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Theorem itg2addnc 33797
Description: Alternate proof of itg2add 23747 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 23696, 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 9460, 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 750 . . . . . . 7 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))) → 𝑥 = (∫1𝑓))
2 itg1cl 23673 . . . . . . . 8 (𝑓 ∈ dom ∫1 → (∫1𝑓) ∈ ℝ)
32adantr 466 . . . . . . 7 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ)
41, 3eqeltrd 2850 . . . . . 6 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))) → 𝑥 ∈ ℝ)
54rexlimiva 3176 . . . . 5 (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓)) → 𝑥 ∈ ℝ)
65abssi 3827 . . . 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 23675 . . . . . 6 (ℝ × {0}) ∈ dom ∫1
9 3nn 11389 . . . . . . . 8 3 ∈ ℕ
10 nnrp 12046 . . . . . . . 8 (3 ∈ ℕ → 3 ∈ ℝ+)
11 ne0i 4070 . . . . . . . 8 (3 ∈ ℝ+ → ℝ+ ≠ ∅)
129, 10, 11mp2b 10 . . . . . . 7 + ≠ ∅
13 itg2addnc.f2 . . . . . . . . . . . . 13 (𝜑𝐹:ℝ⟶(0[,)+∞))
1413ffvelrnda 6503 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ∈ (0[,)+∞))
15 elrege0 12486 . . . . . . . . . . . 12 ((𝐹𝑧) ∈ (0[,)+∞) ↔ ((𝐹𝑧) ∈ ℝ ∧ 0 ≤ (𝐹𝑧)))
1614, 15sylib 208 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℝ) → ((𝐹𝑧) ∈ ℝ ∧ 0 ≤ (𝐹𝑧)))
1716simprd 479 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ≤ (𝐹𝑧))
1817ralrimiva 3115 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧))
19 reex 10230 . . . . . . . . . . 11 ℝ ∈ V
2019a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ∈ V)
21 c0ex 10237 . . . . . . . . . . 11 0 ∈ V
2221a1i 11 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ∈ V)
23 eqidd 2772 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
2413feqmptd 6392 . . . . . . . . . 10 (𝜑𝐹 = (𝑧 ∈ ℝ ↦ (𝐹𝑧)))
2520, 22, 14, 23, 24ofrfval2 7063 . . . . . . . . 9 (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧)))
2618, 25mpbird 247 . . . . . . . 8 (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
2726ralrimivw 3116 . . . . . . 7 (𝜑 → ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
28 r19.2z 4202 . . . . . . 7 ((ℝ+ ≠ ∅ ∧ ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹) → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
2912, 27, 28sylancr 569 . . . . . 6 (𝜑 → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹)
30 fveq2 6333 . . . . . . . . . 10 (𝑓 = (ℝ × {0}) → (∫1𝑓) = (∫1‘(ℝ × {0})))
31 itg10 23676 . . . . . . . . . 10 (∫1‘(ℝ × {0})) = 0
3230, 31syl6req 2822 . . . . . . . . 9 (𝑓 = (ℝ × {0}) → 0 = (∫1𝑓))
3332biantrud 517 . . . . . . . 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 6332 . . . . . . . . . . . . 13 (𝑓 = (ℝ × {0}) → (𝑓𝑧) = ((ℝ × {0})‘𝑧))
3521fvconst2 6614 . . . . . . . . . . . . 13 (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
3634, 35sylan9eq 2825 . . . . . . . . . . . 12 ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) = 0)
3736iftrued 4234 . . . . . . . . . . 11 ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) = 0)
3837mpteq2dva 4879 . . . . . . . . . 10 (𝑓 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ 0))
3938breq1d 4797 . . . . . . . . 9 (𝑓 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹))
4039rexbidv 3200 . . . . . . . 8 (𝑓 = (ℝ × {0}) → (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹))
4133, 40bitr3d 270 . . . . . . 7 (𝑓 = (ℝ × {0}) → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ 0 = (∫1𝑓)) ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐹))
4241rspcev 3461 . . . . . 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 569 . . . . 5 (𝜑 → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ∧ 0 = (∫1𝑓)))
44 eqeq1 2775 . . . . . . . 8 (𝑥 = 0 → (𝑥 = (∫1𝑓) ↔ 0 = (∫1𝑓)))
4544anbi2d 608 . . . . . . 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 3200 . . . . . 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 3502 . . . . 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 224 . . . 4 (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))})
49 ne0i 4070 . . . 4 (0 ∈ {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))} ≠ ∅)
5048, 49syl 17 . . 3 (𝜑 → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ≠ ∅)
51 icossicc 12467 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
52 fss 6197 . . . . . . 7 ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
5351, 52mpan2 665 . . . . . 6 (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶(0[,]+∞))
54 eqid 2771 . . . . . . 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))}
5554itg2addnclem 33794 . . . . . 6 (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
5613, 53, 553syl 18 . . . . 5 (𝜑 → (∫2𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ))
57 itg2addnc.f3 . . . . 5 (𝜑 → (∫2𝐹) ∈ ℝ)
5856, 57eqeltrrd 2851 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ)
59 ressxr 10286 . . . . . . 7 ℝ ⊆ ℝ*
606, 59sstri 3762 . . . . . 6 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*
61 supxrub 12360 . . . . . 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*, < ))
6260, 61mpan 664 . . . . 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*, < ))
6362rgen 3071 . . . 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*, < )
64 breq2 4791 . . . . . 6 (𝑎 = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) → (𝑏𝑎𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ*, < )))
