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Theorem itg2addnc 36132
Description: Alternate proof of itg2add 25124 using the "buffer zone" definition from the first lemma, in which every simple function in the set is divided into to by dividing its buffer by a third and finding the largest allowable function locked to a grid laid out in increments of the new, smaller buffer up to the original simple function. The measurability of this function follows from that of the augend, and subtracting it from the original simple function yields another simple function by i1fsub 25073, which is allowable by the fact that the grid must have a mark between one third and two thirds the original buffer. This has two advantages over the current approach: first, eliminating ax-cc 10371, and second, weakening the measurability hypothesis to only the augend. (Contributed by Brendan Leahy, 31-Oct-2017.) (Revised by Brendan Leahy, 13-Mar-2018.)
Hypotheses
Ref Expression
itg2addnc.f1 (𝜑𝐹 ∈ MblFn)
itg2addnc.f2 (𝜑𝐹:ℝ⟶(0[,)+∞))
itg2addnc.f3 (𝜑 → (∫2𝐹) ∈ ℝ)
itg2addnc.g2 (𝜑𝐺:ℝ⟶(0[,)+∞))
itg2addnc.g3 (𝜑 → (∫2𝐺) ∈ ℝ)
Assertion
Ref Expression
itg2addnc (𝜑 → (∫2‘(𝐹f + 𝐺)) = ((∫2𝐹) + (∫2𝐺)))

Proof of Theorem itg2addnc
Dummy variables 𝑡 𝑠 𝑢 𝑥 𝑦 𝑧 𝑓 𝑔 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprr 771 . . . . . . 7 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))) → 𝑥 = (∫1𝑓))
2 itg1cl 25049 . . . . . . . 8 (𝑓 ∈ dom ∫1 → (∫1𝑓) ∈ ℝ)
32adantr 481 . . . . . . 7 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ)
41, 3eqeltrd 2838 . . . . . 6 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))) → 𝑥 ∈ ℝ)
54rexlimiva 3144 . . . . 5 (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓)) → 𝑥 ∈ ℝ)
65abssi 4027 . . . 4 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ⊆ ℝ
76a1i 11 . . 3 (𝜑 → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ⊆ ℝ)
8 i1f0 25051 . . . . . 6 (ℝ × {0}) ∈ dom ∫1
9 3nn 12232 . . . . . . . 8 3 ∈ ℕ
10 nnrp 12926 . . . . . . . 8 (3 ∈ ℕ → 3 ∈ ℝ+)
11 ne0i 4294 . . . . . . . 8 (3 ∈ ℝ+ → ℝ+ ≠ ∅)
129, 10, 11mp2b 10 . . . . . . 7 + ≠ ∅
13 itg2addnc.f2 . . . . . . . . . . . . 13 (𝜑𝐹:ℝ⟶(0[,)+∞))
1413ffvelcdmda 7035 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ∈ (0[,)+∞))
15 elrege0 13371 . . . . . . . . . . . 12 ((𝐹𝑧) ∈ (0[,)+∞) ↔ ((𝐹𝑧) ∈ ℝ ∧ 0 ≤ (𝐹𝑧)))
1614, 15sylib 217 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℝ) → ((𝐹𝑧) ∈ ℝ ∧ 0 ≤ (𝐹𝑧)))
1716simprd 496 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ≤ (𝐹𝑧))
1817ralrimiva 3143 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧))
19 reex 11142 . . . . . . . . . . 11 ℝ ∈ V
2019a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ∈ V)
21 c0ex 11149 . . . . . . . . . . 11 0 ∈ V
2221a1i 11 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ∈ V)
23 eqidd 2737 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
2413feqmptd 6910 . . . . . . . . . 10 (𝜑𝐹 = (𝑧 ∈ ℝ ↦ (𝐹𝑧)))
2520, 22, 14, 23, 24ofrfval2 7638 . . . . . . . . 9 (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘r𝐹 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧)))
2618, 25mpbird 256 . . . . . . . 8 (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘r𝐹)
2726ralrimivw 3147 . . . . . . 7 (𝜑 → ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹)
28 r19.2z 4452 . . . . . . 7 ((ℝ+ ≠ ∅ ∧ ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹) → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹)
2912, 27, 28sylancr 587 . . . . . 6 (𝜑 → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹)
30 fveq2 6842 . . . . . . . . . 10 (𝑓 = (ℝ × {0}) → (∫1𝑓) = (∫1‘(ℝ × {0})))
31 itg10 25052 . . . . . . . . . 10 (∫1‘(ℝ × {0})) = 0
3230, 31eqtr2di 2793 . . . . . . . . 9 (𝑓 = (ℝ × {0}) → 0 = (∫1𝑓))
3332biantrud 532 . . . . . . . 8 (𝑓 = (ℝ × {0}) → (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ 0 = (∫1𝑓))))
34 fveq1 6841 . . . . . . . . . . . . 13 (𝑓 = (ℝ × {0}) → (𝑓𝑧) = ((ℝ × {0})‘𝑧))
3521fvconst2 7153 . . . . . . . . . . . . 13 (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
3634, 35sylan9eq 2796 . . . . . . . . . . . 12 ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) = 0)
3736iftrued 4494 . . . . . . . . . . 11 ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) = 0)
3837mpteq2dva 5205 . . . . . . . . . 10 (𝑓 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ 0))
3938breq1d 5115 . . . . . . . . 9 (𝑓 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹))
4039rexbidv 3175 . . . . . . . 8 (𝑓 = (ℝ × {0}) → (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹))
4133, 40bitr3d 280 . . . . . . 7 (𝑓 = (ℝ × {0}) → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ 0 = (∫1𝑓)) ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹))
4241rspcev 3581 . . . . . 6 (((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹) → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ 0 = (∫1𝑓)))
438, 29, 42sylancr 587 . . . . 5 (𝜑 → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ 0 = (∫1𝑓)))
44 eqeq1 2740 . . . . . . . 8 (𝑥 = 0 → (𝑥 = (∫1𝑓) ↔ 0 = (∫1𝑓)))
4544anbi2d 629 . . . . . . 7 (𝑥 = 0 → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓)) ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ 0 = (∫1𝑓))))
4645rexbidv 3175 . . . . . 6 (𝑥 = 0 → (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓)) ↔ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ 0 = (∫1𝑓))))
4721, 46elab 3630 . . . . 5 (0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ↔ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ 0 = (∫1𝑓)))
4843, 47sylibr 233 . . . 4 (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))})
4948ne0d 4295 . . 3 (𝜑 → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ≠ ∅)
50 icossicc 13353 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
51 fss 6685 . . . . . . 7 ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
5250, 51mpan2 689 . . . . . 6 (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶(0[,]+∞))
53 eqid 2736 . . . . . . 7 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} = {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}
5453itg2addnclem 36129 . . . . . 6 (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
5513, 52, 543syl 18 . . . . 5 (𝜑 → (∫2𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
56 itg2addnc.f3 . . . . 5 (𝜑 → (∫2𝐹) ∈ ℝ)
5755, 56eqeltrrd 2839 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ)
58 ressxr 11199 . . . . . . 7 ℝ ⊆ ℝ*
596, 58sstri 3953 . . . . . 6 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*
60 supxrub 13243 . . . . . 6 (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}) → 𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
6159, 60mpan 688 . . . . 5 (𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} → 𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
6261rgen 3066 . . . 4 𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < )
63 brralrspcev 5165 . . . 4 ((sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ ∧ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < )) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}𝑏𝑎)
6457, 62, 63sylancl 586 . . 3 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}𝑏𝑎)
65 simprr 771 . . . . . . 7 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))) → 𝑥 = (∫1𝑔))
66 itg1cl 25049 . . . . . . . 8 (𝑔 ∈ dom ∫1 → (∫1𝑔) ∈ ℝ)
6766adantr 481 . . . . . . 7 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))) → (∫1𝑔) ∈ ℝ)
6865, 67eqeltrd 2838 . . . . . 6 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))) → 𝑥 ∈ ℝ)
6968rexlimiva 3144 . . . . 5 (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔)) → 𝑥 ∈ ℝ)
7069abssi 4027 . . . 4 {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} ⊆ ℝ
7170a1i 11 . . 3 (𝜑 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} ⊆ ℝ)
72 itg2addnc.g2 . . . . . . . . . . . . 13 (𝜑𝐺:ℝ⟶(0[,)+∞))
7372ffvelcdmda 7035 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ∈ (0[,)+∞))
74 elrege0 13371 . . . . . . . . . . . 12 ((𝐺𝑧) ∈ (0[,)+∞) ↔ ((𝐺𝑧) ∈ ℝ ∧ 0 ≤ (𝐺𝑧)))
7573, 74sylib 217 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℝ) → ((𝐺𝑧) ∈ ℝ ∧ 0 ≤ (𝐺𝑧)))
7675simprd 496 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ≤ (𝐺𝑧))
7776ralrimiva 3143 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐺𝑧))
7872feqmptd 6910 . . . . . . . . . 10 (𝜑𝐺 = (𝑧 ∈ ℝ ↦ (𝐺𝑧)))
7920, 22, 73, 23, 78ofrfval2 7638 . . . . . . . . 9 (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘r𝐺 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐺𝑧)))
8077, 79mpbird 256 . . . . . . . 8 (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘r𝐺)
8180ralrimivw 3147 . . . . . . 7 (𝜑 → ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺)
82 r19.2z 4452 . . . . . . 7 ((ℝ+ ≠ ∅ ∧ ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺) → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺)
8312, 81, 82sylancr 587 . . . . . 6 (𝜑 → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺)
84 fveq2 6842 . . . . . . . . . 10 (𝑔 = (ℝ × {0}) → (∫1𝑔) = (∫1‘(ℝ × {0})))
8584, 31eqtr2di 2793 . . . . . . . . 9 (𝑔 = (ℝ × {0}) → 0 = (∫1𝑔))
8685biantrud 532 . . . . . . . 8 (𝑔 = (ℝ × {0}) → (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ↔ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ∧ 0 = (∫1𝑔))))
87 fveq1 6841 . . . . . . . . . . . . 13 (𝑔 = (ℝ × {0}) → (𝑔𝑧) = ((ℝ × {0})‘𝑧))
8887, 35sylan9eq 2796 . . . . . . . . . . . 12 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = 0)
8988iftrued 4494 . . . . . . . . . . 11 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) = 0)
9089mpteq2dva 5205 . . . . . . . . . 10 (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ 0))
9190breq1d 5115 . . . . . . . . 9 (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺))
9291rexbidv 3175 . . . . . . . 8 (𝑔 = (ℝ × {0}) → (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺))
9386, 92bitr3d 280 . . . . . . 7 (𝑔 = (ℝ × {0}) → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ∧ 0 = (∫1𝑔)) ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺))
9493rspcev 3581 . . . . . 6 (((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺) → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ∧ 0 = (∫1𝑔)))
958, 83, 94sylancr 587 . . . . 5 (𝜑 → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ∧ 0 = (∫1𝑔)))
96 eqeq1 2740 . . . . . . . 8 (𝑥 = 0 → (𝑥 = (∫1𝑔) ↔ 0 = (∫1𝑔)))
9796anbi2d 629 . . . . . . 7 (𝑥 = 0 → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔)) ↔ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ∧ 0 = (∫1𝑔))))
9897rexbidv 3175 . . . . . 6 (𝑥 = 0 → (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ∧ 0 = (∫1𝑔))))
9921, 98elab 3630 . . . . 5 (0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} ↔ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ∧ 0 = (∫1𝑔)))
10095, 99sylibr 233 . . . 4 (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})
101100ne0d 4295 . . 3 (𝜑 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} ≠ ∅)
102 fss 6685 . . . . . . 7 ((𝐺:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐺:ℝ⟶(0[,]+∞))
10350, 102mpan2 689 . . . . . 6 (𝐺:ℝ⟶(0[,)+∞) → 𝐺:ℝ⟶(0[,]+∞))
104 eqid 2736 . . . . . . 7 {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}
105104itg2addnclem 36129 . . . . . 6 (𝐺:ℝ⟶(0[,]+∞) → (∫2𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
10672, 103, 1053syl 18 . . . . 5 (𝜑 → (∫2𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
107 itg2addnc.g3 . . . . 5 (𝜑 → (∫2𝐺) ∈ ℝ)
108106, 107eqeltrrd 2839 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) ∈ ℝ)
10970, 58sstri 3953 . . . . . 6 {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} ⊆ ℝ*
110 supxrub 13243 . . . . . 6 (({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} ⊆ ℝ*𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) → 𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
111109, 110mpan 688 . . . . 5 (𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} → 𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
112111rgen 3066 . . . 4 𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )
113 brralrspcev 5165 . . . 4 ((sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) ∈ ℝ ∧ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏𝑎)
114108, 112, 113sylancl 586 . . 3 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏𝑎)
115 eqid 2736 . . 3 {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} = {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}
1167, 49, 64, 71, 101, 114, 115supadd 12123 . 2 (𝜑 → (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ, < )) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < ))
117 supxrre 13246 . . . . 5 (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ⊆ ℝ ∧ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ≠ ∅ ∧ ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}𝑏𝑎) → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ, < ))
1187, 49, 64, 117syl3anc 1371 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ, < ))
11955, 118eqtrd 2776 . . 3 (𝜑 → (∫2𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ, < ))
120 supxrre 13246 . . . . 5 (({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} ⊆ ℝ ∧ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} ≠ ∅ ∧ ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏𝑎) → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ, < ))
12171, 101, 114, 120syl3anc 1371 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ, < ))
122106, 121eqtrd 2776 . . 3 (𝜑 → (∫2𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ, < ))
123119, 122oveq12d 7375 . 2 (𝜑 → ((∫2𝐹) + (∫2𝐺)) = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ, < )))
124 ge0addcl 13377 . . . . . . 7 ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞))
12550, 124sselid 3942 . . . . . 6 ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,]+∞))
126125adantl 482 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,]+∞))
127 inidm 4178 . . . . 5 (ℝ ∩ ℝ) = ℝ
128126, 13, 72, 20, 20, 127off 7635 . . . 4 (𝜑 → (𝐹f + 𝐺):ℝ⟶(0[,]+∞))
129 eqid 2736 . . . . 5 {𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1))} = {𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1))}
130129itg2addnclem 36129 . . . 4 ((𝐹f + 𝐺):ℝ⟶(0[,]+∞) → (∫2‘(𝐹f + 𝐺)) = sup({𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1))}, ℝ*, < ))
131128, 130syl 17 . . 3 (𝜑 → (∫2‘(𝐹f + 𝐺)) = sup({𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1))}, ℝ*, < ))
132 itg2addnc.f1 . . . . . . . 8 (𝜑𝐹 ∈ MblFn)
133132, 13, 56, 72, 107itg2addnclem3 36131 . . . . . . 7 (𝜑 → (∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1)) → ∃𝑡𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))))
134 simpl 483 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓 ∈ dom ∫1)
135 simpr 485 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑔 ∈ dom ∫1)
136134, 135i1fadd 25059 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑓f + 𝑔) ∈ dom ∫1)
137136ad3antlr 729 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → (𝑓f + 𝑔) ∈ dom ∫1)
138 reeanv 3217 . . . . . . . . . . . . . . . . 17 (∃𝑐 ∈ ℝ+𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺) ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺))
139138biimpri 227 . . . . . . . . . . . . . . . 16 ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺) → ∃𝑐 ∈ ℝ+𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺))
140139ad2ant2r 745 . . . . . . . . . . . . . . 15 (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) → ∃𝑐 ∈ ℝ+𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺))
141 ifcl 4531 . . . . . . . . . . . . . . . . . . 19 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ+)
142141ad2antlr 725 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺)) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ+)
143 breq1 5108 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (0 ≤ (𝐹𝑧) ↔ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
144143anbi1d 630 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → ((0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) ↔ (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
145144imbi1d 341 . . . . . . . . . . . . . . . . . . . . . 22 (0 = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (((0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))) ↔ ((if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))))
146 breq1 5108 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓𝑧) + 𝑐) = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ↔ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
147146anbi1d 630 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓𝑧) + 𝑐) = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) ↔ (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
148147imbi1d 341 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓𝑧) + 𝑐) = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))) ↔ ((if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))))
149 breq1 5108 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (0 ≤ (𝐺𝑧) ↔ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
150149anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((0 ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) ↔ (0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
151150imbi1d 341 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((0 ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))) ↔ ((0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))))
152 breq1 5108 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) ↔ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
153152anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) ↔ (0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
154153imbi1d 341 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))) ↔ ((0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))))
155 oveq12 7366 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (0 + 0))
156 00id 11330 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 + 0) = 0
157155, 156eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = 0)
158157iftrued 4494 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) = 0)
159158adantll 712 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) = 0)
160 simpll 765 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝜑)
16115simplbi 498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹𝑧) ∈ (0[,)+∞) → (𝐹𝑧) ∈ ℝ)
16214, 161syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ∈ ℝ)
16374simplbi 498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐺𝑧) ∈ (0[,)+∞) → (𝐺𝑧) ∈ ℝ)
16473, 163syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ℝ)
165162, 164, 17, 76addge0d 11731 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑧 ∈ ℝ) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
166160, 165sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
167166ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
168159, 167eqbrtrd 5127 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
169168a1d 25 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → ((0 ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
170166ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
171 oveq1 7364 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓𝑧) = 0 → ((𝑓𝑧) + (𝑔𝑧)) = (0 + (𝑔𝑧)))
172 simplrr 776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑔 ∈ dom ∫1)
173 i1ff 25040 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑔 ∈ dom ∫1𝑔:ℝ⟶ℝ)
174173ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑔 ∈ dom ∫1𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℝ)
175172, 174sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℝ)
176175recnd 11183 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℂ)
177176addid2d 11356 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (0 + (𝑔𝑧)) = (𝑔𝑧))
178171, 177sylan9eqr 2798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (𝑔𝑧))
179178oveq1d 7372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
180179adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
181141rpred 12957 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ)
182181ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ)
183175, 182readdcld 11184 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
184183adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
185160, 164sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ℝ)
186185adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (𝐺𝑧) ∈ ℝ)
187160, 162sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ∈ ℝ)
188187, 185readdcld 11184 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
189188adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
190 simplrr 776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ+)
191190rpred 12957 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ)
192 rpre 12923 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ ℝ+𝑐 ∈ ℝ)
193 rpre 12923 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑑 ∈ ℝ+𝑑 ∈ ℝ)
194 min2 13109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
195192, 193, 194syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
196195ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
197182, 191, 175, 196leadd2dd 11770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑔𝑧) + 𝑑))
198175, 191readdcld 11184 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + 𝑑) ∈ ℝ)
199 letr 11249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ ∧ ((𝑔𝑧) + 𝑑) ∈ ℝ ∧ (𝐺𝑧) ∈ ℝ) → ((((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑔𝑧) + 𝑑) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)))
200183, 198, 185, 199syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑔𝑧) + 𝑑) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)))
201197, 200mpand 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)))
202201imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧))
203164, 162addge02d 11744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ ℝ) → (0 ≤ (𝐹𝑧) ↔ (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧))))
20417, 203mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
205160, 204sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
206205adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
207184, 186, 189, 202, 206letrd 11312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
208207adantlr 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
209180, 208eqbrtrd 5127 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
210 breq1 5108 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 = if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) → (0 ≤ ((𝐹𝑧) + (𝐺𝑧)) ↔ if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
211 breq1 5108 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) → ((((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)) ↔ if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
212210, 211ifboth 4525 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ≤ ((𝐹𝑧) + (𝐺𝑧)) ∧ (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧))) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
213170, 209, 212syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
214213ex 413 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
215214adantld 491 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → ((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
216215adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ¬ (𝑔𝑧) = 0) → ((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
217151, 154, 169, 216ifbothda 4524 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → ((0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
218149anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) ↔ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
219218imbi1d 341 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))) ↔ ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))))
220152anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) ↔ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
221220imbi1d 341 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))) ↔ ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))))
222166ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
223 oveq2 7365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑔𝑧) = 0 → ((𝑓𝑧) + (𝑔𝑧)) = ((𝑓𝑧) + 0))
224 simplrl 775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑓 ∈ dom ∫1)
225 i1ff 25040 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑓 ∈ dom ∫1𝑓:ℝ⟶ℝ)
226225ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑓 ∈ dom ∫1𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
227224, 226sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
228227recnd 11183 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℂ)
229228addid1d 11355 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + 0) = (𝑓𝑧))
230223, 229sylan9eqr 2798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (𝑓𝑧))
231230oveq1d 7372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
232231adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
233227, 182readdcld 11184 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
234233adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
235187adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝐹𝑧) ∈ ℝ)
236188adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
237 simplrl 775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ+)
238237rpred 12957 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ)
239 min1 13108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
240192, 193, 239syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
241240ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
242182, 238, 227, 241leadd2dd 11770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑓𝑧) + 𝑐))
243227, 238readdcld 11184 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + 𝑐) ∈ ℝ)
244 letr 11249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ ∧ ((𝑓𝑧) + 𝑐) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → ((((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧)))
245233, 243, 187, 244syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧)))
246242, 245mpand 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧)))
247246imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧))
248162, 164addge01d 11743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ ℝ) → (0 ≤ (𝐺𝑧) ↔ (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧))))
24976, 248mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
250160, 249sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
251250adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
252234, 235, 236, 247, 251letrd 11312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
253252adantlr 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
254232, 253eqbrtrd 5127 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
255222, 254, 212syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
256255ex 413 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
257256adantlr 713 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
258257adantrd 492 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
259166adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
260182recnd 11183 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℂ)
261228, 176, 260addassd 11177 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))))
262261adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))))
263227, 237ltaddrpd 12990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) < ((𝑓𝑧) + 𝑐))
264227, 243, 263ltled 11303 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐))
265 letr 11249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓𝑧) ∈ ℝ ∧ ((𝑓𝑧) + 𝑐) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧)))
266227, 243, 187, 265syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧)))
267264, 266mpand 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → (𝑓𝑧) ≤ (𝐹𝑧)))
268 le2add 11637 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑓𝑧) ∈ ℝ ∧ ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ) ∧ ((𝐹𝑧) ∈ ℝ ∧ (𝐺𝑧) ∈ ℝ)) → (((𝑓𝑧) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)) → ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
269227, 183, 187, 185, 268syl22anc 837 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)) → ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
270267, 201, 269syl2and 608 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
271270imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
272262, 271eqbrtrd 5127 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
273259, 272, 212syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
274273ex 413 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
275274ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓𝑧) = 0) ∧ ¬ (𝑔𝑧) = 0) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
276219, 221, 258, 275ifbothda 4524 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓𝑧) = 0) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
277145, 148, 217, 276ifbothda 4524 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
278277ralimdva 3164 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (∀𝑧 ∈ ℝ (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) → ∀𝑧 ∈ ℝ if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
279 ovex 7390 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓𝑧) + 𝑐) ∈ V
28021, 279ifex 4536 . . . . . . . . . . . . . . . . . . . . . . . . 25 if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ∈ V
281280a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ∈ V)
282 eqidd 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))))
28320, 281, 14, 282, 24ofrfval2 7638 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ↔ ∀𝑧 ∈ ℝ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
284 ovex 7390 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔𝑧) + 𝑑) ∈ V
28521, 284ifex 4536 . . . . . . . . . . . . . . . . . . . . . . . . 25 if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ∈ V
286285a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧 ∈ ℝ) → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ∈ V)
287 eqidd 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))))
28820, 286, 73, 287, 78ofrfval2 7638 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ↔ ∀𝑧 ∈ ℝ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
289283, 288anbi12d 631 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺) ↔ (∀𝑧 ∈ ℝ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ ∀𝑧 ∈ ℝ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
290 r19.26 3114 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑧 ∈ ℝ (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)) ↔ (∀𝑧 ∈ ℝ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ ∀𝑧 ∈ ℝ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
291289, 290bitr4di 288 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺) ↔ ∀𝑧 ∈ ℝ (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
292291ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺) ↔ ∀𝑧 ∈ ℝ (if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
29319a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → ℝ ∈ V)
294 ovex 7390 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ V
29521, 294ifex 4536 . . . . . . . . . . . . . . . . . . . . . 22 if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ∈ V
296295a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ∈ V)
297 ovexd 7392 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) + (𝐺𝑧)) ∈ V)
298225ffnd 6669 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 ∈ dom ∫1𝑓 Fn ℝ)
299298adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓 Fn ℝ)
300299ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑓 Fn ℝ)
301173ffnd 6669 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔 ∈ dom ∫1𝑔 Fn ℝ)
302301adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑔 Fn ℝ)
303302ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑔 Fn ℝ)
304 eqidd 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) = (𝑓𝑧))
305 eqidd 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = (𝑔𝑧))
306300, 303, 293, 293, 127, 304, 305ofval 7628 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓f + 𝑔)‘𝑧) = ((𝑓𝑧) + (𝑔𝑧)))
307306eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓f + 𝑔)‘𝑧) = 0 ↔ ((𝑓𝑧) + (𝑔𝑧)) = 0))
308306oveq1d 7372 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) = (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)))
309307, 308ifbieq2d 4512 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑))) = if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))))
310309mpteq2dva 5205 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) = (𝑧 ∈ ℝ ↦ if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)))))
31113ffnd 6669 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐹 Fn ℝ)
31272ffnd 6669 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐺 Fn ℝ)
313 eqidd 2737 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
314 eqidd 2737 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
315311, 312, 20, 20, 127, 313, 314offval 7626 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐹f + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹𝑧) + (𝐺𝑧))))
316315ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (𝐹f + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹𝑧) + (𝐺𝑧))))
317293, 296, 297, 310, 316ofrfval2 7638 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → ((𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹f + 𝐺) ↔ ∀𝑧 ∈ ℝ if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
318278, 292, 3173imtr4d 293 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺) → (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹f + 𝐺)))
319318imp 407 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺)) → (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹f + 𝐺))
320 oveq2 7365 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = if(𝑐𝑑, 𝑐, 𝑑) → (((𝑓f + 𝑔)‘𝑧) + 𝑦) = (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
321320ifeq2d 4506 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = if(𝑐𝑑, 𝑐, 𝑑) → if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦)) = if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑))))
322321mpteq2dv 5207 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = if(𝑐𝑑, 𝑐, 𝑑) → (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))))
323322breq1d 5115 . . . . . . . . . . . . . . . . . . 19 (𝑦 = if(𝑐𝑑, 𝑐, 𝑑) → ((𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ↔ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹f + 𝐺)))
324323rspcev 3581 . . . . . . . . . . . . . . . . . 18 ((if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ+ ∧ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹f + 𝐺)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺))
325142, 319, 324syl2anc 584 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺))
326325ex 413 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺)))
327326rexlimdvva 3205 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∃𝑐 ∈ ℝ+𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺)))
328140, 327syl5 34 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺)))
329328a1dd 50 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺))))
330329imp31 418 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺))
331 oveq12 7366 . . . . . . . . . . . . . . 15 ((𝑡 = (∫1𝑓) ∧ 𝑢 = (∫1𝑔)) → (𝑡 + 𝑢) = ((∫1𝑓) + (∫1𝑔)))
332331ad2ant2l 744 . . . . . . . . . . . . . 14 (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) → (𝑡 + 𝑢) = ((∫1𝑓) + (∫1𝑔)))
333134, 135itg1add 25066 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∫1‘(𝑓f + 𝑔)) = ((∫1𝑓) + (∫1𝑔)))
334333eqcomd 2742 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((∫1𝑓) + (∫1𝑔)) = (∫1‘(𝑓f + 𝑔)))
335334adantl 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((∫1𝑓) + (∫1𝑔)) = (∫1‘(𝑓f + 𝑔)))
336332, 335sylan9eqr 2798 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)))) → (𝑡 + 𝑢) = (∫1‘(𝑓f + 𝑔)))
337 eqtr 2759 . . . . . . . . . . . . . 14 ((𝑠 = (𝑡 + 𝑢) ∧ (𝑡 + 𝑢) = (∫1‘(𝑓f + 𝑔))) → 𝑠 = (∫1‘(𝑓f + 𝑔)))
338337ancoms 459 . . . . . . . . . . . . 13 (((𝑡 + 𝑢) = (∫1‘(𝑓f + 𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓f + 𝑔)))
339336, 338sylan 580 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓f + 𝑔)))
340 fveq1 6841 . . . . . . . . . . . . . . . . . . 19 ( = (𝑓f + 𝑔) → (𝑧) = ((𝑓f + 𝑔)‘𝑧))
341340eqeq1d 2738 . . . . . . . . . . . . . . . . . 18 ( = (𝑓f + 𝑔) → ((𝑧) = 0 ↔ ((𝑓f + 𝑔)‘𝑧) = 0))
342340oveq1d 7372 . . . . . . . . . . . . . . . . . 18 ( = (𝑓f + 𝑔) → ((𝑧) + 𝑦) = (((𝑓f + 𝑔)‘𝑧) + 𝑦))
343341, 342ifbieq2d 4512 . . . . . . . . . . . . . . . . 17 ( = (𝑓f + 𝑔) → if((𝑧) = 0, 0, ((𝑧) + 𝑦)) = if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦)))
344343mpteq2dv 5207 . . . . . . . . . . . . . . . 16 ( = (𝑓f + 𝑔) → (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))))
345344breq1d 5115 . . . . . . . . . . . . . . 15 ( = (𝑓f + 𝑔) → ((𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ↔ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺)))
346345rexbidv 3175 . . . . . . . . . . . . . 14 ( = (𝑓f + 𝑔) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺)))
347 fveq2 6842 . . . . . . . . . . . . . . 15 ( = (𝑓f + 𝑔) → (∫1) = (∫1‘(𝑓f + 𝑔)))
348347eqeq2d 2747 . . . . . . . . . . . . . 14 ( = (𝑓f + 𝑔) → (𝑠 = (∫1) ↔ 𝑠 = (∫1‘(𝑓f + 𝑔))))
349346, 348anbi12d 631 . . . . . . . . . . . . 13 ( = (𝑓f + 𝑔) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1‘(𝑓f + 𝑔)))))
350349rspcev 3581 . . . . . . . . . . . 12 (((𝑓f + 𝑔) ∈ dom ∫1 ∧ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1‘(𝑓f + 𝑔)))) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1)))
351137, 330, 339, 350syl12anc 835 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1)))
352351exp31 420 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1)))))
353352rexlimdvva 3205 . . . . . . . . 9 (𝜑 → (∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1)))))
354353impd 411 . . . . . . . 8 (𝜑 → ((∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1))))
355354exlimdvv 1937 . . . . . . 7 (𝜑 → (∃𝑡𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1))))
356133, 355impbid 211 . . . . . 6 (𝜑 → (∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1)) ↔ ∃𝑡𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))))
357 eqeq1 2740 . . . . . . . . . 10 (𝑥 = 𝑡 → (𝑥 = (∫1𝑓) ↔ 𝑡 = (∫1𝑓)))
358357anbi2d 629 . . . . . . . . 9 (𝑥 = 𝑡 → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓)) ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓))))
359358rexbidv 3175 . . . . . . . 8 (𝑥 = 𝑡 → (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓)) ↔ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓))))
360359rexab 3652 . . . . . . 7 (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)))
361 eqeq1 2740 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → (𝑥 = (∫1𝑔) ↔ 𝑢 = (∫1𝑔)))
362361anbi2d 629 . . . . . . . . . . . 12 (𝑥 = 𝑢 → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔)) ↔ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))))
363362rexbidv 3175 . . . . . . . . . . 11 (𝑥 = 𝑢 → (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))))
364363rexab 3652 . . . . . . . . . 10 (∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)))
365364anbi2i 623 . . . . . . . . 9 ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))))
366 19.42v 1957 . . . . . . . . 9 (∃𝑢(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))))
367 reeanv 3217 . . . . . . . . . . . 12 (∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))))
368367anbi1i 624 . . . . . . . . . . 11 ((∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) ↔ ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
369 anass 469 . . . . . . . . . . 11 (((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))))
370368, 369bitr2i 275 . . . . . . . . . 10 ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) ↔ (∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
371370exbii 1850 . . . . . . . . 9 (∃𝑢(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) ↔ ∃𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
372365, 366, 3713bitr2i 298 . . . . . . . 8 ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ ∃𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
373372exbii 1850 . . . . . . 7 (∃𝑡(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ ∃𝑡𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
374360, 373bitri 274 . . . . . 6 (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
375356, 374bitr4di 288 . . . . 5 (𝜑 → (∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1)) ↔ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)))
376375abbidv 2805 . . . 4 (𝜑 → {𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1))} = {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)})
377376supeq1d 9382 . . 3 (𝜑 → sup({𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1))}, ℝ*, < ) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, < ))
378 simpr 485 . . . . . . . . 9 (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (𝑡 + 𝑢))
3796sseli 3940 . . . . . . . . . . 11 (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} → 𝑡 ∈ ℝ)
380379ad2antrr 724 . . . . . . . . . 10 (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑡 ∈ ℝ)
38170sseli 3940 . . . . . . . . . . 11 (𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} → 𝑢 ∈ ℝ)
382381ad2antlr 725 . . . . . . . . . 10 (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑢 ∈ ℝ)
383380, 382readdcld 11184 . . . . . . . . 9 (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → (𝑡 + 𝑢) ∈ ℝ)
384378, 383eqeltrd 2838 . . . . . . . 8 (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 ∈ ℝ)
385384ex 413 . . . . . . 7 ((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) → (𝑠 = (𝑡 + 𝑢) → 𝑠 ∈ ℝ))
386385rexlimivv 3196 . . . . . 6 (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢) → 𝑠 ∈ ℝ)
387386abssi 4027 . . . . 5 {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ
388387a1i 11 . . . 4 (𝜑 → {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ)
389156eqcomi 2745 . . . . . . . 8 0 = (0 + 0)
390 rspceov 7404 . . . . . . . 8 ((0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} ∧ 0 = (0 + 0)) → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢))
391389, 390mp3an3 1450 . . . . . . 7 ((0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢))
39248, 100, 391syl2anc 584 . . . . . 6 (𝜑 → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢))
393 eqeq1 2740 . . . . . . . 8 (𝑠 = 0 → (𝑠 = (𝑡 + 𝑢) ↔ 0 = (𝑡 + 𝑢)))
3943932rexbidv 3213 . . . . . . 7 (𝑠 = 0 → (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢)))
39521, 394spcev 3565 . . . . . 6 (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢) → ∃𝑠𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢))
396392, 395syl 17 . . . . 5 (𝜑 → ∃𝑠𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢))
397 abn0 4340 . . . . 5 ({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅ ↔ ∃𝑠𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢))
398396, 397sylibr 233 . . . 4 (𝜑 → {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅)
39957, 108readdcld 11184 . . . . 5 (𝜑 → (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) ∈ ℝ)
400 simpr 485 . . . . . . . . 9 (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → 𝑏 = (𝑡 + 𝑢))
401379ad2antrl 726 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → 𝑡 ∈ ℝ)
402381ad2antll 727 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → 𝑢 ∈ ℝ)
40357adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ)
404108adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) ∈ ℝ)
405 supxrub 13243 . . . . . . . . . . . . 13 (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}) → 𝑡 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
40659, 405mpan 688 . . . . . . . . . . . 12 (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} → 𝑡 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
407406ad2antrl 726 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → 𝑡 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
408 supxrub 13243 . . . . . . . . . . . . 13 (({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} ⊆ ℝ*𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) → 𝑢 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
409109, 408mpan 688 . . . . . . . . . . . 12 (𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} → 𝑢 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
410409ad2antll 727 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → 𝑢 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
411401, 402, 403, 404, 407, 410le2addd 11774 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → (𝑡 + 𝑢) ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))
412411adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → (𝑡 + 𝑢) ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))
413400, 412eqbrtrd 5127 . . . . . . . 8 (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))
414413ex 413 . . . . . . 7 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → (𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
415414rexlimdvva 3205 . . . . . 6 (𝜑 → (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
416415alrimiv 1930 . . . . 5 (𝜑 → ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
417 breq2 5109 . . . . . . . 8 (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) → (𝑏𝑎𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
418417ralbidv 3174 . . . . . . 7 (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) → (∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏𝑎 ↔ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
419 eqeq1 2740 . . . . . . . . 9 (𝑠 = 𝑏 → (𝑠 = (𝑡 + 𝑢) ↔ 𝑏 = (𝑡 + 𝑢)))
4204192rexbidv 3213 . . . . . . . 8 (𝑠 = 𝑏 → (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢)))
421420ralab 3649 . . . . . . 7 (∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) ↔ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
422418, 421bitrdi 286 . . . . . 6 (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) → (∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏𝑎 ↔ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))))
423422rspcev 3581 . . . . 5 (((sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) ∈ ℝ ∧ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏𝑎)
424399, 416, 423syl2anc 584 . . . 4 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏𝑎)
425 supxrre 13246 . . . 4 (({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ ∧ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅ ∧ ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏𝑎) → sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, < ) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < ))
426388, 398, 424, 425syl3anc 1371 . . 3 (𝜑 → sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, < ) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < ))
427131, 377, 4263eqtrd 2780 . 2 (𝜑 → (∫2‘(𝐹f + 𝐺)) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < ))
428116, 123, 4273eqtr4rd 2787 1 (𝜑 → (∫2‘(𝐹f + 𝐺)) = ((∫2𝐹) + (∫2𝐺)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  Vcvv 3445  wss 3910  c0 4282  ifcif 4486  {csn 4586   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  r cofr 7616  supcsup 9376  cr 11050  0cc0 11051   + caddc 11054  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cn 12153  3c3 12209  +crp 12915  [,)cico 13266  [,]cicc 13267  MblFncmbf 24978  1citg1 24979  2citg2 24980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829  df-mbf 24983  df-itg1 24984  df-itg2 24985
This theorem is referenced by:  ibladdnclem  36134  itgaddnclem1  36136  iblabsnclem  36141  iblabsnc  36142  iblmulc2nc  36143  ftc1anclem4  36154  ftc1anclem5  36155  ftc1anclem6  36156  ftc1anclem8  36158
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