Mathbox for Brendan Leahy < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  itg2addnc Structured version   Visualization version   GIF version

 Description: Alternate proof of itg2add 24275 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 24224, 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 9846, 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.f3 (𝜑 → (∫2𝐹) ∈ ℝ)
itg2addnc.g3 (𝜑 → (∫2𝐺) ∈ ℝ)
Assertion
Ref Expression
itg2addnc (𝜑 → (∫2‘(𝐹f + 𝐺)) = ((∫2𝐹) + (∫2𝐺)))

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 769 . . . . . . 7 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))) → 𝑥 = (∫1𝑓))
2 itg1cl 24201 . . . . . . . 8 (𝑓 ∈ dom ∫1 → (∫1𝑓) ∈ ℝ)
32adantr 481 . . . . . . 7 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))) → (∫1𝑓) ∈ ℝ)
41, 3eqeltrd 2918 . . . . . 6 ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))) → 𝑥 ∈ ℝ)
54rexlimiva 3286 . . . . 5 (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓)) → 𝑥 ∈ ℝ)
65abssi 4050 . . . 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 24203 . . . . . 6 (ℝ × {0}) ∈ dom ∫1
9 3nn 11705 . . . . . . . 8 3 ∈ ℕ
10 nnrp 12390 . . . . . . . 8 (3 ∈ ℕ → 3 ∈ ℝ+)
11 ne0i 4304 . . . . . . . 8 (3 ∈ ℝ+ → ℝ+ ≠ ∅)
129, 10, 11mp2b 10 . . . . . . 7 + ≠ ∅
13 itg2addnc.f2 . . . . . . . . . . . . 13 (𝜑𝐹:ℝ⟶(0[,)+∞))
1413ffvelrnda 6847 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ∈ (0[,)+∞))
15 elrege0 12832 . . . . . . . . . . . 12 ((𝐹𝑧) ∈ (0[,)+∞) ↔ ((𝐹𝑧) ∈ ℝ ∧ 0 ≤ (𝐹𝑧)))
1614, 15sylib 219 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℝ) → ((𝐹𝑧) ∈ ℝ ∧ 0 ≤ (𝐹𝑧)))
1716simprd 496 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ≤ (𝐹𝑧))
1817ralrimiva 3187 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧))
19 reex 10617 . . . . . . . . . . 11 ℝ ∈ V
2019a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ∈ V)
21 c0ex 10624 . . . . . . . . . . 11 0 ∈ V
2221a1i 11 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ∈ V)
23 eqidd 2827 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
2413feqmptd 6730 . . . . . . . . . 10 (𝜑𝐹 = (𝑧 ∈ ℝ ↦ (𝐹𝑧)))
2520, 22, 14, 23, 24ofrfval2 7417 . . . . . . . . 9 (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘r𝐹 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐹𝑧)))
2618, 25mpbird 258 . . . . . . . 8 (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘r𝐹)
2726ralrimivw 3188 . . . . . . 7 (𝜑 → ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹)
28 r19.2z 4443 . . . . . . 7 ((ℝ+ ≠ ∅ ∧ ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹) → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹)
2912, 27, 28sylancr 587 . . . . . 6 (𝜑 → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹)
30 fveq2 6667 . . . . . . . . . 10 (𝑓 = (ℝ × {0}) → (∫1𝑓) = (∫1‘(ℝ × {0})))
31 itg10 24204 . . . . . . . . . 10 (∫1‘(ℝ × {0})) = 0
3230, 31syl6req 2878 . . . . . . . . 9 (𝑓 = (ℝ × {0}) → 0 = (∫1𝑓))
3332biantrud 532 . . . . . . . 8 (𝑓 = (ℝ × {0}) → (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ 0 = (∫1𝑓))))
34 fveq1 6666 . . . . . . . . . . . . 13 (𝑓 = (ℝ × {0}) → (𝑓𝑧) = ((ℝ × {0})‘𝑧))
3521fvconst2 6962 . . . . . . . . . . . . 13 (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
3634, 35sylan9eq 2881 . . . . . . . . . . . 12 ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) = 0)
3736iftrued 4478 . . . . . . . . . . 11 ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) = 0)
3837mpteq2dva 5158 . . . . . . . . . 10 (𝑓 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ 0))
3938breq1d 5073 . . . . . . . . 9 (𝑓 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹))
4039rexbidv 3302 . . . . . . . 8 (𝑓 = (ℝ × {0}) → (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹))
4133, 40bitr3d 282 . . . . . . 7 (𝑓 = (ℝ × {0}) → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ 0 = (∫1𝑓)) ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹))
4241rspcev 3627 . . . . . 6 (((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐹) → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ 0 = (∫1𝑓)))
438, 29, 42sylancr 587 . . . . 5 (𝜑 → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ 0 = (∫1𝑓)))
44 eqeq1 2830 . . . . . . . 8 (𝑥 = 0 → (𝑥 = (∫1𝑓) ↔ 0 = (∫1𝑓)))
4544anbi2d 628 . . . . . . 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 3302 . . . . . 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 3671 . . . . 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 235 . . . 4 (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))})
4948ne0d 4305 . . 3 (𝜑 → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ≠ ∅)
50 icossicc 12814 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
51 fss 6524 . . . . . . 7 ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
5250, 51mpan2 687 . . . . . 6 (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶(0[,]+∞))
53 eqid 2826 . . . . . . 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 34810 . . . . . 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 2919 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ)
58 ressxr 10674 . . . . . . 7 ℝ ⊆ ℝ*
596, 58sstri 3980 . . . . . 6 {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ⊆ ℝ*
60 supxrub 12707 . . . . . 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 686 . . . . 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 3153 . . . 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 5123 . . . 4 ((sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ ∧ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < )) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}𝑏𝑎)
6457, 62, 63sylancl 586 . . 3 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}𝑏𝑎)
65 simprr 769 . . . . . . 7 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))) → 𝑥 = (∫1𝑔))
66 itg1cl 24201 . . . . . . . 8 (𝑔 ∈ dom ∫1 → (∫1𝑔) ∈ ℝ)
6766adantr 481 . . . . . . 7 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))) → (∫1𝑔) ∈ ℝ)
6865, 67eqeltrd 2918 . . . . . 6 ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))) → 𝑥 ∈ ℝ)
6968rexlimiva 3286 . . . . 5 (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔)) → 𝑥 ∈ ℝ)
7069abssi 4050 . . . 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 6847 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ∈ (0[,)+∞))
74 elrege0 12832 . . . . . . . . . . . 12 ((𝐺𝑧) ∈ (0[,)+∞) ↔ ((𝐺𝑧) ∈ ℝ ∧ 0 ≤ (𝐺𝑧)))
7573, 74sylib 219 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℝ) → ((𝐺𝑧) ∈ ℝ ∧ 0 ≤ (𝐺𝑧)))
7675simprd 496 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℝ) → 0 ≤ (𝐺𝑧))
7776ralrimiva 3187 . . . . . . . . 9 (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐺𝑧))
7872feqmptd 6730 . . . . . . . . . 10 (𝜑𝐺 = (𝑧 ∈ ℝ ↦ (𝐺𝑧)))
7920, 22, 73, 23, 78ofrfval2 7417 . . . . . . . . 9 (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘r𝐺 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐺𝑧)))
8077, 79mpbird 258 . . . . . . . 8 (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘r𝐺)
8180ralrimivw 3188 . . . . . . 7 (𝜑 → ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺)
82 r19.2z 4443 . . . . . . 7 ((ℝ+ ≠ ∅ ∧ ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺) → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺)
8312, 81, 82sylancr 587 . . . . . 6 (𝜑 → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺)
84 fveq2 6667 . . . . . . . . . 10 (𝑔 = (ℝ × {0}) → (∫1𝑔) = (∫1‘(ℝ × {0})))
8584, 31syl6req 2878 . . . . . . . . 9 (𝑔 = (ℝ × {0}) → 0 = (∫1𝑔))
8685biantrud 532 . . . . . . . 8 (𝑔 = (ℝ × {0}) → (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ↔ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ∧ 0 = (∫1𝑔))))
87 fveq1 6666 . . . . . . . . . . . . 13 (𝑔 = (ℝ × {0}) → (𝑔𝑧) = ((ℝ × {0})‘𝑧))
8887, 35sylan9eq 2881 . . . . . . . . . . . 12 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = 0)
8988iftrued 4478 . . . . . . . . . . 11 ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) = 0)
9089mpteq2dva 5158 . . . . . . . . . 10 (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ 0))
9190breq1d 5073 . . . . . . . . 9 (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺))
9291rexbidv 3302 . . . . . . . 8 (𝑔 = (ℝ × {0}) → (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺))
9386, 92bitr3d 282 . . . . . . 7 (𝑔 = (ℝ × {0}) → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ∧ 0 = (∫1𝑔)) ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺))
9493rspcev 3627 . . . . . 6 (((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r𝐺) → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ∧ 0 = (∫1𝑔)))
958, 83, 94sylancr 587 . . . . 5 (𝜑 → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ∧ 0 = (∫1𝑔)))
96 eqeq1 2830 . . . . . . . 8 (𝑥 = 0 → (𝑥 = (∫1𝑔) ↔ 0 = (∫1𝑔)))
9796anbi2d 628 . . . . . . 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 3302 . . . . . 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 3671 . . . . 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 235 . . . 4 (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})
101100ne0d 4305 . . 3 (𝜑 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} ≠ ∅)
102 fss 6524 . . . . . . 7 ((𝐺:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐺:ℝ⟶(0[,]+∞))
10350, 102mpan2 687 . . . . . 6 (𝐺:ℝ⟶(0[,)+∞) → 𝐺:ℝ⟶(0[,]+∞))
104 eqid 2826 . . . . . . 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 34810 . . . . . 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 2919 . . . 4 (𝜑 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) ∈ ℝ)
10970, 58sstri 3980 . . . . . 6 {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} ⊆ ℝ*
110 supxrub 12707 . . . . . 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 686 . . . . 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 3153 . . . 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 5123 . . . 4 ((sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) ∈ ℝ ∧ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏𝑎)
114108, 112, 113sylancl 586 . . 3 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏𝑎)
115 eqid 2826 . . 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 11598 . 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 12710 . . . . 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 1365 . . . 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 2861 . . 3 (𝜑 → (∫2𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ, < ))
120 supxrre 12710 . . . . 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 1365 . . . 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 2861 . . 3 (𝜑 → (∫2𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ, < ))
123119, 122oveq12d 7166 . 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 12838 . . . . . . 7 ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞))
12550, 124sseldi 3969 . . . . . 6 ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,]+∞))
126125adantl 482 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,]+∞))
127 inidm 4199 . . . . 5 (ℝ ∩ ℝ) = ℝ
128126, 13, 72, 20, 20, 127off 7414 . . . 4 (𝜑 → (𝐹f + 𝐺):ℝ⟶(0[,]+∞))
129 eqid 2826 . . . . 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 34810 . . . 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 34812 . . . . . . 7 (𝜑 → (∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1)) → ∃𝑡𝑢(∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))))
134 simpl 483 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓 ∈ dom ∫1)
135 simpr 485 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑔 ∈ dom ∫1)
136134, 135i1fadd 24211 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑓f + 𝑔) ∈ dom ∫1)
137136ad3antlr 727 . . . . . . . . . . . 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 3373 . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . . 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 743 . . . . . . . . . . . . . . 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 4514 . . . . . . . . . . . . . . . . . . 19 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ+)
142141ad2antlr 723 . . . . . . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (0 ≤ (𝐹𝑧) ↔ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
144143anbi1d 629 . . . . . . . . . . . . . . . . . . . . . . 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 343 . . . . . . . . . . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓𝑧) + 𝑐) = if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ↔ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
147146anbi1d 629 . . . . . . . . . . . . . . . . . . . . . . 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 343 . . . . . . . . . . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (0 ≤ (𝐺𝑧) ↔ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
150149anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((0 ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) ↔ (0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
151150imbi1d 343 . . . . . . . . . . . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) ↔ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
153152anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑔𝑧) + 𝑑) = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) ↔ (0 ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
154153imbi1d 343 . . . . . . . . . . . . . . . . . . . . . . 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 7157 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (0 + 0))
156 00id 10804 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 + 0) = 0
157155, 156syl6eq 2877 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = 0)
158157iftrued 4478 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓𝑧) = 0 ∧ (𝑔𝑧) = 0) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) = 0)
159158adantll 710 . . . . . . . . . . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝜑)
16115simplbi 498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹𝑧) ∈ (0[,)+∞) → (𝐹𝑧) ∈ ℝ)
16214, 161syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ∈ ℝ)
16374simplbi 498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐺𝑧) ∈ (0[,)+∞) → (𝐺𝑧) ∈ ℝ)
16473, 163syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ℝ)
165162, 164, 17, 76addge0d 11205 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑧 ∈ ℝ) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
166160, 165sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
167166ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
168159, 167eqbrtrd 5085 . . . . . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
171 oveq1 7155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓𝑧) = 0 → ((𝑓𝑧) + (𝑔𝑧)) = (0 + (𝑔𝑧)))
172 simplrr 774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑔 ∈ dom ∫1)
173 i1ff 24192 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑔 ∈ dom ∫1𝑔:ℝ⟶ℝ)
174173ffvelrnda 6847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑔 ∈ dom ∫1𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℝ)
175172, 174sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℝ)
176175recnd 10658 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) ∈ ℂ)
177176addid2d 10830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (0 + (𝑔𝑧)) = (𝑔𝑧))
178171, 177sylan9eqr 2883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (𝑔𝑧))
179178oveq1d 7163 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
180179adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
181141rpred 12421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ)
182181ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ)
183175, 182readdcld 10659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
184183adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
185160, 164sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ℝ)
186185adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (𝐺𝑧) ∈ ℝ)
187160, 162sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ∈ ℝ)
188187, 185readdcld 10659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
189188adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
190 simplrr 774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ+)
191190rpred 12421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ)
192 rpre 12387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ ℝ+𝑐 ∈ ℝ)
193 rpre 12387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑑 ∈ ℝ+𝑑 ∈ ℝ)
194 min2 12573 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
195192, 193, 194syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
196195ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑑)
197182, 191, 175, 196leadd2dd 11244 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑔𝑧) + 𝑑))
198175, 191readdcld 10659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔𝑧) + 𝑑) ∈ ℝ)
199 letr 10723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧)))
202201imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐺𝑧))
203164, 162addge02d 11218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ ℝ) → (0 ≤ (𝐹𝑧) ↔ (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧))))
20417, 203mpbid 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
205160, 204sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
206205adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → (𝐺𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
207184, 186, 189, 202, 206letrd 10786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
208207adantlr 711 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5085 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4508 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ≤ ((𝐹𝑧) + (𝐺𝑧)) ∧ (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧))) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
213170, 209, 212syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
214213ex 413 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → (((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
215214adantld 491 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) → ((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
216215adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓𝑧) = 0) ∧ ¬ (𝑔𝑧) = 0) → ((0 ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
217151, 154, 169, 216ifbothda 4507 . . . . . . . . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) ↔ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧))))
219218imbi1d 343 . . . . . . . . . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . . . . . . . . . . . . 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 343 . . . . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
223 oveq2 7156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑔𝑧) = 0 → ((𝑓𝑧) + (𝑔𝑧)) = ((𝑓𝑧) + 0))
224 simplrl 773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑓 ∈ dom ∫1)
225 i1ff 24192 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑓 ∈ dom ∫1𝑓:ℝ⟶ℝ)
226225ffvelrnda 6847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑓 ∈ dom ∫1𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
227224, 226sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℝ)
228227recnd 10658 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ∈ ℂ)
229228addid1d 10829 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + 0) = (𝑓𝑧))
230223, 229sylan9eqr 2883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) → ((𝑓𝑧) + (𝑔𝑧)) = (𝑓𝑧))
231230oveq1d 7163 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
232231adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
233227, 182readdcld 10659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
234233adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ ℝ)
235187adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝐹𