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Theorem itg2addnclem 35107
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 2801 . . 3 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓r𝐹𝑥 = (∫1𝑓))} = {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓r𝐹𝑥 = (∫1𝑓))}
21itg2val 24336 . 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 8899 . . 3 sup(𝐿, ℝ*, < ) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘r𝐹𝑥 = (∫1𝑔))}, ℝ*, < )
5 xrltso 12526 . . . . 5 < Or ℝ*
65a1i 11 . . . 4 (𝐹:ℝ⟶(0[,]+∞) → < Or ℝ*)
7 simprr 772 . . . . . . . 8 ((𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑥 = (∫1𝑓))) → 𝑥 = (∫1𝑓))
8 itg1cl 24293 . . . . . . . . . 10 (𝑓 ∈ dom ∫1 → (∫1𝑓) ∈ ℝ)
98rexrd 10684 . . . . . . . . 9 (𝑓 ∈ dom ∫1 → (∫1𝑓) ∈ ℝ*)
109adantr 484 . . . . . . . 8 ((𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑥 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ*)
117, 10eqeltrd 2893 . . . . . . 7 ((𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑥 = (∫1𝑓))) → 𝑥 ∈ ℝ*)
1211rexlimiva 3243 . . . . . 6 (∃𝑓 ∈ dom ∫1(𝑓r𝐹𝑥 = (∫1𝑓)) → 𝑥 ∈ ℝ*)
1312abssi 4000 . . . . 5 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓r𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*
14 supxrcl 12700 . . . . 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 6648 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (𝑔𝑧) = (𝑓𝑧))
1716eqeq1d 2803 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((𝑔𝑧) = 0 ↔ (𝑓𝑧) = 0))
1816oveq1d 7154 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((𝑔𝑧) + 𝑦) = ((𝑓𝑧) + 𝑦))
1917, 18ifbieq2d 4453 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦)) = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)))
2019mpteq2dv 5129 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))))
2120breq1d 5043 . . . . . . . . . . . 12 (𝑔 = 𝑓 → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘r𝐹 ↔ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘r𝐹))
2221rexbidv 3259 . . . . . . . . . . 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 6649 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (∫1𝑔) = (∫1𝑓))
2423eqeq2d 2812 . . . . . . . . . . 11 (𝑔 = 𝑓 → (𝑥 = (∫1𝑔) ↔ 𝑥 = (∫1𝑓)))
2522, 24anbi12d 633 . . . . . . . . . 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))))
2625cbvrexvw 3400 . . . . . . . . 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 5037 . . . . . . . . . . . . . . . . 17 (0 = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) → ((𝑓𝑧) ≤ 0 ↔ (𝑓𝑧) ≤ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))))
28 breq2 5037 . . . . . . . . . . . . . . . . 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 11730 . . . . . . . . . . . . . . . . . . 19 0 ≤ 0
3129, 30eqbrtrdi 5072 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑧) = 0 → (𝑓𝑧) ≤ 0)
3231adantl 485 . . . . . . . . . . . . . . . . 17 ((((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → (𝑓𝑧) ≤ 0)
33 rpge0 12394 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
3433ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 (((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 0 ≤ 𝑦)
35 i1ff 24284 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 ∈ dom ∫1𝑓:ℝ⟶ℝ)
3635ffvelrnda 6832 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ dom ∫1𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
3736adantlr 714 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
38 rpre 12389 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
3938ad2antlr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
4037, 39addge01d 11221 . . . . . . . . . . . . . . . . . . 19 (((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (0 ≤ 𝑦 ↔ (𝑓𝑧) ≤ ((𝑓𝑧) + 𝑦)))
4134, 40mpbid 235 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ≤ ((𝑓𝑧) + 𝑦))
4241adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓𝑧) = 0) → (𝑓𝑧) ≤ ((𝑓𝑧) + 𝑦))
4327, 28, 32, 42ifbothda 4465 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ≤ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)))
4443adantlll 717 . . . . . . . . . . . . . . 15 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ≤ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)))
4535ad2antlr 726 . . . . . . . . . . . . . . . . . 18 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝑓:ℝ⟶ℝ)
4645ffvelrnda 6832 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
4746rexrd 10684 . . . . . . . . . . . . . . . 16 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ*)
48 0re 10636 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
4938ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
5046, 49readdcld 10663 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + 𝑦) ∈ ℝ)
51 ifcl 4472 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ ∧ ((𝑓𝑧) + 𝑦) ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ∈ ℝ)
5248, 50, 51sylancr 590 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ∈ ℝ)
5352rexrd 10684 . . . . . . . . . . . . . . . 16 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ∈ ℝ*)
54 iccssxr 12812 . . . . . . . . . . . . . . . . . . 19 (0[,]+∞) ⊆ ℝ*
55 fss 6505 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℝ⟶(0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐹:ℝ⟶ℝ*)
5654, 55mpan2 690 . . . . . . . . . . . . . . . . . 18 (𝐹:ℝ⟶(0[,]+∞) → 𝐹:ℝ⟶ℝ*)
5756ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝐹:ℝ⟶ℝ*)
5857ffvelrnda 6832 . . . . . . . . . . . . . . . 16 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ∈ ℝ*)
59 xrletr 12543 . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . 14 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ≤ (𝐹𝑧) → (𝑓𝑧) ≤ (𝐹𝑧)))
6261ralimdva 3147 . . . . . . . . . . . . 13 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (∀𝑧 ∈ ℝ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦)) ≤ (𝐹𝑧) → ∀𝑧 ∈ ℝ (𝑓𝑧) ≤ (𝐹𝑧)))
63 reex 10621 . . . . . . . . . . . . . . 15 ℝ ∈ V
6463a1i 11 . . . . . . . . . . . . . 14 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ℝ ∈ V)
65 eqidd 2802 . . . . . . . . . . . . . 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 6712 . . . . . . . . . . . . . . 15 (𝐹:ℝ⟶(0[,]+∞) → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹𝑧)))
6867ad2antrr 725 . . . . . . . . . . . . . 14 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹𝑧)))
6964, 52, 58, 65, 68ofrfval2 7411 . . . . . . . . . . . . 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 6712 . . . . . . . . . . . . . . 15 (𝑓 ∈ dom ∫1𝑓 = (𝑧 ∈ ℝ ↦ (𝑓𝑧)))
7170ad2antlr 726 . . . . . . . . . . . . . 14 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓𝑧)))
7264, 46, 58, 71, 68ofrfval2 7411 . . . . . . . . . . . . 13 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (𝑓r𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓𝑧) ≤ (𝐹𝑧)))
7362, 69, 723imtr4d 297 . . . . . . . . . . . 12 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘r𝐹𝑓r𝐹))
7473rexlimdva 3246 . . . . . . . . . . 11 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘r𝐹𝑓r𝐹))
7574anim1d 613 . . . . . . . . . 10 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑦))) ∘r𝐹𝑥 = (∫1𝑓)) → (𝑓r𝐹𝑥 = (∫1𝑓))))
7675reximdva 3236 . . . . . . . . 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 245 . . . . . . . 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 3994 . . . . . . 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 3917 . . . . . 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 1199 . . . . . . . . . . 11 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑥 = (∫1𝑓))) → 𝑥 = (∫1𝑓))
