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Theorem itg2addnc 33782
Description: Alternate proof of itg2add 23750 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 23699, 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 9549, 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 780 . . . . . . 7 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))) → 𝑥 = (∫1𝑓))
2 itg1cl 23676 . . . . . . . 8 (𝑓 ∈ dom ∫1 → (∫1𝑓) ∈ ℝ)
32adantr 468 . . . . . . 7 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ)
41, 3eqeltrd 2896 . . . . . 6 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))) → 𝑥 ∈ ℝ)
54rexlimiva 3227 . . . . 5 (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓)) → 𝑥 ∈ ℝ)
65abssi 3885 . . . 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 23678 . . . . . 6 (ℝ × {0}) ∈ dom ∫1
9 3nn 11470 . . . . . . . 8 3 ∈ ℕ
10 nnrp 12063 . . . . . . . 8 (3 ∈ ℕ → 3 ∈ ℝ+)
11 ne0i 4133 . . . . . . . 8 (3 ∈ ℝ+ → ℝ+ ≠ ∅)
129, 10, 11mp2b 10 . . . . . . 7 + ≠ ∅
13 itg2addnc.f2 . . . . . . . . . . . . 13 (𝜑𝐹:ℝ⟶(0[,)+∞))
1413ffvelrnda 6588 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ∈ (0[,)+∞))
15 elrege0 12505 . . . . . . . . . . . 12 ((𝐹𝑧) ∈ (0[,)+∞) ↔ ((𝐹𝑧) ∈ ℝ ∧ 0 ≤ (𝐹𝑧)))
1614, 15sylib 209 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℝ) → ((𝐹𝑧) ∈ ℝ ∧ 0 ≤ (𝐹𝑧)))
1716simprd 485 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ≤ (𝐹𝑧))
1817ralrimiva 3165 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧))
19 reex 10319 . . . . . . . . . . 11 ℝ ∈ V
2019a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ∈ V)
21 c0ex 10326 . . . . . . . . . . 11 0 ∈ V
2221a1i 11 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ∈ V)
23 eqidd 2818 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
2413feqmptd 6477 . . . . . . . . . 10 (𝜑𝐹 = (𝑧 ∈ ℝ ↦ (𝐹𝑧)))
2520, 22, 14, 23, 24ofrfval2 7152 . . . . . . . . 9 (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧)))
2618, 25mpbird 248 . . . . . . . 8 (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
2726ralrimivw 3166 . . . . . . 7 (𝜑 → ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
28 r19.2z 4266 . . . . . . 7 ((ℝ+ ≠ ∅ ∧ ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹) → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
2912, 27, 28sylancr 577 . . . . . 6 (𝜑 → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
30 fveq2 6415 . . . . . . . . . 10 (𝑓 = (ℝ × {0}) → (∫1𝑓) = (∫1‘(ℝ × {0})))
31 itg10 23679 . . . . . . . . . 10 (∫1‘(ℝ × {0})) = 0
3230, 31syl6req 2868 . . . . . . . . 9 (𝑓 = (ℝ × {0}) → 0 = (∫1𝑓))
3332biantrud 523 . . . . . . . 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 6414 . . . . . . . . . . . . 13 (𝑓 = (ℝ × {0}) → (𝑓𝑧) = ((ℝ × {0})‘𝑧))
3521fvconst2 6701 . . . . . . . . . . . . 13 (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
3634, 35sylan9eq 2871 . . . . . . . . . . . 12 ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) = 0)
3736iftrued 4298 . . . . . . . . . . 11 ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) = 0)
3837mpteq2dva 4949 . . . . . . . . . 10 (𝑓 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ 0))
3938breq1d 4865 . . . . . . . . 9 (𝑓 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹))
4039rexbidv 3251 . . . . . . . 8 (𝑓 = (ℝ × {0}) → (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹))
4133, 40bitr3d 272 . . . . . . 7 (𝑓 = (ℝ × {0}) → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ 0 = (∫1𝑓)) ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹))
4241rspcev 3513 . . . . . 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 577 . . . . 5 (𝜑 → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ 0 = (∫1𝑓)))
44 eqeq1 2821 . . . . . . . 8 (𝑥 = 0 → (𝑥 = (∫1𝑓) ↔ 0 = (∫1𝑓)))
4544anbi2d 616 . . . . . . 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 3251 . . . . . 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 3556 . . . . 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 225 . . . 4 (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))})
4948ne0d 4134 . . 3 (𝜑 → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ≠ ∅)
50 icossicc 12486 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
51 fss 6276 . . . . . . 7 ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
5250, 51mpan2 674 . . . . . 6 (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶(0[,]+∞))
53 eqid 2817 . . . . . . 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 33779 . . . . . 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 2897 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ)
58 ressxr 10375 . . . . . . 7 ℝ ⊆ ℝ*
596, 58sstri 3818 . . . . . 6 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*
60 supxrub 12379 . . . . . 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 673 . . . . 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 3121 . . . 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 4915 . . . 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 576 . . 3 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}𝑏𝑎)
65 simprr 780 . . . . . . 7 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))) → 𝑥 = (∫1𝑔))
66 itg1cl 23676 . . . . . . . 8 (𝑔 ∈ dom ∫1 → (∫1𝑔) ∈ ℝ)
