 r19.2z 4012 . . . . . . . . . . . . . . . . . . . 20 ((ℝ+ ≠ ∅ ∧ ∀𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
11298, 110, 111sylancr 694 . . . . . . . . . . . . . . . . . . 19 (𝐹:ℝ⟶(0[,]+∞) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
113 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 = (ℝ × {0}) → (∫1𝑔) = (∫1‘(ℝ × {0})))
114 itg10 23261 . . . . . . . . . . . . . . . . . . . . . . 23 (∫1‘(ℝ × {0})) = 0
115113, 114syl6req 2661 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 = (ℝ × {0}) → 0 = (∫1𝑔))
116115biantrud 527 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = (ℝ × {0}) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ 0 = (∫1𝑔))))
117 fveq1 6102 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔 = (ℝ × {0}) → (𝑔𝑧) = ((ℝ × {0})‘𝑧))
118105fvconst2 6374 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
119117, 118sylan9eq 2664 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = 0)
120 iftrue 4042 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔𝑧) = 0 → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦)) = 0)
121119, 120syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦)) = 0)
122121mpteq2dva 4672 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ 0))
123122breq1d 4593 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹))
124123rexbidv 3034 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = (ℝ × {0}) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹))
125116, 124bitr3d 269 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = (ℝ × {0}) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ 0 = (∫1𝑔)) ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹))
126125rspcev 3282 . . . . . . . . . . . . . . . . . . 19 (((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ 0 = (∫1𝑔)))
12796, 112, 126sylancr 694 . . . . . . . . . . . . . . . . . 18 (𝐹:ℝ⟶(0[,]+∞) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ 0 = (∫1𝑔)))
128 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑏 = -∞ → 𝑏 = -∞)
129 mnflt 11833 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ ℝ → -∞ < 0)
13048, 129mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑏 = -∞ → -∞ < 0)
131128, 130eqbrtrd 4605 . . . . . . . . . . . . . . . . . 18 (𝑏 = -∞ → 𝑏 < 0)
132 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 0 → (𝑎 = (∫1𝑔) ↔ 0 = (∫1𝑔)))
133132anbi2d 736 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 0 → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ 0 = (∫1𝑔))))
134133rexbidv 3034 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 0 → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ 0 = (∫1𝑔))))
135 breq2 4587 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 0 → (𝑏 < 𝑎𝑏 < 0))
136134, 135anbi12d 743 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 0 → ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎) ↔ (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ 0 = (∫1𝑔)) ∧ 𝑏 < 0)))
137105, 136spcev 3273 . . . . . . . . . . . . . . . . . 18 ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ 0 = (∫1𝑔)) ∧ 𝑏 < 0) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎))
138127, 131, 137syl2an 493 . . . . . . . . . . . . . . . . 17 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 = -∞) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎))
13995, 138sylan 487 . . . . . . . . . . . . . . . 16 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 = -∞) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎))
140 simp-4r 803 . . . . . . . . . . . . . . . . . 18 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ∈ ℝ*)
1418adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ)
142141ad3antlr 763 . . . . . . . . . . . . . . . . . 18 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 ≠ -∞) → (∫1𝑓) ∈ ℝ)
143 simpllr 795 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) → 𝑏 ∈ ℝ*)
144 ngtmnft 11872 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 ∈ ℝ* → (𝑏 = -∞ ↔ ¬ -∞ < 𝑏))
145144biimprd 237 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 ∈ ℝ* → (¬ -∞ < 𝑏𝑏 = -∞))
146145necon1ad 2799 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ ℝ* → (𝑏 ≠ -∞ → -∞ < 𝑏))
147146imp 444 . . . . . . . . . . . . . . . . . . 19 ((𝑏 ∈ ℝ*𝑏 ≠ -∞) → -∞ < 𝑏)
148143, 147sylan 487 . . . . . . . . . . . . . . . . . 18 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 ≠ -∞) → -∞ < 𝑏)
149 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → 𝑏 ∈ ℝ*)
1509adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ*)
151149, 150anim12i 588 . . . . . . . . . . . . . . . . . . . 20 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) → (𝑏 ∈ ℝ* ∧ (∫1𝑓) ∈ ℝ*))
152 xrltle 11858 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 ∈ ℝ* ∧ (∫1𝑓) ∈ ℝ*) → (𝑏 < (∫1𝑓) → 𝑏 ≤ (∫1𝑓)))
153152imp 444 . . . . . . . . . . . . . . . . . . . 20 (((𝑏 ∈ ℝ* ∧ (∫1𝑓) ∈ ℝ*) ∧ 𝑏 < (∫1𝑓)) → 𝑏 ≤ (∫1𝑓))
154151, 153sylan 487 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) → 𝑏 ≤ (∫1𝑓))
155154adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ≤ (∫1𝑓))
156 xrre 11874 . . . . . . . . . . . . . . . . . 18 (((𝑏 ∈ ℝ* ∧ (∫1𝑓) ∈ ℝ) ∧ (-∞ < 𝑏𝑏 ≤ (∫1𝑓))) → 𝑏 ∈ ℝ)
157140, 142, 148, 155, 156syl22anc 1319 . . . . . . . . . . . . . . . . 17 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ∈ ℝ)
158127ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ 0 = (∫1𝑔)))
159 simplrl 796 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑏 < (∫1𝑓))
160 simplrl 796 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑓 ∈ dom ∫1)
161 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 ∈ dom ∫1 ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑓 ∈ dom ∫1)
162 cnvimass 5404 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 “ (ran 𝑓 ∖ {0})) ⊆ dom 𝑓
163 fdm 5964 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓:ℝ⟶ℝ → dom 𝑓 = ℝ)
16435, 163syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 ∈ dom ∫1 → dom 𝑓 = ℝ)
165162, 164syl5sseq 3616 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 ∈ dom ∫1 → (𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
166165adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 ∈ dom ∫1 ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
167 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 ∈ dom ∫1 ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0)
168163eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓:ℝ⟶ℝ → ℝ = dom 𝑓)
169 ffun 5961 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑓:ℝ⟶ℝ → Fun 𝑓)
170 difpreima 6251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (Fun 𝑓 → (𝑓 “ (ran 𝑓 ∖ {0})) = ((𝑓 “ ran 𝑓) ∖ (𝑓 “ {0})))
171169, 170syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓:ℝ⟶ℝ → (𝑓 “ (ran 𝑓 ∖ {0})) = ((𝑓 “ ran 𝑓) ∖ (𝑓 “ {0})))
172 cnvimarndm 5405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑓 “ ran 𝑓) = dom 𝑓
173172difeq1i 3686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑓 “ ran 𝑓) ∖ (𝑓 “ {0})) = (dom 𝑓 ∖ (𝑓 “ {0}))
174171, 173syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓:ℝ⟶ℝ → (𝑓 “ (ran 𝑓 ∖ {0})) = (dom 𝑓 ∖ (𝑓 “ {0})))
175168, 174difeq12d 3691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓:ℝ⟶ℝ → (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0}))) = (dom 𝑓 ∖ (dom 𝑓 ∖ (𝑓 “ {0}))))
176 cnvimass 5404 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 “ {0}) ⊆ dom 𝑓
177 dfss4 3820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓 “ {0}) ⊆ dom 𝑓 ↔ (dom 𝑓 ∖ (dom 𝑓 ∖ (𝑓 “ {0}))) = (𝑓 “ {0}))
178176, 177mpbi 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (dom 𝑓 ∖ (dom 𝑓 ∖ (𝑓 “ {0}))) = (𝑓 “ {0})
179175, 178syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓:ℝ⟶ℝ → (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0}))) = (𝑓 “ {0}))
180179eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0}))) ↔ 𝑧 ∈ (𝑓 “ {0})))
181 ffn 5958 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓:ℝ⟶ℝ → 𝑓 Fn ℝ)
182 fniniseg 6246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 Fn ℝ → (𝑧 ∈ (𝑓 “ {0}) ↔ (𝑧 ∈ ℝ ∧ (𝑓𝑧) = 0)))
183 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑧 ∈ ℝ ∧ (𝑓𝑧) = 0) → (𝑓𝑧) = 0)
184182, 183syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 Fn ℝ → (𝑧 ∈ (𝑓 “ {0}) → (𝑓𝑧) = 0))
185181, 184syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (𝑓 “ {0}) → (𝑓𝑧) = 0))
186180, 185sylbid 229 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓𝑧) = 0))
18735, 186syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 ∈ dom ∫1 → (𝑧 ∈ (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓𝑧) = 0))
188187imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑧 ∈ (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0})))) → (𝑓𝑧) = 0)
189188adantlr 747 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1 ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) ∧ 𝑧 ∈ (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0})))) → (𝑓𝑧) = 0)
190161, 166, 167, 189itg10a 23283 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓 ∈ dom ∫1 ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (∫1𝑓) = 0)
191160, 190sylan 487 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (∫1𝑓) = 0)
192159, 191breqtrd 4609 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑏 < 0)
193158, 192, 137syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎))
194 simprl 790 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) → 𝑓 ∈ dom ∫1)
195 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
196194, 195anim12i 588 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → (𝑓 ∈ dom ∫1𝑏 ∈ ℝ))
19763a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ℝ ∈ V)
198 fvex 6113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓𝑢) ∈ V
199198a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑓𝑢) ∈ V)
200 ovex 6577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V
201200, 105ifex 4106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V
202201a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V)
20335feqmptd 6159 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓 ∈ dom ∫1𝑓 = (𝑢 ∈ ℝ ↦ (𝑓𝑢)))
204203ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 = (𝑢 ∈ ℝ ↦ (𝑓𝑢)))
205 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
206197, 199, 202, 204, 205offval2 6812 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓𝑓 − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑢 ∈ ℝ ↦ ((𝑓𝑢) − if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))))
207 ovif2 6636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓𝑢) − if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑢) − 0))
208172, 163syl5eq 2656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑓:ℝ⟶ℝ → (𝑓 “ ran 𝑓) = ℝ)
209208difeq1d 3689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑓:ℝ⟶ℝ → ((𝑓 “ ran 𝑓) ∖ (𝑓 “ {0})) = (ℝ ∖ (𝑓 “ {0})))
210171, 209eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑓:ℝ⟶ℝ → (𝑓 “ (ran 𝑓 ∖ {0})) = (ℝ ∖ (𝑓 “ {0})))
211210eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑓:ℝ⟶ℝ → (𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (𝑓 “ {0}))))
21235, 211syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑓 ∈ dom ∫1 → (𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (𝑓 “ {0}))))
213212ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (𝑓 “ {0}))))
214 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
215214biantrurd 528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ ¬ 𝑢 ∈ (𝑓 “ {0}))))
216 eldif 3550 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑢 ∈ (ℝ ∖ (𝑓 “ {0})) ↔ (𝑢 ∈ ℝ ∧ ¬ 𝑢 ∈ (𝑓 “ {0})))
217215, 216syl6bbr 277 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (𝑓 “ {0}) ↔ 𝑢 ∈ (ℝ ∖ (𝑓 “ {0}))))
218213, 217bitr4d 270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ 𝑢 ∈ (𝑓 “ {0})))
219218con2bid 343 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (𝑓 “ {0}) ↔ ¬ 𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))))
220 fniniseg 6246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑓 Fn ℝ → (𝑢 ∈ (𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓𝑢) = 0)))
22135, 181, 2203syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑓 ∈ dom ∫1 → (𝑢 ∈ (𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓𝑢) = 0)))
222221ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓𝑢) = 0)))
223219, 222bitr3d 269 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ (𝑢 ∈ ℝ ∧ (𝑓𝑢) = 0)))
224 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑓𝑢) = 0 → ((𝑓𝑢) − 0) = (0 − 0))
225 0m0e0 11007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (0 − 0) = 0
226224, 225syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑓𝑢) = 0 → ((𝑓𝑢) − 0) = 0)
227226adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑢 ∈ ℝ ∧ (𝑓𝑢) = 0) → ((𝑓𝑢) − 0) = 0)
228223, 227syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓𝑢) − 0) = 0))
229228imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) ∧ ¬ 𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓𝑢) − 0) = 0)
230229ifeq2da 4067 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑢) − 0)) = if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
231207, 230syl5eq 2656 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → ((𝑓𝑢) − if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
232231mpteq2dva 4672 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ ((𝑓𝑢) − if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
233206, 232eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓𝑓 − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
234 simpll 786 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 ∈ dom ∫1)
235200a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V)
236 1ex 9914 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1 ∈ V
237236, 105ifex 4106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V
238237a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V)
239 fconstmpt 5085 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (ℝ × {(((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑢 ∈ ℝ ↦ (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))
240239a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (ℝ × {(((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑢 ∈ ℝ ↦ (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
241 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑏 ∈ 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)))
242197, 235, 238, 240, 241offval2 6812 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑢 ∈ ℝ ↦ ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))))
243 ovif2 6636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · 0))
244 resubcl 10224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((∫1𝑓) ∈ ℝ ∧ 𝑏 ∈ ℝ) → ((∫1𝑓) − 𝑏) ∈ ℝ)
2458, 244sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((∫1𝑓) − 𝑏) ∈ ℝ)
246245adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1𝑓) − 𝑏) ∈ ℝ)
247 2re 10967 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2 ∈ ℝ
248 i1fima 23251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑓 ∈ dom ∫1 → (𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol)
249 mblvol 23105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
250248, 249syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑓 ∈ dom ∫1 → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
251 neldifsn 4262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ¬ 0 ∈ (ran 𝑓 ∖ {0})
252 i1fima2 23252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑓 ∈ dom ∫1 ∧ ¬ 0 ∈ (ran 𝑓 ∖ {0})) → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
253251, 252mpan2 703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑓 ∈ dom ∫1 → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
254250, 253eqeltrrd 2689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑓 ∈ dom ∫1 → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
255 remulcl 9900 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((2 ∈ ℝ ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
256247, 254, 255sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑓 ∈ dom ∫1 → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
257256ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
258 2cnd 10970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 2 ∈ ℂ)
259254ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
260259recnd 9947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
261 2ne0 10990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2 ≠ 0
262261a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 2 ≠ 0)
263 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)
264258, 260, 262, 263mulne0d 10558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ≠ 0)
265246, 257, 264redivcld 10732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
266265recnd 9947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
267266mulid1d 9936 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · 1) = (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))
268266mul01d 10114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · 0) = 0)
269267, 268ifeq12d 4056 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · 0)) = if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
270243, 269syl5eq 2656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
271270mpteq2dv 4673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
272242, 271eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1𝑓) − 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})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
273 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))
274273i1f1 23263 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
275248, 253, 274syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 ∈ dom ∫1 → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
276275ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
277276, 265i1fmulc 23276 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) ∈ dom ∫1)
278272, 277eqeltrrd 2689 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1)
279 i1fsub 23281 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ∈ dom ∫1 ∧ (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1) → (𝑓𝑓 − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
280234, 278, 279syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓𝑓 − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
281233, 280eqeltrrd 2689 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
282 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑢 = 𝑧 → (𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))))
283 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
284283oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑢 = 𝑧 → ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) = ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
285282, 284ifbieq1d 4059 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑢 = 𝑧 → if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
286 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
287 ovex 6577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ V
288287, 105ifex 4106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V
289285, 286, 288fvmpt 6191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 ∈ ℝ → ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) = if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
290289breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ ℝ → (0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) ↔ 0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
291290, 289ifbieq1d 4059 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℝ → if(0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), 0) = if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
292 iftrue 4042 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
293 iftrue 4042 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
294293breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → (0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))))
295294, 293ifbieq1d 4059 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
296 iftrue 4042 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
297295, 296sylan9eqr 2666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
298292, 297eqtr4d 2647 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
299 iffalse 4045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
300 ianor 508 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (¬ (0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) ↔ (¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∨ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))))
301294ifbid 4058 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
302 iffalse 4045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
303301, 302sylan9eqr 2666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
304303ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) → (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0))
305 iffalse 4045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
306 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 0 = 0
307 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) → (if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 ↔ if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0))
308 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (0 = if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) → (0 = 0 ↔ if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0))
309307, 308ifboth 4074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 ∧ 0 = 0) → if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
310305, 306, 309sylancl 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
311304, 310pm2.61d1 170 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
312311, 310jaoi 393 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∨ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
313300, 312sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
314299, 313eqtr4d 2647 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (¬ (0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
315298, 314pm2.61i 175 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0)
316291, 315syl6reqr 2663 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ ℝ → if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), 0))
317316mpteq2ia 4668 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑧 ∈ ℝ ↦ if(0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), 0))
318317i1fpos 23279 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
319281, 318syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
320196, 319sylan 487 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
321196, 265sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
3228ad2antrl 760 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) → (∫1𝑓) ∈ ℝ)
323322, 195, 244syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → ((∫1𝑓) − 𝑏) ∈ ℝ)
324323adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1𝑓) − 𝑏) ∈ ℝ)
325256adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓))) → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
326325ad3antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
327 simprl 790 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑏 < (∫1𝑓))
328 simprr 792 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑏 ∈ ℝ)
329141ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → (∫1𝑓) ∈ ℝ)
330328, 329posdifd 10493 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → (𝑏 < (∫1𝑓) ↔ 0 < ((∫1𝑓) − 𝑏)))
331327, 330mpbid 221 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → 0 < ((∫1𝑓) − 𝑏))
332331adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < ((∫1𝑓) − 𝑏))
333254adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓))) → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
334333ad3antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
335 mblss 23106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol → (𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
336 ovolge0 23056 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ → 0 ≤ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
337248, 335, 3363syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 ∈ dom ∫1 → 0 ≤ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
338 ltlen 10017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
33948, 254, 338sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 ∈ dom ∫1 → (0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ↔ (0 ≤ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)))
340339biimprd 237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 ∈ dom ∫1 → ((0 ≤ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
341337, 340mpand 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 ∈ dom ∫1 → ((vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0 → 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
342341ad2antrl 760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) → ((vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0 → 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
343342imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
344343adantlr 747 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
345 2pos 10989 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 < 2
346 mulgt0 9994 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2 ∈ ℝ ∧ 0 < 2) ∧ ((vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ ∧ 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) → 0 < (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
347247, 345, 346mpanl12 714 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ ∧ 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) → 0 < (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
348334, 344, 347syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
349324, 326, 332, 348divgt0d 10838 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))
350321, 349elrpd 11745 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ+)
351 simprl 790 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓))) → 𝑓𝑟𝐹)
352351ad3antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓𝑟𝐹)
353 ffn 5958 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹:ℝ⟶(0[,]+∞) → 𝐹 Fn ℝ)
35435, 181syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 ∈ dom ∫1𝑓 Fn ℝ)
355354adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓))) → 𝑓 Fn ℝ)
356 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → 𝑓 Fn ℝ)
357 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → 𝐹 Fn ℝ)
35863a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → ℝ ∈ V)
359 inidm 3784 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (ℝ ∩ ℝ) = ℝ
360 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) = (𝑓𝑧))
361 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
362356, 357, 358, 358, 359, 360, 361ofrfval 6803 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → (𝑓𝑟𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓𝑧) ≤ (𝐹𝑧)))
363353, 355, 362syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) → (𝑓𝑟𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓𝑧) ≤ (𝐹𝑧)))
364363ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓𝑟𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓𝑧) ≤ (𝐹𝑧)))
365 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓))) → 𝑓 ∈ dom ∫1)
366365anim2i 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) → (𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1))
367366, 195anim12i 588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ))
368 breq1 4586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0 = if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) → (0 ≤ (𝐹𝑧) ↔ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹𝑧)))
369 breq1 4586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) = if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹𝑧) ↔ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹𝑧)))
370 simplll 794 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝐹:ℝ⟶(0[,]+∞))
371370ffvelrnda 6267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ∈ (0[,]+∞))
372371, 100sylib 207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹𝑧)))
373372simprd 478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹𝑧))
374373ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) ≤ (𝐹𝑧)) ∧ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) → 0 ≤ (𝐹𝑧))
375 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) = if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → (((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) = (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
376375breq1d 4593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) = if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → ((((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹𝑧) ↔ (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹𝑧)))
377 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (0 = if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → (0 + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) = (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
378377breq1d 4593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0 = if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → ((0 + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹𝑧) ↔ (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹𝑧)))
37935ad3antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓:ℝ⟶ℝ)
380379ffvelrnda 6267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
381380recnd 9947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℂ)
382245recnd 9947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((∫1𝑓) − 𝑏) ∈ ℂ)
383382adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1𝑓) − 𝑏) ∈ ℂ)
384256recnd 9947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑓 ∈ dom ∫1 → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℂ)
385384ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℂ)
386383, 385, 264divcld 10680 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
387386adantlll 750 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
388387adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
389381, 388npcand 10275 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) = (𝑓𝑧))
390389adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) ≤ (𝐹𝑧)) → (((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) = (𝑓𝑧))
391 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧))
392390, 391eqbrtrd 4605 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) ≤ (𝐹𝑧)) → (((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹𝑧))
393392ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) ≤ (𝐹𝑧)) ∧ ¬ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) ∧ (0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})))) → (((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹𝑧))
394299pm2.24d 146 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (¬ (0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → (¬ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 → (0 + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹𝑧)))
395394impcom 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((¬ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 ∧ ¬ (0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})))) → (0 + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹𝑧))
396395adantll 746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) ≤ (𝐹𝑧)) ∧ ¬ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) ∧ ¬ (0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})))) → (0 + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹𝑧))
397376, 378, 393, 396ifbothda 4073 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) ≤ (𝐹𝑧)) ∧ ¬ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) → (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹𝑧))
398368, 369, 374, 397ifbothda 4073 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) ≤ (𝐹𝑧)) → if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹𝑧))
399398ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) ≤ (𝐹𝑧) → if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹𝑧)))
400367, 399sylanl1 680 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) ≤ (𝐹𝑧) → if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹𝑧)))
401400ralimdva 2945 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∀𝑧 ∈ ℝ (𝑓𝑧) ≤ (𝐹𝑧) → ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹𝑧)))
402364, 401sylbid 229 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓𝑟𝐹 → ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹𝑧)))
403352, 402mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹𝑧))
404 ovex 6577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ V
405105, 404ifex 4106 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) ∈ V
406405a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) ∈ V)
407 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))) = (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))))
408104, 406, 99, 407, 67ofrfval2 6813 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹:ℝ⟶(0[,]+∞) → ((𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟𝐹 ↔ ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹𝑧)))
409408ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟𝐹 ↔ ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹𝑧)))
410403, 409mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟𝐹)
411 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) → (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦) = (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
412411ifeq2d 4055 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) → if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦)) = if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))))
413412mpteq2dv 4673 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) → (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))))
414413breq1d 4593 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) → ((𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟𝐹 ↔ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟𝐹))
415414rspcev 3282 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ+ ∧ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟𝐹) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟𝐹)
416350, 410, 415syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟𝐹)
417 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = 𝑔 → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1𝑔))
418417eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1𝑔))
419418biantrud 527 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1𝑔))))
420 nfmpt1 4675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑧(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
421420nfeq2 2766 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑧 𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
422 fveq1 6102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (𝑔𝑧) = ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧))
423287, 105ifex 4106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V
424 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
425424fvmpt2 6200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑧 ∈ ℝ ∧ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V) → ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) = if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
426423, 425mpan2 703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 ∈ ℝ → ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) = if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
427422, 426sylan9eq 2664 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
428427eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) = 0 ↔ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0))
429427oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + 𝑦) = (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))
430428, 429ifbieq2d 4061 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦)) = if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦)))
431421, 430mpteq2da 4671 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))))
432431breq1d 4593 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ↔ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟𝐹))
433432rexbidv 3034 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟𝐹))
434419, 433bitr3d 269 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1𝑔)) ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟𝐹))
435434rspcev 3282 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 ∧ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟𝐹) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1𝑔)))
436320, 416, 435syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1𝑔)))
437 simplrr 797 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 ∈ ℝ)
438200a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V)
439236, 105ifex 4106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V
440439a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V)
441 fconstmpt 5085 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (ℝ × {(((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑧 ∈ ℝ ↦ (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))
442441a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (ℝ × {(((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑧 ∈ ℝ ↦ (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
443 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓 ∈ dom ∫1𝑏 ∈ 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)))
444197, 438, 440, 442, 443offval2 6812 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑧 ∈ ℝ ↦ ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))))
445 ovif2 6636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · 0))
446267, 268ifeq12d 4056 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · 0)) = if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
447445, 446syl5eq 2656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
448447mpteq2dv 4673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
449444, 448eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1𝑓) − 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})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
450 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))
451450i1f1 23263 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
452248, 253, 451syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 ∈ dom ∫1 → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
453452ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
454453, 265i1fmulc 23276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) ∈ dom ∫1)
455449, 454eqeltrrd 2689 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1)
456 i1fsub 23281 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓 ∈ dom ∫1 ∧ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1) → (𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
457234, 455, 456syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
458 itg1cl 23258 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1 → (∫1‘(𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ∈ ℝ)
459457, 458syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ∈ ℝ)
460459adantlrl 752 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ∈ ℝ)
461319adantlrl 752 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
462 itg1cl 23258 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) ∈ ℝ)
463461, 462syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) ∈ ℝ)
464 simplrl 796 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1𝑓))
465 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
4668adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → (∫1𝑓) ∈ ℝ)
46797a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → 2 ∈ ℝ+)
468465, 466, 467ltdiv1d 11793 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → (𝑏 < (∫1𝑓) ↔ (𝑏 / 2) < ((∫1𝑓) / 2)))
469 recn 9905 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 ∈ ℝ → 𝑏 ∈ ℂ)
4704692halvesd 11155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 ∈ ℝ → ((𝑏 / 2) + (𝑏 / 2)) = 𝑏)
471470oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑏 ∈ ℝ → (((𝑏 / 2) + (𝑏 / 2)) − (𝑏 / 2)) = (𝑏 − (𝑏 / 2)))
472469halfcld 11154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 ∈ ℝ → (𝑏 / 2) ∈ ℂ)
473472, 472pncand 10272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑏 ∈ ℝ → (((𝑏 / 2) + (𝑏 / 2)) − (𝑏 / 2)) = (𝑏 / 2))
474471, 473eqtr3d 2646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ∈ ℝ → (𝑏 − (𝑏 / 2)) = (𝑏 / 2))
475474breq1d 4593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 ∈ ℝ → ((𝑏 − (𝑏 / 2)) < ((∫1𝑓) / 2) ↔ (𝑏 / 2) < ((∫1𝑓) / 2)))
476475adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((𝑏 − (𝑏 / 2)) < ((∫1𝑓) / 2) ↔ (𝑏 / 2) < ((∫1𝑓) / 2)))
477 rehalfcl 11135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ∈ ℝ → (𝑏 / 2) ∈ ℝ)
478477adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → (𝑏 / 2) ∈ ℝ)
4798rehalfcld 11156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 ∈ dom ∫1 → ((∫1𝑓) / 2) ∈ ℝ)
480479adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((∫1𝑓) / 2) ∈ ℝ)
481465, 478, 480ltsubaddd 10502 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((𝑏 − (𝑏 / 2)) < ((∫1𝑓) / 2) ↔ 𝑏 < (((∫1𝑓) / 2) + (𝑏 / 2))))
482468, 476, 4813bitr2d 295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → (𝑏 < (∫1𝑓) ↔ 𝑏 < (((∫1𝑓) / 2) + (𝑏 / 2))))
483482adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑏 < (∫1𝑓) ↔ 𝑏 < (((∫1𝑓) / 2) + (𝑏 / 2))))
484483adantlrl 752 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑏 < (∫1𝑓) ↔ 𝑏 < (((∫1𝑓) / 2) + (𝑏 / 2))))
485464, 484mpbid 221 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (((∫1𝑓) / 2) + (𝑏 / 2)))
486453, 265itg1mulc 23277 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘((ℝ × {(((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))))
487449fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘((ℝ × {(((∫1𝑓) − 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})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))))
488450itg11 23264 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))
489248, 253, 488syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 ∈ dom ∫1 → (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))))
490489oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 ∈ dom ∫1 → ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · (vol‘(𝑓 “ (ran 𝑓 ∖ {0})))))
491490ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · (vol‘(𝑓 “ (ran 𝑓 ∖ {0})))))
492253recnd 9947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 ∈ dom ∫1 → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
493492ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
494266, 493mulcomd 9940 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · (vol‘(𝑓 “ (ran 𝑓 ∖ {0})))) = ((vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) · (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
495250ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
496495oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1𝑓) − 𝑏)) = ((vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1𝑓) − 𝑏)))
497260, 383mulcomd 9940 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1𝑓) − 𝑏)) = (((∫1𝑓) − 𝑏) · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
498496, 497eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1𝑓) − 𝑏)) = (((∫1𝑓) − 𝑏) · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
499498oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1𝑓) − 𝑏)) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) = ((((∫1𝑓) − 𝑏) · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))
500493, 383, 385, 264divassd 10715 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1𝑓) − 𝑏)) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) = ((vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) · (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