8193ad2ant2 1131 . . . . . . . . . . 11 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑥 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ*)
8280, 81eqeltrd 2893 . . . . . . . . . 10 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑥 = (∫1𝑓))) → 𝑥 ∈ ℝ*)
8382rexlimdv3a 3248 . . . . . . . . 9 (𝐹:ℝ⟶(0[,]+∞) → (∃𝑓 ∈ dom ∫1(𝑓r𝐹𝑥 = (∫1𝑓)) → 𝑥 ∈ ℝ*))
8483abssdv 3999 . . . . . . . 8 (𝐹:ℝ⟶(0[,]+∞) → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓r𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*)
85 xrsupss 12694 . . . . . . . 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 8911 . . . . . 6 (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓r𝐹𝑥 = (∫1𝑓))} → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) < 𝑏))
8879, 87syld 47 . . . . 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 410 . . . 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 12710 . . . . . . . 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 689 . . . . . . 7 (𝑏 ∈ ℝ* → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓r𝐹𝑥 = (∫1𝑓))}𝑏 < 𝑠))
9291adantl 485 . . . . . 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 781 . . . . . . . . . . . . . . 15 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) → 𝑠 = (∫1𝑓))
9493breq2d 5045 . . . . . . . . . . . . . 14 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) → (𝑏 < 𝑠𝑏 < (∫1𝑓)))
95 simplll 774 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) → 𝐹:ℝ⟶(0[,]+∞))
96 i1f0 24295 . . . . . . . . . . . . . . . . . . 19 (ℝ × {0}) ∈ dom ∫1
97 2rp 12386 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ+
9897ne0ii 4256 . . . . . . . . . . . . . . . . . . . 20 + ≠ ∅
99 ffvelrn 6830 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ∈ (0[,]+∞))
100 elxrge0 12839 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹𝑧) ∈ (0[,]+∞) ↔ ((𝐹𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹𝑧)))
10199, 100sylib 221 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹𝑧)))
102101simprd 499 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹𝑧))
103102ralrimiva 3152 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:ℝ⟶(0[,]+∞) → ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧))
10463a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:ℝ⟶(0[,]+∞) → ℝ ∈ V)
105 c0ex 10628 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
106105a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → 0 ∈ V)
107 eqidd 2802 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
108104, 106, 99, 107, 67ofrfval2 7411 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:ℝ⟶(0[,]+∞) → ((𝑧 ∈ ℝ ↦ 0) ∘r𝐹 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧)))
109103, 108mpbird 260 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ 0) ∘r𝐹)
110109ralrimivw 3153 . . . . . . . . . . . . . . . . . . . 20 (𝐹:ℝ⟶(0[,]+∞) → ∀𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹)
111 r19.2z 4401 . . . . . . . . . . . . . . . . . . . 20 ((ℝ+ ≠ ∅ ∧ ∀𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹)
11298, 110, 111sylancr 590 . . . . . . . . . . . . . . . . . . 19 (𝐹:ℝ⟶(0[,]+∞) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹)
113 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 = (ℝ × {0}) → (∫1𝑔) = (∫1‘(ℝ × {0})))
114 itg10 24296 . . . . . . . . . . . . . . . . . . . . . . 23 (∫1‘(ℝ × {0})) = 0
115113, 114eqtr2di 2853 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 = (ℝ × {0}) → 0 = (∫1𝑔))
116115biantrud 535 . . . . . . . . . . . . . . . . . . . . 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 6648 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔 = (ℝ × {0}) → (𝑔𝑧) = ((ℝ × {0})‘𝑧))
118105fvconst2 6947 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
119117, 118sylan9eq 2856 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = 0)
120 iftrue 4434 . . . . . . . . . . . . . . . . . . . . . . . . 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 5128 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ 0))
123122breq1d 5043 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘r𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹))
124123rexbidv 3259 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = (ℝ × {0}) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘r𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹))
125116, 124bitr3d 284 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = (ℝ × {0}) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑦))) ∘r𝐹 ∧ 0 = (∫1𝑔)) ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹))
126125rspcev 3574 . . . . . . . . . . . . . . . . . . 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 590 . . . . . . . . . . . . . . . . . 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 12510 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ ℝ → -∞ < 0)
13048, 129mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑏 = -∞ → -∞ < 0)
131128, 130eqbrtrd 5055 . . . . . . . . . . . . . . . . . 18 (𝑏 = -∞ → 𝑏 < 0)
132 eqeq1 2805 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 0 → (𝑎 = (∫1𝑔) ↔ 0 = (∫1𝑔)))
133132anbi2d 631 . . . . . . . . . . . . . . . . . . . . 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 3259 . . . . . . . . . . . . . . . . . . . 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 5037 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 0 → (𝑏 < 𝑎𝑏 < 0))
136134, 135anbi12d 633 . . . . . . . . . . . . . . . . . . 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 3558 . . . . . . . . . . . . . . . . . 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 598 . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . 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 783 . . . . . . . . . . . . . . . . . 18 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ∈ ℝ*)
1418adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ)
142141ad3antlr 730 . . . . . . . . . . . . . . . . . 18 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 ≠ -∞) → (∫1𝑓) ∈ ℝ)
143 simpllr 775 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) → 𝑏 ∈ ℝ*)
144 ngtmnft 12551 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 ∈ ℝ* → (𝑏 = -∞ ↔ ¬ -∞ < 𝑏))
145144biimprd 251 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 ∈ ℝ* → (¬ -∞ < 𝑏𝑏 = -∞))
146145necon1ad 3007 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ ℝ* → (𝑏 ≠ -∞ → -∞ < 𝑏))
147146imp 410 . . . . . . . . . . . . . . . . . . 19 ((𝑏 ∈ ℝ*𝑏 ≠ -∞) → -∞ < 𝑏)
148143, 147sylan 583 . . . . . . . . . . . . . . . . . 18 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 ≠ -∞) → -∞ < 𝑏)
149 simpr 488 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → 𝑏 ∈ ℝ*)
1509adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ*)
151149, 150anim12i 615 . . . . . . . . . . . . . . . . . . . 20 (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) → (𝑏 ∈ ℝ* ∧ (∫1𝑓) ∈ ℝ*))
152 xrltle 12534 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 ∈ ℝ* ∧ (∫1𝑓) ∈ ℝ*) → (𝑏 < (∫1𝑓) → 𝑏 ≤ (∫1𝑓)))
153152imp 410 . . . . . . . . . . . . . . . . . . . 20 (((𝑏 ∈ ℝ* ∧ (∫1𝑓) ∈ ℝ*) ∧ 𝑏 < (∫1𝑓)) → 𝑏 ≤ (∫1𝑓))
154151, 153sylan 583 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) → 𝑏 ≤ (∫1𝑓))
155154adantr 484 . . . . . . . . . . . . . . . . . 18 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ≤ (∫1𝑓))
156 xrre 12554 . . . . . . . . . . . . . . . . . 18 (((𝑏 ∈ ℝ* ∧ (∫1𝑓) ∈ ℝ) ∧ (-∞ < 𝑏𝑏 ≤ (∫1𝑓))) → 𝑏 ∈ ℝ)
157140, 142, 148, 155, 156syl22anc 837 . . . . . . . . . . . . . . . . 17 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ 𝑏 < (∫1𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ∈ ℝ)
158127ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑏 < (∫1𝑓))
160 simplrl 776 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑓 ∈ dom ∫1)
161 simpl 486 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 ∈ dom ∫1 ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑓 ∈ dom ∫1)
162 cnvimass 5920 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 “ (ran 𝑓 ∖ {0})) ⊆ dom 𝑓
163162, 35fssdm 6508 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 ∈ dom ∫1 → (𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
164163adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 ∈ dom ∫1 ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
165 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 ∈ dom ∫1 ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0)
166 fdm 6499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓:ℝ⟶ℝ → dom 𝑓 = ℝ)
167166eqcomd 2807 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓:ℝ⟶ℝ → ℝ = dom 𝑓)
168 ffun 6494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑓:ℝ⟶ℝ → Fun 𝑓)
169 difpreima 6816 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (Fun 𝑓 → (𝑓 “ (ran 𝑓 ∖ {0})) = ((𝑓 “ ran 𝑓) ∖ (𝑓 “ {0})))
170168, 169syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓:ℝ⟶ℝ → (𝑓 “ (ran 𝑓 ∖ {0})) = ((𝑓 “ ran 𝑓) ∖ (𝑓 “ {0})))
171 cnvimarndm 5921 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑓 “ ran 𝑓) = dom 𝑓
172171difeq1i 4049 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑓 “ ran 𝑓) ∖ (𝑓 “ {0})) = (dom 𝑓 ∖ (𝑓 “ {0}))
173170, 172eqtrdi 2852 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓:ℝ⟶ℝ → (𝑓 “ (ran 𝑓 ∖ {0})) = (dom 𝑓 ∖ (𝑓 “ {0})))
174167, 173difeq12d 4054 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓:ℝ⟶ℝ → (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0}))) = (dom 𝑓 ∖ (dom 𝑓 ∖ (𝑓 “ {0}))))
175 cnvimass 5920 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 “ {0}) ⊆ dom 𝑓
176 dfss4 4188 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓 “ {0}) ⊆ dom 𝑓 ↔ (dom 𝑓 ∖ (dom 𝑓 ∖ (𝑓 “ {0}))) = (𝑓 “ {0}))
177175, 176mpbi 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (dom 𝑓 ∖ (dom 𝑓 ∖ (𝑓 “ {0}))) = (𝑓 “ {0})
178174, 177eqtrdi 2852 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓:ℝ⟶ℝ → (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0}))) = (𝑓 “ {0}))
179178eleq2d 2878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0}))) ↔ 𝑧 ∈ (𝑓 “ {0})))
180 ffn 6491 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓:ℝ⟶ℝ → 𝑓 Fn ℝ)
181 fniniseg 6811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 Fn ℝ → (𝑧 ∈ (𝑓 “ {0}) ↔ (𝑧 ∈ ℝ ∧ (𝑓𝑧) = 0)))
182 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑧 ∈ ℝ ∧ (𝑓𝑧) = 0) → (𝑓𝑧) = 0)
183181, 182syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 Fn ℝ → (𝑧 ∈ (𝑓 “ {0}) → (𝑓𝑧) = 0))
184180, 183syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (𝑓 “ {0}) → (𝑓𝑧) = 0))
185179, 184sylbid 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓𝑧) = 0))
18635, 185syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 ∈ dom ∫1 → (𝑧 ∈ (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓𝑧) = 0))
187186imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑧 ∈ (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0})))) → (𝑓𝑧) = 0)
188187adantlr 714 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1 ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) ∧ 𝑧 ∈ (ℝ ∖ (𝑓 “ (ran 𝑓 ∖ {0})))) → (𝑓𝑧) = 0)
189161, 164, 165, 188itg10a 24318 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓 ∈ dom ∫1 ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (∫1𝑓) = 0)
190160, 189sylan 583 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (∫1𝑓) = 0)
191159, 190breqtrd 5059 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑏 < 0)
192158, 191, 137syl2anc 587 . . . . . . . . . . . . . . . . . . . . 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))
193 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) → 𝑓 ∈ dom ∫1)
194 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
195193, 194anim12i 615 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → (𝑓 ∈ dom ∫1𝑏 ∈ ℝ))
19663a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ℝ ∈ V)
197 fvex 6662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓𝑢) ∈ V
198197a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑓𝑢) ∈ V)
199 ovex 7172 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V
200199, 105ifex 4476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V
201200a1i 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)
20235feqmptd 6712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓 ∈ dom ∫1𝑓 = (𝑢 ∈ ℝ ↦ (𝑓𝑢)))
203202ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 = (𝑢 ∈ ℝ ↦ (𝑓𝑢)))
204 eqidd 2802 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
205196, 198, 201, 203, 204offval2 7410 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
206 ovif2 7235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
207171, 166syl5eq 2848 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑓:ℝ⟶ℝ → (𝑓 “ ran 𝑓) = ℝ)
208207difeq1d 4052 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑓:ℝ⟶ℝ → ((𝑓 “ ran 𝑓) ∖ (𝑓 “ {0})) = (ℝ ∖ (𝑓 “ {0})))
209170, 208eqtrd 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑓:ℝ⟶ℝ → (𝑓 “ (ran 𝑓 ∖ {0})) = (ℝ ∖ (𝑓 “ {0})))
210209eleq2d 2878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑓:ℝ⟶ℝ → (𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (𝑓 “ {0}))))
21135, 210syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑓 ∈ dom ∫1 → (𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (𝑓 “ {0}))))
212211ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (𝑓 “ {0}))))
213 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
214213biantrurd 536 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ ¬ 𝑢 ∈ (𝑓 “ {0}))))
215 eldif 3894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑢 ∈ (ℝ ∖ (𝑓 “ {0})) ↔ (𝑢 ∈ ℝ ∧ ¬ 𝑢 ∈ (𝑓 “ {0})))
216214, 215syl6bbr 292 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (𝑓 “ {0}) ↔ 𝑢 ∈ (ℝ ∖ (𝑓 “ {0}))))
217212, 216bitr4d 285 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ 𝑢 ∈ (𝑓 “ {0})))
218217con2bid 358 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (𝑓 “ {0}) ↔ ¬ 𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))))
219 fniniseg 6811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑓 Fn ℝ → (𝑢 ∈ (𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓𝑢) = 0)))
22035, 180, 2193syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑓 ∈ dom ∫1 → (𝑢 ∈ (𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓𝑢) = 0)))
221220ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓𝑢) = 0)))
222218, 221bitr3d 284 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ (𝑢 ∈ ℝ ∧ (𝑓𝑢) = 0)))
223 oveq1 7146 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑓𝑢) = 0 → ((𝑓𝑢) − 0) = (0 − 0))
224 0m0e0 11749 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (0 − 0) = 0
225223, 224eqtrdi 2852 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑓𝑢) = 0 → ((𝑓𝑢) − 0) = 0)