6766adantr 468 . . . . . . 7 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))) → (∫1𝑔) ∈ ℝ)
6865, 67eqeltrd 2896 . . . . . 6 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))) → 𝑥 ∈ ℝ)
6968rexlimiva 3227 . . . . 5 (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔)) → 𝑥 ∈ ℝ)
7069abssi 3885 . . . 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[,)+∞))
7372ffvelrnda 6588 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ∈ (0[,)+∞))
74 elrege0 12505 . . . . . . . . . . . 12 ((𝐺𝑧) ∈ (0[,)+∞) ↔ ((𝐺𝑧) ∈ ℝ ∧ 0 ≤ (𝐺𝑧)))
7573, 74sylib 209 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℝ) → ((𝐺𝑧) ∈ ℝ ∧ 0 ≤ (𝐺𝑧)))
7675simprd 485 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ≤ (𝐺𝑧))
7776ralrimiva 3165 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐺𝑧))
7872feqmptd 6477 . . . . . . . . . 10 (𝜑𝐺 = (𝑧 ∈ ℝ ↦ (𝐺𝑧)))
7920, 22, 73, 23, 78ofrfval2 7152 . . . . . . . . 9 (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐺𝑧)))
8077, 79mpbird 248 . . . . . . . 8 (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺)
8180ralrimivw 3166 . . . . . . 7 (𝜑 → ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺)
82 r19.2z 4266 . . . . . . 7 ((ℝ+ ≠ ∅ ∧ ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺) → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺)
8312, 81, 82sylancr 577 . . . . . 6 (𝜑 → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺)
84 fveq2 6415 . . . . . . . . . 10 (𝑔 = (ℝ × {0}) → (∫1𝑔) = (∫1‘(ℝ × {0})))
8584, 31syl6req 2868 . . . . . . . . 9 (𝑔 = (ℝ × {0}) → 0 = (∫1𝑔))
8685biantrud 523 . . . . . . . 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 6414 . . . . . . . . . . . . 13 (𝑔 = (ℝ × {0}) → (𝑔𝑧) = ((ℝ × {0})‘𝑧))
8887, 35sylan9eq 2871 . . . . . . . . . . . 12 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = 0)
8988iftrued 4298 . . . . . . . . . . 11 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) = 0)
9089mpteq2dva 4949 . . . . . . . . . 10 (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ 0))
9190breq1d 4865 . . . . . . . . 9 (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ↔ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺))
9291rexbidv 3251 . . . . . . . 8 (𝑔 = (ℝ × {0}) → (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺))
9386, 92bitr3d 272 . . . . . . 7 (𝑔 = (ℝ × {0}) → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ∧ 0 = (∫1𝑔)) ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺))
9493rspcev 3513 . . . . . 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 577 . . . . 5 (𝜑 → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ∧ 0 = (∫1𝑔)))
96 eqeq1 2821 . . . . . . . 8 (𝑥 = 0 → (𝑥 = (∫1𝑔) ↔ 0 = (∫1𝑔)))
9796anbi2d 616 . . . . . . 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 3251 . . . . . 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 3556 . . . . 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 225 . . . 4 (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})
101100ne0d 4134 . . 3 (𝜑 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ≠ ∅)
102 fss 6276 . . . . . . 7 ((𝐺:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐺:ℝ⟶(0[,]+∞))
10350, 102mpan2 674 . . . . . 6 (𝐺:ℝ⟶(0[,)+∞) → 𝐺:ℝ⟶(0[,]+∞))
104 eqid 2817 . . . . . . 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 33779 . . . . . 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 2897 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) ∈ ℝ)
10970, 58sstri 3818 . . . . . 6 {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ⊆ ℝ*
110 supxrub 12379 . . . . . 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 673 . . . . 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 3121 . . . 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 4915 . . . 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 576 . . 3 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏𝑎)
115 eqid 2817 . . 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 11283 . 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 12382 . . . . 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 1483 . . . 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 2851 . . 3 (𝜑 → (∫2𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ, < ))
120 supxrre 12382 . . . . 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 1483 . . . 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 2851 . . 3 (𝜑 → (∫2𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ, < ))
123119, 122oveq12d 6899 . 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 12511 . . . . . . 7 ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞))
12550, 124sseldi 3807 . . . . . 6 ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,]+∞))
126125adantl 469 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,]+∞))
127 inidm 4030 . . . . 5 (ℝ ∩ ℝ) = ℝ
128126, 13, 72, 20, 20, 127off 7149 . . . 4 (𝜑 → (𝐹𝑓 + 𝐺):ℝ⟶(0[,]+∞))
129 eqid 2817 . . . . 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 33779 . . . 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 33781 . . . . . . 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 470 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓 ∈ dom ∫1)
135 simpr 473 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑔 ∈ dom ∫1)