501383, 258, 260, 262, 263divcan5rd 10707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1𝑓) − 𝑏) · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) = (((∫1𝑓) − 𝑏) / 2))
502499, 500, 5013eqtr3d 2652 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) · (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) = (((∫1𝑓) − 𝑏) / 2))
503491, 494, 5023eqtrd 2648 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = (((∫1𝑓) − 𝑏) / 2))
504486, 487, 5033eqtr3d 2652 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (((∫1𝑓) − 𝑏) / 2))
505504oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1𝑓) − (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = ((∫1𝑓) − (((∫1𝑓) − 𝑏) / 2)))
506 itg1sub 23282 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓 ∈ dom ∫1 ∧ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1) → (∫1‘(𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = ((∫1𝑓) − (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))))
507234, 455, 506syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = ((∫1𝑓) − (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))))
5088recnd 9947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 ∈ dom ∫1 → (∫1𝑓) ∈ ℂ)
509508ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1𝑓) ∈ ℂ)
510469ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 ∈ ℂ)
511509, 510, 258, 262divsubdird 10719 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − 𝑏) / 2) = (((∫1𝑓) / 2) − (𝑏 / 2)))
512511oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1𝑓) − (((∫1𝑓) − 𝑏) / 2)) = ((∫1𝑓) − (((∫1𝑓) / 2) − (𝑏 / 2))))
513508adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → (∫1𝑓) ∈ ℂ)
514513halfcld 11154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((∫1𝑓) / 2) ∈ ℂ)
515472adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → (𝑏 / 2) ∈ ℂ)
516513, 514, 515subsubd 10299 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((∫1𝑓) − (((∫1𝑓) / 2) − (𝑏 / 2))) = (((∫1𝑓) − ((∫1𝑓) / 2)) + (𝑏 / 2)))
517516adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1𝑓) − (((∫1𝑓) / 2) − (𝑏 / 2))) = (((∫1𝑓) − ((∫1𝑓) / 2)) + (𝑏 / 2)))
5185082halvesd 11155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 ∈ dom ∫1 → (((∫1𝑓) / 2) + ((∫1𝑓) / 2)) = (∫1𝑓))
519518oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 ∈ dom ∫1 → ((((∫1𝑓) / 2) + ((∫1𝑓) / 2)) − ((∫1𝑓) / 2)) = ((∫1𝑓) − ((∫1𝑓) / 2)))
520508halfcld 11154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 ∈ dom ∫1 → ((∫1𝑓) / 2) ∈ ℂ)
521520, 520pncand 10272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 ∈ dom ∫1 → ((((∫1𝑓) / 2) + ((∫1𝑓) / 2)) − ((∫1𝑓) / 2)) = ((∫1𝑓) / 2))
522519, 521eqtr3d 2646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 ∈ dom ∫1 → ((∫1𝑓) − ((∫1𝑓) / 2)) = ((∫1𝑓) / 2))
523522oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 ∈ dom ∫1 → (((∫1𝑓) − ((∫1𝑓) / 2)) + (𝑏 / 2)) = (((∫1𝑓) / 2) + (𝑏 / 2)))
524523ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − ((∫1𝑓) / 2)) + (𝑏 / 2)) = (((∫1𝑓) / 2) + (𝑏 / 2)))
525512, 517, 5243eqtrrd 2649 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) / 2) + (𝑏 / 2)) = ((∫1𝑓) − (((∫1𝑓) − 𝑏) / 2)))
526505, 507, 5253eqtr4d 2654 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = (((∫1𝑓) / 2) + (𝑏 / 2)))
527526adantlrl 752 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = (((∫1𝑓) / 2) + (𝑏 / 2)))
528485, 527breqtrrd 4611 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))))
529457adantlrl 752 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
530 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0))
531530adantlrl 752 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0))
532234, 36sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
533265adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
534532, 533resubcld 10337 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ ℝ)
535534leidd 10473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
536535adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
537296breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) → (((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))))
538537adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) → (((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))))
539536, 538mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
540534adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ ℝ)
54148a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) → 0 ∈ ℝ)
54248a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → 0 ∈ ℝ)
543534, 542ltnled 10063 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) < 0 ↔ ¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))))
544543biimpar 501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) < 0)
545540, 541, 544ltled 10064 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ 0)
546 iffalse 4045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
547546breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) → (((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ 0))
548547adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) → (((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ 0))
549545, 548mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
550539, 549pm2.61dan 828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
551531, 550sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
552551adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
553 iftrue 4042 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0) = (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))
554553oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
555 iba 523 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → (0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ↔ (0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})))))
556555bicomd 212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → ((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) ↔ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))))
557556ifbid 4058 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
558554, 557breq12d 4596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → (((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
559558adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → (((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
560552, 559mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
56135ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓:ℝ⟶ℝ)
562171eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑧 ∈ ((𝑓 “ ran 𝑓) ∖ (𝑓 “ {0}))))
563 eldif 3550 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑧 ∈ ((𝑓 “ ran 𝑓) ∖ (𝑓 “ {0})) ↔ (𝑧 ∈ (𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (𝑓 “ {0})))
564562, 563syl6bb 275 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ (𝑧 ∈ (𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (𝑓 “ {0}))))
565564notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑓:ℝ⟶ℝ → (¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ (𝑧 ∈ (𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (𝑓 “ {0}))))
566565adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ (𝑧 ∈ (𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (𝑓 “ {0}))))
567 pm4.53 512 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (¬ (𝑧 ∈ (𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (𝑓 “ {0})) ↔ (¬ 𝑧 ∈ (𝑓 “ ran 𝑓) ∨ 𝑧 ∈ (𝑓 “ {0})))
568208eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (𝑓 “ ran 𝑓) ↔ 𝑧 ∈ ℝ))
569568biimpar 501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ (𝑓 “ ran 𝑓))
570569pm2.24d 146 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (𝑓 “ ran 𝑓) → (𝑓𝑧) = 0))
571182simplbda 652 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑓 Fn ℝ ∧ 𝑧 ∈ (𝑓 “ {0})) → (𝑓𝑧) = 0)
572571ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑓 Fn ℝ → (𝑧 ∈ (𝑓 “ {0}) → (𝑓𝑧) = 0))
573181, 572syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (𝑓 “ {0}) → (𝑓𝑧) = 0))
574573adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (𝑧 ∈ (𝑓 “ {0}) → (𝑓𝑧) = 0))
575570, 574jaod 394 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → ((¬ 𝑧 ∈ (𝑓 “ ran 𝑓) ∨ 𝑧 ∈ (𝑓 “ {0})) → (𝑓𝑧) = 0))
576567, 575syl5bi 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ (𝑧 ∈ (𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (𝑓 “ {0})) → (𝑓𝑧) = 0))
577566, 576sylbid 229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → (𝑓𝑧) = 0))
578577imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓𝑧) = 0)
579561, 578sylanl1 680 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓𝑧) = 0)
580579oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓𝑧) − 0) = (0 − 0))
581580, 225syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓𝑧) − 0) = 0)
582581, 30syl6eqbr 4622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓𝑧) − 0) ≤ 0)
583 iffalse 4045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0) = 0)
584583oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = ((𝑓𝑧) − 0))
585300, 299sylbir 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((¬ 0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∨ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
586585olcs 409 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
587584, 586breq12d 4596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → (((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓𝑧) − 0) ≤ 0))
588587adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → (((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓𝑧) − 0) ≤ 0))
589582, 588mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
590560, 589pm2.61dan 828 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
591590ralrimiva 2949 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∀𝑧 ∈ ℝ ((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
59263a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ℝ ∈ V)
593 ovex 6577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ V
594593a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ V)
595423a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V)
596 fvex 6113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓𝑧) ∈ V
597596a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ V)
598200, 105ifex 4106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V
599598a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V)
60070ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓𝑧)))
601 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
602592, 597, 599, 600, 601offval2 6812 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑧 ∈ ℝ ↦ ((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))))
603 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
604592, 594, 595, 602, 603ofrfval2 6813 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∘𝑟 ≤ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ↔ ∀𝑧 ∈ ℝ ((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
605591, 604mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∘𝑟 ≤ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
606 itg1le 23286 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1 ∧ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 ∧ (𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∘𝑟 ≤ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (∫1‘(𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ≤ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
607529, 461, 605, 606syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ≤ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
608437, 460, 463, 528, 607ltletrd 10076 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
609608adantllr 751 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
610609adantlll 750 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
611 fvex 6113 . . . . . . . . . . . . . . . . . . . . . . 23 (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) ∈ V
612 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (𝑎 = (∫1𝑔) ↔ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1𝑔)))
613612anbi2d 736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1𝑔))))
614613rexbidv 3034 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1𝑔))))
615 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (𝑏 < 𝑎𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))))
616614, 615anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎) ↔ (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1𝑔)) ∧ 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))))
617611, 616spcev 3273 . . . . . . . . . . . . . . . . . . . . . 22 ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1𝑔)) ∧ 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎))
618436, 610, 617syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎))
619193, 618pm2.61dane 2869 . . . . . . . . . . . . . . . . . . . 20 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎))
620619expr 641 . . . . . . . . . . . . . . . . . . 19 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) → (𝑏 ∈ ℝ → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎)))
621620adantllr 751 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) → (𝑏 ∈ ℝ → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎)))
622621adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 ≠ -∞) → (𝑏 ∈ ℝ → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎)))
623157, 622mpd 15 . . . . . . . . . . . . . . . 16 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 ≠ -∞) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎))
624139, 623pm2.61dane 2869 . . . . . . . . . . . . . . 15 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎))
625624ex 449 . . . . . . . . . . . . . 14 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) → (𝑏 < (∫1𝑓) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎)))
62694, 625sylbid 229 . . . . . . . . . . . . 13 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) → (𝑏 < 𝑠 → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎)))
627626imp 444 . . . . . . . . . . . 12 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < 𝑠) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎))
628627an32s 842 . . . . . . . . . . 11 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ 𝑏 < 𝑠) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓𝑟𝐹𝑠 = (∫1𝑓)))) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘𝑟𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎))
629628rexlimdvaa 3014 . . . . . . . . . 10 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ 𝑏 < 𝑠) → (∃𝑓 ∈ dom ∫1(𝑓𝑟𝐹𝑠 = (∫1𝑓)) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