6564ralbidv 3135 . . . . 5 (𝑎 = sup({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))}𝑏𝑎 ↔ ∀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*, < )))
6665rspcev 3461 . . . 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)
6758, 63, 66sylancl 568 . . 3 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}𝑏𝑎)
68 simprr 750 . . . . . . 7 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))) → 𝑥 = (∫1𝑔))
69 itg1cl 23673 . . . . . . . 8 (𝑔 ∈ dom ∫1 → (∫1𝑔) ∈ ℝ)
7069adantr 466 . . . . . . 7 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))) → (∫1𝑔) ∈ ℝ)
7168, 70eqeltrd 2850 . . . . . 6 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))) → 𝑥 ∈ ℝ)
7271rexlimiva 3176 . . . . 5 (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔)) → 𝑥 ∈ ℝ)
7372abssi 3827 . . . 4 {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ⊆ ℝ
7473a1i 11 . . 3 (𝜑 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ⊆ ℝ)
75 itg2addnc.g2 . . . . . . . . . . . . 13 (𝜑𝐺:ℝ⟶(0[,)+∞))
7675ffvelrnda 6503 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ∈ (0[,)+∞))
77 elrege0 12486 . . . . . . . . . . . 12 ((𝐺𝑧) ∈ (0[,)+∞) ↔ ((𝐺𝑧) ∈ ℝ ∧ 0 ≤ (𝐺𝑧)))
7876, 77sylib 208 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℝ) → ((𝐺𝑧) ∈ ℝ ∧ 0 ≤ (𝐺𝑧)))
7978simprd 479 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ≤ (𝐺𝑧))
8079ralrimiva 3115 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐺𝑧))
8175feqmptd 6392 . . . . . . . . . 10 (𝜑𝐺 = (𝑧 ∈ ℝ ↦ (𝐺𝑧)))
8220, 22, 76, 23, 81ofrfval2 7063 . . . . . . . . 9 (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐺𝑧)))
8380, 82mpbird 247 . . . . . . . 8 (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺)
8483ralrimivw 3116 . . . . . . 7 (𝜑 → ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺)
85 r19.2z 4202 . . . . . . 7 ((ℝ+ ≠ ∅ ∧ ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺) → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺)
8612, 84, 85sylancr 569 . . . . . 6 (𝜑 → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺)
87 fveq2 6333 . . . . . . . . . 10 (𝑔 = (ℝ × {0}) → (∫1𝑔) = (∫1‘(ℝ × {0})))
8887, 31syl6req 2822 . . . . . . . . 9 (𝑔 = (ℝ × {0}) → 0 = (∫1𝑔))
8988biantrud 517 . . . . . . . 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))))
90 fveq1 6332 . . . . . . . . . . . . 13 (𝑔 = (ℝ × {0}) → (𝑔𝑧) = ((ℝ × {0})‘𝑧))
9190, 35sylan9eq 2825 . . . . . . . . . . . 12 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = 0)
9291iftrued 4234 . . . . . . . . . . 11 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) = 0)
9392mpteq2dva 4879 . . . . . . . . . 10 (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ 0))
9493breq1d 4797 . . . . . . . . 9 (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ↔ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺))
9594rexbidv 3200 . . . . . . . 8 (𝑔 = (ℝ × {0}) → (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺))
9689, 95bitr3d 270 . . . . . . 7 (𝑔 = (ℝ × {0}) → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ∧ 0 = (∫1𝑔)) ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟𝐺))
9796rspcev 3461 . . . . . 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)))
988, 86, 97sylancr 569 . . . . 5 (𝜑 → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ∧ 0 = (∫1𝑔)))
99 eqeq1 2775 . . . . . . . 8 (𝑥 = 0 → (𝑥 = (∫1𝑔) ↔ 0 = (∫1𝑔)))
10099anbi2d 608 . . . . . . 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))))
101100rexbidv 3200 . . . . . 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))))
10221, 101elab 3502 . . . . 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)))
10398, 102sylibr 224 . . . 4 (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))})
104 ne0i 4070 . . . 4 (0 ∈ {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))} ≠ ∅)
105103, 104syl 17 . . 3 (𝜑 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ≠ ∅)
106 fss 6197 . . . . . . 7 ((𝐺:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐺:ℝ⟶(0[,]+∞))
10751, 106mpan2 665 . . . . . 6 (𝐺:ℝ⟶(0[,)+∞) → 𝐺:ℝ⟶(0[,]+∞))
108 eqid 2771 . . . . . . 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))}
109108itg2addnclem 33794 . . . . . 6 (𝐺:ℝ⟶(0[,]+∞) → (∫2𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
11075, 107, 1093syl 18 . . . . 5 (𝜑 → (∫2𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))
111 itg2addnc.g3 . . . . 5 (𝜑 → (∫2𝐺) ∈ ℝ)
112110, 111eqeltrrd 2851 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) ∈ ℝ)
11373, 59sstri 3762 . . . . . 6 {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} ⊆ ℝ*
114 supxrub 12360 . . . . . 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*, < ))
115113, 114mpan 664 . . . . 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*, < ))
116115rgen 3071 . . . 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*, < )
117 breq2 4791 . . . . . 6 (𝑎 = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) → (𝑏𝑎𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))
118117ralbidv 3135 . . . . 5 (𝑎 = sup({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))}𝑏𝑎 ↔ ∀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*, < )))
119118rspcev 3461 . . . 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)
120112, 116, 119sylancl 568 . . 3 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}𝑏𝑎)
121 eqid 2771 . . 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)}