136134, 135i1fadd 23686 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑓𝑓 + 𝑔) ∈ dom ∫1)
137136ad3antlr 713 . . . . . . . . . . . 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 3306 . . . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . . . 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 744 . . . . . . . . . . . . . . 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 4334 . . . . . . . . . . . . . . . . . . 19 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ+)
142141ad2antlr 709 . . . . . . . . . . . . . . . . . 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 4858 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (0 ≤ (𝐹𝑧) ↔ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
144143anbi1d 617 . . . . . . . . . . . . . . . . . . . . . . 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 332 . . . . . . . . . . . . . . . . . . . . . 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 4858 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓𝑧) + 𝑐) = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ↔ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
147146anbi1d 617 . . . . . . . . . . . . . . . . . . . . . . 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 332 . . . . . . . . . . . . . . . . . . . . . 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 4858 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (0 ≤ (𝐺𝑧) ↔ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
150149anbi2d 616 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((0 ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) ↔ (0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
151150imbi1d 332 . . . . . . . . . . . . . . . . . . . . . . 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 4858 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) ↔ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
153152anbi2d 616 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) ↔ (0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
154153imbi1d 332 . . . . . . . . . . . . . . . . . . . . . . 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 6890 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (0 + 0))
156 00id 10503 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 + 0) = 0
157155, 156syl6eq 2867 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = 0)
158157iftrued 4298 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) = 0)
159158adantll 696 . . . . . . . . . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝜑)
16115simplbi 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹𝑧) ∈ (0[,)+∞) → (𝐹𝑧) ∈ ℝ)
16214, 161syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ∈ ℝ)
16374simplbi 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐺𝑧) ∈ (0[,)+∞) → (𝐺𝑧) ∈ ℝ)
16473, 163syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ℝ)
165162, 164, 17, 76addge0d 10895 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑧 ∈ ℝ) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
166160, 165sylan 571 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
167166ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
168159, 167eqbrtrd 4877 . . . . . . . . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
171 oveq1 6888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓𝑧) = 0 → ((𝑓𝑧) + (𝑔𝑧)) = (0 + (𝑔𝑧)))
172 simplrr 787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑔 ∈ dom ∫1)
173 i1ff 23667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑔 ∈ dom ∫1𝑔:ℝ⟶ℝ)
174173ffvelrnda 6588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑔 ∈ dom ∫1𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℝ)
175172, 174sylan 571 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℝ)
176175recnd 10360 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℂ)
177176addid2d 10529 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (0 + (𝑔𝑧)) = (𝑔𝑧))
178171, 177sylan9eqr 2873 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (𝑔𝑧))
179178oveq1d 6896 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12093 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ)
182181ad2antlr 709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ)
183175, 182readdcld 10361 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
184183adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
185160, 164sylan 571 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ℝ)
186185adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (𝐺𝑧) ∈ ℝ)
187160, 162sylan 571 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ∈ ℝ)
188187, 185readdcld 10361 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
189188adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
190 simplrr 787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ+)
191190rpred 12093 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ)
192 rpre 12060 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ ℝ+𝑐 ∈ ℝ)
193 rpre 12060 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑑 ∈ ℝ+𝑑 ∈ ℝ)
194 min2 12246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
195192, 193, 194syl2an 585 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
196195ad2antlr 709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