226225adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑢 ∈ ℝ ∧ (𝑓𝑢) = 0) → ((𝑓𝑢) − 0) = 0)
227222, 226syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓𝑢) − 0) = 0))
228227imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) ∧ ¬ 𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓𝑢) − 0) = 0)
229228ifeq2da 4459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
230206, 229syl5eq 2848 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
231230mpteq2dva 5128 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
232205, 231eqtrd 2836 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
233 simpll 766 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 ∈ dom ∫1)
234199a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V)
235 1ex 10630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1 ∈ V
236235, 105ifex 4476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V
237236a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V)
238 fconstmpt 5582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (ℝ × {(((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑢 ∈ ℝ ↦ (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))
239238a1i 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})))))))
240 eqidd 2802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
241196, 234, 237, 239, 240offval2 7410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
242 ovif2 7235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
243 resubcl 10943 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((∫1𝑓) ∈ ℝ ∧ 𝑏 ∈ ℝ) → ((∫1𝑓) − 𝑏) ∈ ℝ)
2448, 243sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((∫1𝑓) − 𝑏) ∈ ℝ)
245244adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1𝑓) − 𝑏) ∈ ℝ)
246 2re 11703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2 ∈ ℝ
247 i1fima 24286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑓 ∈ dom ∫1 → (𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol)
248 mblvol 24138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
249247, 248syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑓 ∈ dom ∫1 → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
250 neldifsn 4688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ¬ 0 ∈ (ran 𝑓 ∖ {0})
251 i1fima2 24287 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑓 ∈ dom ∫1 ∧ ¬ 0 ∈ (ran 𝑓 ∖ {0})) → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
252250, 251mpan2 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑓 ∈ dom ∫1 → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
253249, 252eqeltrrd 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑓 ∈ dom ∫1 → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
254 remulcl 10615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((2 ∈ ℝ ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
255246, 253, 254sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑓 ∈ dom ∫1 → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
256255ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
257 2cnd 11707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 2 ∈ ℂ)
258253ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
259258recnd 10662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
260 2ne0 11733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2 ≠ 0
261260a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 2 ≠ 0)
262 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)
263257, 259, 261, 262mulne0d 11285 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ≠ 0)
264245, 256, 263redivcld 11461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
265264recnd 10662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
266265mulid1d 10651 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))))
267265mul01d 10832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) · 0) = 0)
268266, 267ifeq12d 4448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
269242, 268syl5eq 2848 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
270269mpteq2dv 5129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
271241, 270eqtrd 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
272 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))
273272i1f1 24298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
274247, 252, 273syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 ∈ dom ∫1 → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
275274ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
276275, 264i1fmulc 24311 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
277271, 276eqeltrrd 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
278 i1fsub 24316 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
279233, 277, 278syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . 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)
280232, 279eqeltrrd 2894 . . . . . . . . . . . . . . . . . . . . . . . . 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)
281 iftrue 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))
282 iftrue 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))
283282breq2d 5045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))))))
284283, 282ifbieq1d 4451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
285 iftrue 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))
286284, 285sylan9eqr 2858 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))
287281, 286eqtr4d 2839 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
288 iffalse 4437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
289 ianor 979 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))
290283ifbid 4450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
291 iffalse 4437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
292290, 291sylan9eqr 2858 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
293292ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
294 iffalse 4437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
295 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 0 = 0
296 eqeq1 2805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
297 eqeq1 2805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
298296, 297ifboth 4466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
299294, 295, 298sylancl 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
300293, 299pm2.61d1 183 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
301300, 299jaoi 854 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
302289, 301sylbi 220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
303288, 302eqtr4d 2839 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
304287, 303pm2.61i 185 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
305 eleq1w 2875 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑢 = 𝑧 → (𝑢 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))))
306 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
307306oveq1d 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑢 = 𝑧 → ((𝑓𝑢) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) = ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))))
308305, 307ifbieq1d 4451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
309 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
310 ovex 7172 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ V
311310, 105ifex 4476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V
312308, 309, 311fvmpt 6749 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
313312breq2d 5045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
314313, 312ifbieq1d 4451 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
315304, 314eqtr4id 2855 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
316315mpteq2ia 5124 . . . . . . . . . . . . . . . . . . . . . . . . . 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))
317316i1fpos 24314 . . . . . . . . . . . . . . . . . . . . . . . . 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)
318280, 317syl 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)
319195, 318sylan 583 . . . . . . . . . . . . . . . . . . . . . . 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)
320195, 264sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . 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)
3218ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) → (∫1𝑓) ∈ ℝ)
322321, 194, 243syl2an 598 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → ((∫1𝑓) − 𝑏) ∈ ℝ)
323322adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1𝑓) − 𝑏) ∈ ℝ)
324255adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓))) → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
325324ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . . . . . 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)
326 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑏 < (∫1𝑓))
327 simprr 772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑏 ∈ ℝ)
328141ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → (∫1𝑓) ∈ ℝ)
329327, 328posdifd 11220 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → (𝑏 < (∫1𝑓) ↔ 0 < ((∫1𝑓) − 𝑏)))
330326, 329mpbid 235 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → 0 < ((∫1𝑓) − 𝑏))
331330adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < ((∫1𝑓) − 𝑏))
332253adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓))) → (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
333332ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
334 mblss 24139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol → (𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
335 ovolge0 24089 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ → 0 ≤ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
336247, 334, 3353syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 ∈ dom ∫1 → 0 ≤ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
337 ltlen 10734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
33848, 253, 337sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 ∈ dom ∫1 → (0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ↔ (0 ≤ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)))
339338biimprd 251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 ∈ dom ∫1 → ((0 ≤ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
340336, 339mpand 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 ∈ dom ∫1 → ((vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0 → 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
341340ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) → ((vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0 → 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
342341imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
343342adantlr 714 . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))
344 2pos 11732 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 < 2
345 mulgt0 10711 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2 ∈ ℝ ∧ 0 < 2) ∧ ((vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ ∧ 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) → 0 < (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
346246, 344, 345mpanl12 701 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ ∧ 0 < (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) → 0 < (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))
347333, 343, 346syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))
348323, 325, 331, 347divgt0d 11568 . . . . . . . . . . . . . . . . . . . . . . . . 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}))))))
349320, 348elrpd 12420 . . . . . . . . . . . . . . . . . . . . . . . 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+)
350 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓))) → 𝑓r𝐹)
351350ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓r𝐹)
352 ffn 6491 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹:ℝ⟶(0[,]+∞) → 𝐹 Fn ℝ)
35335, 180syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 ∈ dom ∫1𝑓 Fn ℝ)
354353adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓))) → 𝑓 Fn ℝ)
355 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → 𝑓 Fn ℝ)
356 simpl 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → 𝐹 Fn ℝ)
35763a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → ℝ ∈ V)
358 inidm 4148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (ℝ ∩ ℝ) = ℝ
359 eqidd 2802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) = (𝑓𝑧))
360 eqidd 2802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
361355, 356, 357, 357, 358, 359, 360ofrfval 7401 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → (𝑓r𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓𝑧) ≤ (𝐹𝑧)))
362352, 354, 361syl2an 598 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) → (𝑓r𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓𝑧) ≤ (𝐹𝑧)))
363362ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓r𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓𝑧) ≤ (𝐹𝑧)))
364 simpl 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓))) → 𝑓 ∈ dom ∫1)
365364anim2i 619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) → (𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1))
366365, 194anim12i 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓r𝐹𝑠 = (∫1𝑓)))) ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) → ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ))
367 breq1 5036 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
368 breq1 5036 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
369 simplll 774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝐹:ℝ⟶(0[,]+∞))
370369ffvelrnda 6832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ∈ (0[,]+∞))
371370, 100sylib 221 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹𝑧)))
372371simprd 499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹𝑧))
373372ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
374 oveq1 7146 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))
375374breq1d 5043 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
376 oveq1 7146 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))
377376breq1d 5043 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
37835ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓:ℝ⟶ℝ)
379378ffvelrnda 6832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
380379recnd 10662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℂ)
381244recnd 10662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((∫1𝑓) − 𝑏) ∈ ℂ)
382381adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1𝑓) − 𝑏) ∈ ℂ)
383255recnd 10662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑓 ∈ dom ∫1 → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℂ)
384383ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℂ)
385382, 384, 263divcld 11409 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
386385adantlll 717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
387386adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
388380, 387npcand 10994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
389388adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
390 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧))
391389, 390eqbrtrd 5055 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
392391ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
393288pm2.24d 154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
394393impcom 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
395394adantll 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
396375, 377, 392, 395ifbothda 4465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
397367, 368, 373, 396ifbothda 4465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
398397ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
399366, 398sylanl1 679 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
400399ralimdva 3147 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
401363, 400sylbid 243 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
402351, 401mpd 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))
403 ovex 7172 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
404105, 403ifex 4476 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
405404a1i 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)
406 eqidd 2802 . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))))
407104, 405, 99, 406, 67ofrfval2 7411 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
408407ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
409402, 408mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . 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)
410 oveq2 7147 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))
411410ifeq2d 4447 . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))))))
412411mpteq2dv 5129 . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))))
413412breq1d 5043 . . . . . . . . . . . . . . . . . . . . . . . . 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))
414413rspcev 3574 . . . . . . . . . . . . . . . . . . . . . . . 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)
415349, 409, 414syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . 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)
416 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
417416eqcoms 2809 . . . . . . . . . . . . . . . . . . . . . . . . . 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))
418417biantrud 535 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
419 nfmpt1 5131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
420419nfeq2 2975 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
421 fveq1 6648 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
422310, 105ifex 4476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
423 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
424423fvmpt2 6760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
425422, 424mpan2 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
426421, 425sylan9eq 2856 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
427426eqeq1d 2803 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
428426oveq1d 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
429427, 428ifbieq2d 4453 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
430420, 429mpteq2da 5127 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
431430breq1d 5043 . . . . . . . . . . . . . . . . . . . . . . . . . 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))
432431rexbidv 3259 . . . . . . . . . . . . . . . . . . . . . . . . 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))
433418, 432bitr3d 284 . . . . . . . . . . . . . . . . . . . . . . . 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))
434433rspcev 3574 . . . . . . . . . . . . . . . . . . . . . . 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)))
435319, 415, 434syl2anc 587 . . . . . . . . . . . . . . . . . . . . . 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)))
436 simplrr 777 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 ∈ ℝ)
437199a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V)
438235, 105ifex 4476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V
439438a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V)
440 fconstmpt 5582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (ℝ × {(((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑧 ∈ ℝ ↦ (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))))
441440a1i 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})))))))
442 eqidd 2802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
443196, 437, 439, 441, 442offval2 7410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
444 ovif2 7235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
445266, 267ifeq12d 4448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
446444, 445syl5eq 2848 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
447446mpteq2dv 5129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
448443, 447eqtrd 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
449 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))
450449i1f1 24298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
451247, 252, 450syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 ∈ dom ∫1 → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
452451ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
453452, 264i1fmulc 24311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
454448, 453eqeltrrd 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
455 i1fsub 24316 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
456233, 454, 455syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
457 itg1cl 24293 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
458456, 457syl 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)
459458adantlrl 719 . . . . . . . . . . . . . . . . . . . . . . . . 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)
460318adantlrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . 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)
461 itg1cl 24293 . . . . . . . . . . . . . . . . . . . . . . . . . 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)
462460, 461syl 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)
463 simplrl 776 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1𝑓))
464 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
4658adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → (∫1𝑓) ∈ ℝ)
46697a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → 2 ∈ ℝ+)
467464, 465, 466ltdiv1d 12468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → (𝑏 < (∫1𝑓) ↔ (𝑏 / 2) < ((∫1𝑓) / 2)))
468 recn 10620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 ∈ ℝ → 𝑏 ∈ ℂ)
4694682halvesd 11875 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 ∈ ℝ → ((𝑏 / 2) + (𝑏 / 2)) = 𝑏)
470469oveq1d 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑏 ∈ ℝ → (((𝑏 / 2) + (𝑏 / 2)) − (𝑏 / 2)) = (𝑏 − (𝑏 / 2)))
471468halfcld 11874 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 ∈ ℝ → (𝑏 / 2) ∈ ℂ)
472471, 471pncand 10991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑏 ∈ ℝ → (((𝑏 / 2) + (𝑏 / 2)) − (𝑏 / 2)) = (𝑏 / 2))
473470, 472eqtr3d 2838 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ∈ ℝ → (𝑏 − (𝑏 / 2)) = (𝑏 / 2))
474473breq1d 5043 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 ∈ ℝ → ((𝑏 − (𝑏 / 2)) < ((∫1𝑓) / 2) ↔ (𝑏 / 2) < ((∫1𝑓) / 2)))
475474adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((𝑏 − (𝑏 / 2)) < ((∫1𝑓) / 2) ↔ (𝑏 / 2) < ((∫1𝑓) / 2)))
476 rehalfcl 11855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ∈ ℝ → (𝑏 / 2) ∈ ℝ)
477476adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → (𝑏 / 2) ∈ ℝ)
4788rehalfcld 11876 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 ∈ dom ∫1 → ((∫1𝑓) / 2) ∈ ℝ)
479478adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((∫1𝑓) / 2) ∈ ℝ)
480464, 477, 479ltsubaddd 11229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((𝑏 − (𝑏 / 2)) < ((∫1𝑓) / 2) ↔ 𝑏 < (((∫1𝑓) / 2) + (𝑏 / 2))))
481467, 475, 4803bitr2d 310 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → (𝑏 < (∫1𝑓) ↔ 𝑏 < (((∫1𝑓) / 2) + (𝑏 / 2))))
482481adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑏 < (∫1𝑓) ↔ 𝑏 < (((∫1𝑓) / 2) + (𝑏 / 2))))
483482adantlrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑏 < (∫1𝑓) ↔ 𝑏 < (((∫1𝑓) / 2) + (𝑏 / 2))))
484463, 483mpbid 235 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (((∫1𝑓) / 2) + (𝑏 / 2)))
485452, 264itg1mulc 24312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
486448fveq2d 6653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
487449itg11 24299 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))
488247, 252, 487syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 ∈ dom ∫1 → (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))))
489488oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))
490489ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))
491252recnd 10662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 ∈ dom ∫1 → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
492491ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
493265, 492mulcomd 10655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))
494249ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol‘(𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))
495494oveq1d 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
496259, 382mulcomd 10655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))
497495, 496eqtrd 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))
498497oveq1d 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))))
499492, 382, 384, 263divassd 11444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))
500382, 257, 259, 261, 262divcan5rd 11436 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
501498, 499, 5003eqtr3d 2844 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
502490, 493, 5013eqtrd 2840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
503485, 486, 5023eqtr3d 2844 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
504503oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
505 itg1sub 24317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
506233, 454, 505syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
5078recnd 10662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 ∈ dom ∫1 → (∫1𝑓) ∈ ℂ)
508507ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1𝑓) ∈ ℂ)
509468ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 ∈ ℂ)
510508, 509, 257, 261divsubdird 11448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − 𝑏) / 2) = (((∫1𝑓) / 2) − (𝑏 / 2)))
511510oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1𝑓) − (((∫1𝑓) − 𝑏) / 2)) = ((∫1𝑓) − (((∫1𝑓) / 2) − (𝑏 / 2))))
512507adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → (∫1𝑓) ∈ ℂ)
513512halfcld 11874 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((∫1𝑓) / 2) ∈ ℂ)
514471adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → (𝑏 / 2) ∈ ℂ)
515512, 513, 514subsubd 11018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) → ((∫1𝑓) − (((∫1𝑓) / 2) − (𝑏 / 2))) = (((∫1𝑓) − ((∫1𝑓) / 2)) + (𝑏 / 2)))
516515adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1𝑓) − (((∫1𝑓) / 2) − (𝑏 / 2))) = (((∫1𝑓) − ((∫1𝑓) / 2)) + (𝑏 / 2)))
5175072halvesd 11875 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 ∈ dom ∫1 → (((∫1𝑓) / 2) + ((∫1𝑓) / 2)) = (∫1𝑓))
518517oveq1d 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 ∈ dom ∫1 → ((((∫1𝑓) / 2) + ((∫1𝑓) / 2)) − ((∫1𝑓) / 2)) = ((∫1𝑓) − ((∫1𝑓) / 2)))
519507halfcld 11874 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 ∈ dom ∫1 → ((∫1𝑓) / 2) ∈ ℂ)
520519, 519pncand 10991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 ∈ dom ∫1 → ((((∫1𝑓) / 2) + ((∫1𝑓) / 2)) − ((∫1𝑓) / 2)) = ((∫1𝑓) / 2))
521518, 520eqtr3d 2838 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑓 ∈ dom ∫1 → ((∫1𝑓) − ((∫1𝑓) / 2)) = ((∫1𝑓) / 2))
522521oveq1d 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓 ∈ dom ∫1 → (((∫1𝑓) − ((∫1𝑓) / 2)) + (𝑏 / 2)) = (((∫1𝑓) / 2) + (𝑏 / 2)))
523522ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) − ((∫1𝑓) / 2)) + (𝑏 / 2)) = (((∫1𝑓) / 2) + (𝑏 / 2)))
524511, 516, 5233eqtrrd 2841 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1𝑓) / 2) + (𝑏 / 2)) = ((∫1𝑓) − (((∫1𝑓) − 𝑏) / 2)))
525504, 506, 5243eqtr4d 2846 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
526525adantlrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
527484, 526breqtrrd 5061 . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
528456adantlrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . 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)
529 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0))
530529adantlrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0))
531233, 36sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
532264adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
533531, 532resubcld 11061 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) − (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ ℝ)
534533leidd 11199 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))
535534adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))
536285breq2d 5045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))))))
537536adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))))))
538535, 537mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
539533adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
54048a1i 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)
54148a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝑓 ∈ dom ∫1𝑏 ∈ ℝ) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → 0 ∈ ℝ)
542533, 541ltnled 10780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))))))
543542biimpar 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
544539, 540, 543ltled 10781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
545 iffalse 4437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
546545breq2d 5045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
547546adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
548544, 547mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
549538, 548pm2.61dan 812 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
550530, 549sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
551550adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
552 iftrue 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))))
553552oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))))
554 iba 531 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))
555554bicomd 226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))))))
556555ifbid 4450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
557553, 556breq12d 5046 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
558557adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
559551, 558mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
56035ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓:ℝ⟶ℝ)
561170eleq2d 2878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑧 ∈ ((𝑓 “ ran 𝑓) ∖ (𝑓 “ {0}))))
562 eldif 3894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑧 ∈ ((𝑓 “ ran 𝑓) ∖ (𝑓 “ {0})) ↔ (𝑧 ∈ (𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (𝑓 “ {0})))
563561, 562syl6bb 290 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ (𝑧 ∈ (𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (𝑓 “ {0}))))
564563notbid 321 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑓:ℝ⟶ℝ → (¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ (𝑧 ∈ (𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (𝑓 “ {0}))))
565564adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ (𝑧 ∈ (𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (𝑓 “ {0}))))
566 pm4.53 983 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (¬ (𝑧 ∈ (𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (𝑓 “ {0})) ↔ (¬ 𝑧 ∈ (𝑓 “ ran 𝑓) ∨ 𝑧 ∈ (𝑓 “ {0})))
567207eleq2d 2878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (𝑓 “ ran 𝑓) ↔ 𝑧 ∈ ℝ))
568567biimpar 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ (𝑓 “ ran 𝑓))
569568pm2.24d 154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (𝑓 “ ran 𝑓) → (𝑓𝑧) = 0))
570181simplbda 503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑓 Fn ℝ ∧ 𝑧 ∈ (𝑓 “ {0})) → (𝑓𝑧) = 0)
571570ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑓 Fn ℝ → (𝑧 ∈ (𝑓 “ {0}) → (𝑓𝑧) = 0))
572180, 571syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑓:ℝ⟶ℝ → (𝑧 ∈ (𝑓 “ {0}) → (𝑓𝑧) = 0))
573572adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (𝑧 ∈ (𝑓 “ {0}) → (𝑓𝑧) = 0))
574569, 573jaod 856 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → ((¬ 𝑧 ∈ (𝑓 “ ran 𝑓) ∨ 𝑧 ∈ (𝑓 “ {0})) → (𝑓𝑧) = 0))
575566, 574syl5bi 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ (𝑧 ∈ (𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (𝑓 “ {0})) → (𝑓𝑧) = 0))
576565, 575sylbid 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → (𝑓𝑧) = 0))
577576imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓𝑧) = 0)
578560, 577sylanl1 679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓𝑧) = 0)
579578oveq1d 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
580579, 224eqtrdi 2852 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓𝑧) − 0) = 0)
581580, 30eqbrtrdi 5072 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓𝑧) − 0) ≤ 0)
582 iffalse 4437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0) = 0)
583582oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = ((𝑓𝑧) − 0))
584289, 288sylbir 238 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
585584olcs 873 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
586583, 585breq12d 5046 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
587586adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
588581, 587mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
589559, 588pm2.61dan 812 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
590589ralrimiva 3152 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
59163a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ℝ ∈ V)
592 ovex 7172 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓𝑧) − if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ V
593592a1i 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)
594422a1i 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)
595 fvex 6662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑓𝑧) ∈ V
596595a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ V)
597199, 105ifex 4476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 if(𝑧 ∈ (𝑓 “ (ran 𝑓 ∖ {0})), (((∫1𝑓) − 𝑏) / (2 · (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V
598597a1i 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)
59970ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓𝑧)))
600 eqidd 2802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
601591, 596, 598, 599, 600offval2 7410 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
602 eqidd 2802 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
603591, 593, 594, 601, 602ofrfval2 7411 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
604590, 603mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
605 itg1le 24321 . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
606528, 460, 604, 605syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
607436, 459, 462, 527, 606ltletrd 10793 . . . . . . . . . . . . . . . . . . . . . . . 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))))
608607adantllr 718 . . . . . . . . . . . . . . . . . . . . . . 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))))
609608adantlll 717 . . . . . . . . . . . . . . . . . . . . . 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))))
610 fvex 6662 . . . . . . . . . . . . . . . . . . . . . . 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
611 eqeq1 2805 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
612611anbi2d 631 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
613612rexbidv 3259 . . . . . . . . . . . . . . . . . . . . . . . 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))))
614 breq2 5037 . . . . . . . . . . . . . . . . . . . . . . . 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)))))
615613, 614anbi12d 633 . . . . . . . . . . . . . . . . . . . . . . 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))))))
616610, 615spcev 3558 . . . . . . . . . . . . . . . . . . . . . 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))
617435, 609, 616syl2anc 587 . . . . . . . . . . . . . . . . . . . . 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))
618192, 617pm2.61dane 3077 . . . . . . . . . . . . . . . . . . . 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))
619618expr 460 . . . . . . . . . . . . . . . . . . 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)))
620619adantllr 718 . . . . . . . . . . . . . . . . . 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)))
621620adantr 484 . . . . . . . . . . . . . . . . 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)))
622157, 621mpd 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))
623139, 622pm2.61dane 3077 . . . . . . . . . . . . . . 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))
624623ex 416 . . . . . . . . . . . . . 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)))
62594, 624sylbid 243 . . . . . . . . . . . . 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)))
626625imp 410 . . . . . . . . . . . 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))
627626an32s 651 . . . . . . . . . . 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))
628627rexlimdvaa 3247 . . . . . . . . . 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, ((𝑔𝑧) + 𝑦))) ∘r𝐹𝑎 = (∫1𝑔)) ∧ 𝑏 < 𝑎)))
629628expimpd 457 . . . . . . . . 9 ((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