1227, 50, 67, 74, 105, 120, 121supadd 11194 . 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, < ))
123 supxrre 12363 . . . . 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, < ))
1247, 50, 67, 123syl3anc 1476 . . . 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, < ))
12556, 124eqtrd 2805 . . 3 (𝜑 → (∫2𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))}, ℝ, < ))
126 supxrre 12363 . . . . 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, < ))
12774, 105, 120, 126syl3anc 1476 . . . 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, < ))
128110, 127eqtrd 2805 . . 3 (𝜑 → (∫2𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))}, ℝ, < ))
129125, 128oveq12d 6812 . 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, < )))
130 ge0addcl 12492 . . . . . . 7 ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞))
13151, 130sseldi 3751 . . . . . 6 ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,]+∞))
132131adantl 467 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,]+∞))
133 inidm 3972 . . . . 5 (ℝ ∩ ℝ) = ℝ
134132, 13, 75, 20, 20, 133off 7060 . . . 4 (𝜑 → (𝐹𝑓 + 𝐺):ℝ⟶(0[,]+∞))
135 eqid 2771 . . . . 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))}
136135itg2addnclem 33794 . . . 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*, < ))
137134, 136syl 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*, < ))
138 itg2addnc.f1 . . . . . . . 8 (𝜑𝐹 ∈ MblFn)
139138, 13, 57, 75, 111itg2addnclem3 33796 . . . . . . 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))))
140 simpl 468 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓 ∈ dom ∫1)
141 simpr 471 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑔 ∈ dom ∫1)
142140, 141i1fadd 23683 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑓𝑓 + 𝑔) ∈ dom ∫1)
143142ad3antlr 704 . . . . . . . . . . . 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)
144 reeanv 3255 . . . . . . . . . . . . . . . . 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))
145144biimpri 218 . . . . . . . . . . . . . . . 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))
146145ad2ant2r 735 . . . . . . . . . . . . . . 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))
147 ifcl 4270 . . . . . . . . . . . . . . . . . . 19 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ+)
148147ad2antlr 700 . . . . . . . . . . . . . . . . . 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+)
149 breq1 4790 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (0 ≤ (𝐹𝑧) ↔ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
150149anbi1d 609 . . . . . . . . . . . . . . . . . . . . . . 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))))
151150imbi1d 330 . . . . . . . . . . . . . . . . . . . . . 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)))))
152 breq1 4790 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓𝑧) + 𝑐) = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ↔ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
153152anbi1d 609 . . . . . . . . . . . . . . . . . . . . . . 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))))
154153imbi1d 330 . . . . . . . . . . . . . . . . . . . . . 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)))))
155 breq1 4790 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (0 ≤ (𝐺𝑧) ↔ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
156155anbi2d 608 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((0 ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) ↔ (0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
157156imbi1d 330 . . . . . . . . . . . . . . . . . . . . . . 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)))))
158 breq1 4790 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) ↔ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
159158anbi2d 608 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) ↔ (0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
160159imbi1d 330 . . . . . . . . . . . . . . . . . . . . . . 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)))))
161 oveq12 6803 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (0 + 0))
162 00id 10414 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 + 0) = 0
163161, 162syl6eq 2821 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = 0)
164163iftrued 4234 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) = 0)
165164adantll 687 . . . . . . . . . . . . . . . . . . . . . . . . 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)
166 simpll 744 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝜑)
16715simplbi 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹𝑧) ∈ (0[,)+∞) → (𝐹𝑧) ∈ ℝ)
16814, 167syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ∈ ℝ)
16977simplbi 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐺𝑧) ∈ (0[,)+∞) → (𝐺𝑧) ∈ ℝ)
17076, 169syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ℝ)
171168, 170, 17, 79addge0d 10806 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑧 ∈ ℝ) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
172166, 171sylan 563 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
173172ad2antrr 699 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
174165, 173eqbrtrd 4809 . . . . . . . . . . . . . . . . . . . . . . . 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)))
175174a1d 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))))
176172ad2antrr 699 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
177 oveq1 6801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓𝑧) = 0 → ((𝑓𝑧) + (𝑔𝑧)) = (0 + (𝑔𝑧)))
178 simplrr 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑔 ∈ dom ∫1)
179 i1ff 23664 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑔 ∈ dom ∫1𝑔:ℝ⟶ℝ)
180179ffvelrnda 6503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑔 ∈ dom ∫1𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℝ)
181178, 180sylan 563 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℝ)
182181recnd 10271 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℂ)
183182addid2d 10440 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (0 + (𝑔𝑧)) = (𝑔𝑧))
184177, 183sylan9eqr 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (𝑔𝑧))
185184oveq1d 6809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
186185adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
187147rpred 12076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ)
188187ad2antlr 700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ)
189181, 188readdcld 10272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
190189adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
191166, 170sylan 563 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ℝ)
192191adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (𝐺𝑧) ∈ ℝ)
193166, 168sylan 563 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ∈ ℝ)
194193, 191readdcld 10272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
195194adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
196 simplrr 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ+)
197196rpred 12076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ)
198 rpre 12043 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ ℝ+𝑐 ∈ ℝ)
199 rpre 12043 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑑 ∈ ℝ+𝑑 ∈ ℝ)
200 min2 12227 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
201198, 199, 200syl2an 577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
202201ad2antlr 700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
203188, 197, 181, 202leadd2dd 10845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑔𝑧) + 𝑑))
204181, 197readdcld 10272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + 𝑑) ∈ ℝ)
205 letr 10334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
206189, 204, 191, 205syl3anc 1476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
207203, 206mpand 669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)))
208207imp 393 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧))
209170, 168addge02d 10819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ ℝ) → (0 ≤ (𝐹𝑧) ↔ (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧))))
21017, 209mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
211166, 210sylan 563 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
212211adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
213190, 192, 195, 208, 212letrd 10397 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
214213adantlr 688 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
215186, 214eqbrtrd 4809 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
216 breq1 4790 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
217 breq1 4790 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
218216, 217ifboth 4264 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
219176, 215, 218syl2anc 567 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
220219ex 397 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
221220adantld 474 . . . . . . . . . . . . . . . . . . . . . . . 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))))
222221adantr 466 . . . . . . . . . . . . . . . . . . . . . . 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))))
223157, 160, 175, 222ifbothda 4263 . . . . . . . . . . . . . . . . . . . . . 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))))
224155anbi2d 608 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) ↔ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
225224imbi1d 330 . . . . . . . . . . . . . . . . . . . . . . 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)))))
226158anbi2d 608 . . . . . . . . . . . . . . . . . . . . . . . 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))))
227226imbi1d 330 . . . . . . . . . . . . . . . . . . . . . . 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)))))
228172ad2antrr 699 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
229 oveq2 6802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑔𝑧) = 0 → ((𝑓𝑧) + (𝑔𝑧)) = ((𝑓𝑧) + 0))
230 simplrl 756 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑓 ∈ dom ∫1)
231 i1ff 23664 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑓 ∈ dom ∫1𝑓:ℝ⟶ℝ)
232231ffvelrnda 6503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑓 ∈ dom ∫1𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
233230, 232sylan 563 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
234233recnd 10271 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℂ)
235234addid1d 10439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + 0) = (𝑓𝑧))
236229, 235sylan9eqr 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (𝑓𝑧))
237236oveq1d 6809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
238237adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
239233, 188readdcld 10272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
240239adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
241193adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝐹𝑧) ∈ ℝ)
242194adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
243 simplrl 756 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ+)
244243rpred 12076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ)
245 min1 12226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
246198, 199, 245syl2an 577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
247246ad2antlr 700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
248188, 244, 233, 247leadd2dd 10845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑓𝑧) + 𝑐))
249233, 244readdcld 10272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + 𝑐) ∈ ℝ)
250 letr 10334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
251239, 249, 193, 250syl3anc 1476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
252248, 251mpand 669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧)))
253252imp 393 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧))
254168, 170addge01d 10818 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ ℝ) → (0 ≤ (𝐺𝑧) ↔ (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧))))
25579, 254mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
256166, 255sylan 563 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
257256adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
258240, 241, 242, 253, 257letrd 10397 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
259258adantlr 688 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
260238, 259eqbrtrd 4809 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
261228, 260, 218syl2anc 567 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
262261ex 397 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
263262adantlr 688 . . . . . . . . . . . . . . . . . . . . . . . 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))))
264263adantrd 475 . . . . . . . . . . . . . . . . . . . . . . 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))))
265172adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
266188recnd 10271 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℂ)
267234, 182, 266addassd 10265 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
268267adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
269233, 243ltaddrpd 12109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) < ((𝑓𝑧) + 𝑐))
270233, 249, 269ltled 10388 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐))
271 letr 10334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓𝑧) ∈ ℝ ∧ ((𝑓𝑧) + 𝑐) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧)))
272233, 249, 193, 271syl3anc 1476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧)))
273270, 272mpand 669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → (𝑓𝑧) ≤ (𝐹𝑧)))
274 le2add 10713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
275233, 189, 193, 191, 274syl22anc 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
276273, 207, 275syl2and 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
277276imp 393 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
278268, 277eqbrtrd 4809 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
279265, 278, 218syl2anc 567 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
280279ex 397 . . . . . . . . . . . . . . . . . . . . . . . 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))))
281280ad2antrr 699 . . . . . . . . . . . . . . . . . . . . . . 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))))
282225, 227, 264, 281ifbothda 4263 . . . . . . . . . . . . . . . . . . . . . 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))))
283151, 154, 223, 282ifbothda 4263 . . . . . . . . . . . . . . . . . . . . 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))))
284283ralimdva 3111 . . . . . . . . . . . . . . . . . . . 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))))
285 ovex 6824 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓𝑧) + 𝑐) ∈ V
28621, 285ifex 4296 . . . . . . . . . . . . . . . . . . . . . . . . 25 if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ∈ V
287286a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ∈ V)
288 eqidd 2772 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))))
28920, 287, 14, 288, 24ofrfval2 7063 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹 ↔ ∀𝑧 ∈ ℝ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
290 ovex 6824 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔𝑧) + 𝑑) ∈ V
29121, 290ifex 4296 . . . . . . . . . . . . . . . . . . . . . . . . 25 if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ∈ V
292291a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧 ∈ ℝ) → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ∈ V)
293 eqidd 2772 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))))
29420, 292, 76, 293, 81ofrfval2 7063 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺 ↔ ∀𝑧 ∈ ℝ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
295289, 294anbi12d 610 . . . . . . . . . . . . . . . . . . . . . 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))))
296 r19.26 3212 . . . . . . . . . . . . . . . . . . . . . 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)))
297295, 296syl6bbr 278 . . . . . . . . . . . . . . . . . . . . 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))))
298297ad2antrr 699 . . . . . . . . . . . . . . . . . . . 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))))
29919a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → ℝ ∈ V)
300 ovex 6824 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ V
30121, 300ifex 4296 . . . . . . . . . . . . . . . . . . . . . 22 if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ∈ V
302301a1i 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)
303 ovexd 6826 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) + (𝐺𝑧)) ∈ V)
304 ffn 6186 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓:ℝ⟶ℝ → 𝑓 Fn ℝ)
305231, 304syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 ∈ dom ∫1𝑓 Fn ℝ)
306305adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓 Fn ℝ)
307306ad2antlr 700 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑓 Fn ℝ)
308 ffn 6186 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑔:ℝ⟶ℝ → 𝑔 Fn ℝ)
309179, 308syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔 ∈ dom ∫1𝑔 Fn ℝ)
310309adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑔 Fn ℝ)
311310ad2antlr 700 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑔 Fn ℝ)
312 eqidd 2772 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) = (𝑓𝑧))
313 eqidd 2772 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = (𝑔𝑧))
314307, 311, 299, 299, 133, 312, 313ofval 7054 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑓 + 𝑔)‘𝑧) = ((𝑓𝑧) + (𝑔𝑧)))
315314eqeq1d 2773 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑓 + 𝑔)‘𝑧) = 0 ↔ ((𝑓𝑧) + (𝑔𝑧)) = 0))
316314oveq1d 6809 . . . . . . . . . . . . . . . . . . . . . . 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)))
317315, 316ifbieq2d 4251 . . . . . . . . . . . . . . . . . . . . . 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))))
318317mpteq2dva 4879 . . . . . . . . . . . . . . . . . . . . 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)))))
319 ffn 6186 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:ℝ⟶(0[,)+∞) → 𝐹 Fn ℝ)
32013, 319syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐹 Fn ℝ)
321 ffn 6186 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐺:ℝ⟶(0[,)+∞) → 𝐺 Fn ℝ)
32275, 321syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐺 Fn ℝ)
323 eqidd 2772 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
324 eqidd 2772 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
325320, 322, 20, 20, 133, 323, 324offval 7052 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐹𝑓 + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹𝑧) + (𝐺𝑧))))
326325ad2antrr 699 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (𝐹𝑓 + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹𝑧) + (𝐺𝑧))))
327299, 302, 303, 318, 326ofrfval2 7063 . . . . . . . . . . . . . . . . . . . 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))))
328284, 298, 3273imtr4d 283 . . . . . . . . . . . . . . . . . . 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)))
329328imp 393 . . . . . . . . . . . . . . . . . 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))
330 oveq2 6802 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = if(𝑐𝑑, 𝑐, 𝑑) → (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦) = (((𝑓𝑓 + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
331330ifeq2d 4245 . . . . . . . . . . . . . . . . . . . . 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))))
332331mpteq2dv 4880 . . . . . . . . . . . . . . . . . . . 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)))))
333332breq1d 4797 . . . . . . . . . . . . . . . . . . 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)))
334333rspcev 3461 . . . . . . . . . . . . . . . . . 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))
335148, 329, 334syl2anc 567 . . . . . . . . . . . . . . . . 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))
336335ex 397 . . . . . . . . . . . . . . . 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)))
337336rexlimdvva 3186 . . . . . . . . . . . . . . 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)))
338146, 337syl5 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)))
339338a1dd 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))))
340339imp31 404 . . . . . . . . . . . 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))
341 oveq12 6803 . . . . . . . . . . . . . . 15 ((𝑡 = (∫1𝑓) ∧ 𝑢 = (∫1𝑔)) → (𝑡 + 𝑢) = ((∫1𝑓) + (∫1𝑔)))
342341ad2ant2l 734 . . . . . . . . . . . . . 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)))
343140, 141itg1add 23689 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∫1‘(𝑓𝑓 + 𝑔)) = ((∫1𝑓) + (∫1𝑔)))
344343eqcomd 2777 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((∫1𝑓) + (∫1𝑔)) = (∫1‘(𝑓𝑓 + 𝑔)))
345344adantl 467 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((∫1𝑓) + (∫1𝑔)) = (∫1‘(𝑓𝑓 + 𝑔)))
346342, 345sylan9eqr 2827 . . . . . . . . . . . . 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)))
347 eqtr 2790 . . . . . . . . . . . . . 14 ((𝑠 = (𝑡 + 𝑢) ∧ (𝑡 + 𝑢) = (∫1‘(𝑓𝑓 + 𝑔))) → 𝑠 = (∫1‘(𝑓𝑓 + 𝑔)))
348347ancoms 455 . . . . . . . . . . . . 13 (((𝑡 + 𝑢) = (∫1‘(𝑓𝑓 + 𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓𝑓 + 𝑔)))
349346, 348sylan 563 . . . . . . . . . . . 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)))
350 fveq1 6332 . . . . . . . . . . . . . . . . . . 19 ( = (𝑓𝑓 + 𝑔) → (𝑧) = ((𝑓𝑓 + 𝑔)‘𝑧))
351350eqeq1d 2773 . . . . . . . . . . . . . . . . . 18 ( = (𝑓𝑓 + 𝑔) → ((𝑧) = 0 ↔ ((𝑓𝑓 + 𝑔)‘𝑧) = 0))
352350oveq1d 6809 . . . . . . . . . . . . . . . . . 18 ( = (𝑓𝑓 + 𝑔) → ((𝑧) + 𝑦) = (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))
353351, 352ifbieq2d 4251 . . . . . . . . . . . . . . . . 17 ( = (𝑓𝑓 + 𝑔) → if((𝑧) = 0, 0, ((𝑧) + 𝑦)) = if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦)))
354353mpteq2dv 4880 . . . . . . . . . . . . . . . 16 ( = (𝑓𝑓 + 𝑔) → (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(((𝑓𝑓 + 𝑔)‘𝑧) = 0, 0, (((𝑓𝑓 + 𝑔)‘𝑧) + 𝑦))))
355354breq1d 4797 . . . . . . . . . . . . . . 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)))
356355rexbidv 3200 . . . . . . . . . . . . . 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)))
357 fveq2 6333 . . . . . . . . . . . . . . 15 ( = (𝑓𝑓 + 𝑔) → (∫1) = (∫1‘(𝑓𝑓 + 𝑔)))
358357eqeq2d 2781 . . . . . . . . . . . . . 14 ( = (𝑓𝑓 + 𝑔) → (𝑠 = (∫1) ↔ 𝑠 = (∫1‘(𝑓𝑓 + 𝑔))))
359356, 358anbi12d 610 . . . . . . . . . . . . 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)))))
360359rspcev 3461 . . . . . . . . . . . 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)))
361143, 340, 349, 360syl12anc 1474 . . . . . . . . . . 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)))
362361exp31 406 . . . . . . . . . 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)))))
363362rexlimdvva 3186 . . . . . . . . 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)))))
364363impd 396 . . . . . . . 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))))
365364exlimdvv 2014 . . . . . . 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))))
366139, 365impbid 202 . . . . . 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))))
367 eqeq1 2775 . . . . . . . . . 10 (𝑥 = 𝑡 → (𝑥 = (∫1𝑓) ↔ 𝑡 = (∫1𝑓)))
368367anbi2d 608 . . . . . . . . 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))))
369368rexbidv 3200 . . . . . . . 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))))
370369rexab 3522 . . . . . . 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)))
371 eqeq1 2775 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → (𝑥 = (∫1𝑔) ↔ 𝑢 = (∫1𝑔)))
372371anbi2d 608 . . . . . . . . . . . 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))))
373372rexbidv 3200 . . . . . . . . . . 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))))
374373rexab 3522 . . . . . . . . . 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)))
375374anbi2i 603 . . . . . . . . 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))))
376 19.42v 2033 . . . . . . . . 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))))
377 reeanv 3255 . . . . . . . . . . . 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))))
378377anbi1i 604 . . . . . . . . . . 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)))
379 anass 459 . . . . . . . . . . 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))))
380378, 379bitr2i 265 . . . . . . . . . 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)))
381380exbii 1924 . . . . . . . . 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)))
382375, 376, 3813bitr2i 288 . . . . . . . 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)))
383382exbii 1924 . . . . . . 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)))
384370, 383bitri 264 . . . . . 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)))
385366, 384syl6bbr 278 . . . . 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)))
386385abbidv 2890 . . . 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)})
387386supeq1d 8509 . . 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*, < ))
388 simpr 471 . . . . . . . . 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))
3896sseli 3749 . . . . . . . . . . 11 (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘𝑟𝐹𝑥 = (∫1𝑓))} → 𝑡 ∈ ℝ)
390389ad2antrr 699 . . . . . . . . . 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)
39173sseli 3749 . . . . . . . . . . 11 (𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘𝑟𝐺𝑥 = (∫1𝑔))} → 𝑢 ∈ ℝ)
392391ad2antlr 700 . . . . . . . . . 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)
393390, 392readdcld 10272 . . . . . . . . 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)
394388, 393eqeltrd 2850 . . . . . . . 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)
395394ex 397 . . . . . . 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))
396395rexlimivv 3184 . . . . . 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)
397396abssi 3827 . . . . 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
398397a1i 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)
399162eqcomi 2780 . . . . . . . 8 0 = (0 + 0)
400 rspceov 6838 . . . . . . . 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))
401399, 400mp3an3 1561 . . . . . . 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))
40248, 103, 401syl2anc 567 . . . . . 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))
403 eqeq1 2775 . . . . . . . 8 (𝑠 = 0 → (𝑠 = (𝑡 + 𝑢) ↔ 0 = (𝑡 + 𝑢)))
4044032rexbidv 3205 . . . . . . 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)))
40521, 404spcev 3452 . . . . . 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))
406402, 405syl 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))
407 abn0 4102 . . . . 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))
408406, 407sylibr 224 . . . 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)} ≠ ∅)
40958, 112readdcld 10272 . . . . 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)
410 simpr 471 . . . . . . . . 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))
411389ad2antrl 701 . . . . . . . . . . 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)
412391ad2antll 702 . . . . . . . . . . 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)
41358adantr 466 . . . . . . . . . . 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)
414112adantr 466 . . . . . . . . . . 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)
415 supxrub 12360 . . . . . . . . . . . . 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*, < ))
41660, 415mpan 664 . . . . . . . . . . . 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*, < ))
417416ad2antrl 701 . . . . . . . . . . 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*, < ))
418 supxrub 12360 . . . . . . . . . . . . 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*, < ))
419113, 418mpan 664 . . . . . . . . . . . 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*, < ))
420419ad2antll 702 . . . . . . . . . . 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*, < ))
421411, 412, 413, 414, 417, 420le2addd 10849 . . . . . . . . . 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*, < )))
422421adantr 466 . . . . . . . . 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*, < )))
423410, 422eqbrtrd 4809 . . . . . . . 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*, < )))
424423ex 397 . . . . . . 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*, < ))))
425424rexlimdvva 3186 . . . . . 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*, < ))))
426425alrimiv 2007 . . . . 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*, < ))))
427 breq2 4791 . . . . . . . 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*, < ))))
428427ralbidv 3135 . . . . . . 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*, < ))))
429 eqeq1 2775 . . . . . . . . 9 (𝑠 = 𝑏 → (𝑠 = (𝑡 + 𝑢) ↔ 𝑏 = (𝑡 + 𝑢)))
4304292rexbidv 3205 . . . . . . . 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)))
431430ralab 3520 . . . . . . 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*, < ))))
432428, 431syl6bb 276 . . . . . 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*, < )))))
433432rspcev 3461 . . . . 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)
434409, 426, 433syl2anc 567 . . . 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)
435 supxrre 12363 . . . 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, < ))
436398, 408, 434, 435syl3anc 1476 . . 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, < ))
437137, 387, 4363eqtrd 2809 . 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, < ))
438122, 129, 4373eqtr4rd 2816 1 (𝜑 → (∫2‘(𝐹𝑓 + 𝐺)) = ((∫2𝐹) + (∫2𝐺)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  wral 3061  wrex 3062  Vcvv 3351  wss 3724  c0 4064  ifcif 4226  {csn 4317   class class class wbr 4787  cmpt 4864   × cxp 5248  dom cdm 5250   Fn wfn 6027  wf 6028  cfv 6032  (class class class)co 6794  𝑓 cof 7043  𝑟 cofr 7044  supcsup 8503  cr 10138  0cc0 10139   + caddc 10142  +∞cpnf 10274  *cxr 10276   < clt 10277  cle 10278  cn 11223  3c3 11274  +crp 12036  [,)cico 12383  [,]cicc 12384  MblFncmbf 23603  1citg1 23604  2citg2 23605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-inf2 8703  ax-cnex 10195  ax-resscn 10196  ax-1cn 10197  ax-icn 10198  ax-addcl 10199  ax-addrcl 10200  ax-mulcl 10201  ax-mulrcl 10202  ax-mulcom 10203  ax-addass 10204  ax-mulass 10205  ax-distr 10206  ax-i2m1 10207  ax-1ne0 10208  ax-1rid 10209  ax-rnegex 10210  ax-rrecex 10211  ax-cnre 10212  ax-pre-lttri 10213  ax-pre-lttrn 10214  ax-pre-ltadd 10215  ax-pre-mulgt0 10216  ax-pre-sup 10217  ax-addf 10218
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-disj 4756  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-isom 6041  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-of 7045  df-ofr 7046  df-om 7214  df-1st 7316  df-2nd 7317  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-2o 7715  df-oadd 7718  df-er 7897  df-map 8012  df-pm 8013  df-en 8111  df-dom 8112  df-sdom 8113  df-fin 8114  df-fi 8474  df-sup 8505  df-inf 8506  df-oi 8572  df-card 8966  df-cda 9193  df-pnf 10279  df-mnf 10280  df-xr 10281  df-ltxr 10282  df-le 10283  df-sub 10471  df-neg 10472  df-div 10888  df-nn 11224  df-2 11282  df-3 11283  df-n0 11496  df-z 11581  df-uz 11890  df-q 11993  df-rp 12037  df-xneg 12152  df-xadd 12153  df-xmul 12154  df-ioo 12385  df-ico 12387  df-icc 12388  df-fz 12535  df-fzo 12675  df-fl 12802  df-seq 13010  df-exp 13069  df-hash 13323  df-cj 14048  df-re 14049  df-im 14050  df-sqrt 14184  df-abs 14185  df-clim 14428  df-sum 14626  df-rest 16292  df-topgen 16313  df-psmet 19954  df-xmet 19955  df-met 19956  df-bl 19957  df-mopn 19958  df-top 20920  df-topon 20937  df-bases 20972  df-cmp 21412  df-ovol 23453  df-vol 23454  df-mbf 23608  df-itg1 23609  df-itg2 23610
This theorem is referenced by:  ibladdnclem  33799  itgaddnclem1  33801  iblabsnclem  33806  iblabsnc  33807  iblmulc2nc  33808  ftc1anclem4  33821  ftc1anclem5  33822  ftc1anclem6  33823  ftc1anclem8  33825
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