197182, 191, 175, 196leadd2dd 10934 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑔𝑧) + 𝑑))
198175, 191readdcld 10361 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + 𝑑) ∈ ℝ)
199 letr 10423 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)))
202201imp 395 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧))
203164, 162addge02d 10908 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ ℝ) → (0 ≤ (𝐹𝑧) ↔ (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧))))
20417, 203mpbid 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
205160, 204sylan 571 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
206205adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
207184, 186, 189, 202, 206letrd 10486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
208207adantlr 697 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4877 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4858 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4858 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4328 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . . . . . . . . . . . . . . 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 399 . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . . . . 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 4327 . . . . . . . . . . . . . . . . . . . . . 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 616 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) ↔ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
219218imbi1d 332 . . . . . . . . . . . . . . . . . . . . . . 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 616 . . . . . . . . . . . . . . . . . . . . . . . 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 332 . . . . . . . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
223 oveq2 6889 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑔𝑧) = 0 → ((𝑓𝑧) + (𝑔𝑧)) = ((𝑓𝑧) + 0))
224 simplrl 786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑓 ∈ dom ∫1)
225 i1ff 23667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑓 ∈ dom ∫1𝑓:ℝ⟶ℝ)
226225ffvelrnda 6588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑓 ∈ dom ∫1𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
227224, 226sylan 571 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
228227recnd 10360 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℂ)
229228addid1d 10528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + 0) = (𝑓𝑧))
230223, 229sylan9eqr 2873 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (𝑓𝑧))
231230oveq1d 6896 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10361 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
234233adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
235187adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝐹𝑧) ∈ ℝ)
236188adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
237 simplrl 786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ+)
238237rpred 12093 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ)
239 min1 12245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
240192, 193, 239syl2an 585 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
241240ad2antlr 709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
242182, 238, 227, 241leadd2dd 10934 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑓𝑧) + 𝑐))
243227, 238readdcld 10361 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + 𝑐) ∈ ℝ)
244 letr 10423 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧)))
247246imp 395 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧))
248162, 164addge01d 10907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ ℝ) → (0 ≤ (𝐺𝑧) ↔ (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧))))
24976, 248mpbid 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
250160, 249sylan 571 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
251250adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
252234, 235, 236, 247, 251letrd 10486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
253252adantlr 697 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4877 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . . . . . . . . . . . . . . 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 399 . . . . . . . . . . . . . . . . . . . . . . . . 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 697 . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
260182recnd 10360 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℂ)
261228, 176, 260addassd 10354 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) < ((𝑓𝑧) + 𝑐))
264227, 243, 263ltled 10477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐))
265 letr 10423 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓𝑧) ∈ ℝ ∧ ((𝑓𝑧) + 𝑐) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧)))
266227, 243, 187, 265syl3anc 1483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧)))
267264, 266mpand 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → (𝑓𝑧) ≤ (𝐹𝑧)))
268 le2add 10802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 858 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 597 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4877 . . . . . . . . . . . . . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . . . . . . . . . . . . . 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 399 . . . . . . . . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . . . . . . . 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 4327 . . . . . . . . . . . . . . . . . . . . . 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 4327 . . . . . . . . . . . . . . . . . . . . 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 3161 . . . . . . . . . . . . . . . . . . . 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 6913 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓𝑧) + 𝑐) ∈ V
28021, 279ifex 4338 . . . . . . . . . . . . . . . . . . . . . . . . 25 if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ∈ V
281280a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ∈ V)
282 eqidd 2818 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))))
28320, 281, 14, 282, 24ofrfval2 7152 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ↔ ∀𝑧 ∈ ℝ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
284 ovex 6913 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔𝑧) + 𝑑) ∈ V
28521, 284ifex 4338 . . . . . . . . . . . . . . . . . . . . . . . . 25 if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ∈ V
286285a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧 ∈ ℝ) → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ∈ V)
287 eqidd 2818 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))))
28820, 286, 73, 287, 78ofrfval2 7152 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ↔ ∀𝑧 ∈ ℝ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
289283, 288anbi12d 618 . . . . . . . . . . . . . . . . . . . . . 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 3263 . . . . . . . . . . . . . . . . . . . . . 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, 290syl6bbr 280 . . . . . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . . . . 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 6913 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ V
29521, 294ifex 4338 . . . . . . . . . . . . . . . . . . . . . 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 6915 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) + (𝐺𝑧)) ∈ V)
298225ffnd 6264 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 ∈ dom ∫1𝑓 Fn ℝ)
299298adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓 Fn ℝ)
300299ad2antlr 709 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑓 Fn ℝ)
301173ffnd 6264 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔 ∈ dom ∫1𝑔 Fn ℝ)
302301adantl 469 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑔 Fn ℝ)
303302ad2antlr 709 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑔 Fn ℝ)
304 eqidd 2818 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) = (𝑓𝑧))
305 eqidd 2818 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = (𝑔𝑧))
306300, 303, 293, 293, 127, 304, 305ofval 7143 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑓 + 𝑔)‘𝑧) = ((𝑓𝑧) + (𝑔𝑧)))
307306eqeq1d 2819 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑓 + 𝑔)‘𝑧) = 0 ↔ ((𝑓𝑧) + (𝑔𝑧)) = 0))
308306oveq1d 6896 . . . . . . . . . . . . . . . . . . . . . . 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 4315 . . . . . . . . . . . . . . . . . . . . . 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 4949 . . . . . . . . . . . . . . . . . . . . 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 6264 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐹 Fn ℝ)
31272ffnd 6264 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐺 Fn ℝ)
313 eqidd 2818 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
314 eqidd 2818 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
315311, 312, 20, 20, 127, 313, 314offval 7141 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐹𝑓 + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹𝑧) + (𝐺𝑧))))
316315ad2antrr 708 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (𝐹𝑓 + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹𝑧) + (𝐺𝑧))))
317293, 296, 297, 310, 316ofrfval2 7152 . . . . . . . . . . . . . . . . . . . 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 285 . . . . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . . . . 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 6889 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = if(𝑐𝑑, 𝑐, 𝑑) → (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦) = (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
321320ifeq2d 4309 . . . . . . . . . . . . . . . . . . . . 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 4950 . . . . . . . . . . . . . . . . . . . 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 4865 . . . . . . . . . . . . . . . . . . 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 3513 . . . . . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . . . . . 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 399 . . . . . . . . . . . . . . . 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 3237 . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . 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 6890 . . . . . . . . . . . . . . 15 ((𝑡 = (∫1𝑓) ∧ 𝑢 = (∫1𝑔)) → (𝑡 + 𝑢) = ((∫1𝑓) + (∫1𝑔)))
332331ad2ant2l 743 . . . . . . . . . . . . . 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 23692 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∫1‘(𝑓𝑓 + 𝑔)) = ((∫1𝑓) + (∫1𝑔)))
334333eqcomd 2823 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((∫1𝑓) + (∫1𝑔)) = (∫1‘(𝑓𝑓 + 𝑔)))
335334adantl 469 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((∫1𝑓) + (∫1𝑔)) = (∫1‘(𝑓𝑓 + 𝑔)))
336332, 335sylan9eqr 2873 . . . . . . . . . . . . 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 2836 . . . . . . . . . . . . . 14 ((𝑠 = (𝑡 + 𝑢) ∧ (𝑡 + 𝑢) = (∫1‘(𝑓𝑓 + 𝑔))) → 𝑠 = (∫1‘(𝑓𝑓 + 𝑔)))
338337ancoms 448 . . . . . . . . . . . . 13 (((𝑡 + 𝑢) = (∫1‘(𝑓𝑓 + 𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓𝑓 + 𝑔)))
339336, 338sylan 571 . . . . . . . . . . . 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 6414 . . . . . . . . . . . . . . . . . . 19 ( = (𝑓𝑓 + 𝑔) → (𝑧) = ((𝑓𝑓 + 𝑔)‘𝑧))
341340eqeq1d 2819 . . . . . . . . . . . . . . . . . 18 ( = (𝑓𝑓 + 𝑔) → ((𝑧) = 0 ↔ ((𝑓𝑓 + 𝑔)‘𝑧) = 0))
342340oveq1d 6896 . . . . . . . . . . . . . . . . . 18 ( = (𝑓𝑓 + 𝑔) → ((𝑧) + 𝑦) = (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))
343341, 342ifbieq2d 4315 . . . . . . . . . . . . . . . . 17 ( = (𝑓𝑓 + 𝑔) → if((𝑧) = 0, 0, ((𝑧) + 𝑦)) = if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦)))
344343mpteq2dv 4950 . . . . . . . . . . . . . . . 16 ( = (𝑓𝑓 + 𝑔) → (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))))
345344breq1d 4865 . . . . . . . . . . . . . . 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 3251 . . . . . . . . . . . . . 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 6415 . . . . . . . . . . . . . . 15 ( = (𝑓𝑓 + 𝑔) → (∫1) = (∫1‘(𝑓𝑓 + 𝑔)))
348347eqeq2d 2827 . . . . . . . . . . . . . 14 ( = (𝑓𝑓 + 𝑔) → (𝑠 = (∫1) ↔ 𝑠 = (∫1‘(𝑓𝑓 + 𝑔))))
349346, 348anbi12d 618 . . . . . . . . . . . . 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 3513 . . . . . . . . . . . 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 856 . . . . . . . . . . 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 408 . . . . . . . . . 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 3237 . . . . . . . . 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 398 . . . . . . . 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 2025 . . . . . . 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 203 . . . . . 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 2821 . . . . . . . . . 10 (𝑥 = 𝑡 → (𝑥 = (∫1𝑓) ↔ 𝑡 = (∫1𝑓)))
358357anbi2d 616 . . . . . . . . 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 3251 . . . . . . . 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 3576 . . . . . . 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 2821 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → (𝑥 = (∫1𝑔) ↔ 𝑢 = (∫1𝑔)))
362361anbi2d 616 . . . . . . . . . . . 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 3251 . . . . . . . . . . 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 3576 . . . . . . . . . 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 611 . . . . . . . . 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 2044 . . . . . . . . 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 3306 . . . . . . . . . . . 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 612 . . . . . . . . . . 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 456 . . . . . . . . . . 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 267 . . . . . . . . . 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 1933 . . . . . . . . 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 290 . . . . . . . 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 1933 . . . . . . 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 266 . . . . . 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, 374syl6bbr 280 . . . . 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 2936 . . . 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 8598 . . 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 473 . . . . . . . . 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 3805 . . . . . . . . . . 11 (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} → 𝑡 ∈ ℝ)
380379ad2antrr 708 . . . . . . . . . 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 3805 . . . . . . . . . . 11 (𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} → 𝑢 ∈ ℝ)
382381ad2antlr 709 . . . . . . . . . 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 10361 . . . . . . . . 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 2896 . . . . . . . 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 399 . . . . . . 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 3235 . . . . . 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 3885 . . . . 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 2826 . . . . . . . 8 0 = (0 + 0)
390 rspceov 6927 . . . . . . . 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 1567 . . . . . . 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 575 . . . . . 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 2821 . . . . . . . 8 (𝑠 = 0 → (𝑠 = (𝑡 + 𝑢) ↔ 0 = (𝑡 + 𝑢)))
3943932rexbidv 3256 . . . . . . 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 3504 . . . . . 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 4166 . . . . 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 225 . . . 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 10361 . . . . 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 473 . . . . . . . . 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 710 . . . . . . . . . . 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 711 . . . . . . . . . . 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 468 . . . . . . . . . . 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 468 . . . . . . . . . . 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 12379 . . . . . . . . . . . . 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 673 . . . . . . . . . . . 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 710 . . . . . . . . . . 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 12379 . . . . . . . . . . . . 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 673 . . . . . . . . . . . 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 711 . . . . . . . . . . 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 10938 . . . . . . . . . 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 468 . . . . . . . . 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 4877 . . . . . . . 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 399 . . . . . . 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 3237 . . . . . 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 2018 . . . . 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 4859 . . . . . . . 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 3185 . . . . . . 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 2821 . . . . . . . . 9 (𝑠 = 𝑏 → (𝑠 = (𝑡 + 𝑢) ↔ 𝑏 = (𝑡 + 𝑢)))
4204192rexbidv 3256 . . . . . . . 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 3574 . . . . . . 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, 421syl6bb 278 . . . . . 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 3513 . . . . 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 575 . . . 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 12382 . . . 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 1483 . . 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 2855 . 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 2862 1 (𝜑 → (∫2‘(𝐹𝑓 + 𝐺)) = ((∫2𝐹) + (∫2𝐺)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wex 1859  wcel 2157  {cab 2803  wne 2989  wral 3107  wrex 3108  Vcvv 3402  wss 3780  c0 4127  ifcif 4290  {csn 4381   class class class wbr 4855  cmpt 4934   × cxp 5320  dom cdm 5322   Fn wfn 6103  wf 6104  cfv 6108  (class class class)co 6881  𝑓 cof 7132  𝑟 cofr 7133  supcsup 8592  cr 10227  0cc0 10228   + caddc 10231  +∞cpnf 10363  *cxr 10365   < clt 10366  cle 10367  cn 11312  3c3 11364  +crp 12053  [,)cico 12402  [,]cicc 12403  MblFncmbf 23605  1citg1 23606  2citg2 23607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4975  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186  ax-inf2 8792  ax-cnex 10284  ax-resscn 10285  ax-1cn 10286  ax-icn 10287  ax-addcl 10288  ax-addrcl 10289  ax-mulcl 10290  ax-mulrcl 10291  ax-mulcom 10292  ax-addass 10293  ax-mulass 10294  ax-distr 10295  ax-i2m1 10296  ax-1ne0 10297  ax-1rid 10298  ax-rnegex 10299  ax-rrecex 10300  ax-cnre 10301  ax-pre-lttri 10302  ax-pre-lttrn 10303  ax-pre-ltadd 10304  ax-pre-mulgt0 10305  ax-pre-sup 10306  ax-addf 10307
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-disj 4824  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5230  df-eprel 5235  df-po 5243  df-so 5244  df-fr 5281  df-se 5282  df-we 5283  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-pred 5904  df-ord 5950  df-on 5951  df-lim 5952  df-suc 5953  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-f1 6113  df-fo 6114  df-f1o 6115  df-fv 6116  df-isom 6117  df-riota 6842  df-ov 6884  df-oprab 6885  df-mpt2 6886  df-of 7134  df-ofr 7135  df-om 7303  df-1st 7405  df-2nd 7406  df-wrecs 7649  df-recs 7711  df-rdg 7749  df-1o 7803  df-2o 7804  df-oadd 7807  df-er 7986  df-map 8101  df-pm 8102  df-en 8200  df-dom 8201  df-sdom 8202  df-fin 8203  df-fi 8563  df-sup 8594  df-inf 8595  df-oi 8661  df-card 9055  df-cda 9282  df-pnf 10368  df-mnf 10369  df-xr 10370  df-ltxr 10371  df-le 10372  df-sub 10560  df-neg 10561  df-div 10977  df-nn 11313  df-2 11371  df-3 11372  df-n0 11567  df-z 11651  df-uz 11912  df-q 12015  df-rp 12054  df-xneg 12169  df-xadd 12170  df-xmul 12171  df-ioo 12404  df-ico 12406  df-icc 12407  df-fz 12557  df-fzo 12697  df-fl 12824  df-seq 13032  df-exp 13091  df-hash 13345  df-cj 14069  df-re 14070  df-im 14071  df-sqrt 14205  df-abs 14206  df-clim 14449  df-sum 14647  df-rest 16295  df-topgen 16316  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-top 20920  df-topon 20937  df-bases 20972  df-cmp 21412  df-ovol 23455  df-vol 23456  df-mbf 23610  df-itg1 23611  df-itg2 23612
This theorem is referenced by:  ibladdnclem  33784  itgaddnclem1  33786  iblabsnclem  33791  iblabsnc  33792  iblmulc2nc  33793  ftc1anclem4  33806  ftc1anclem5  33807  ftc1anclem6  33808  ftc1anclem8  33810
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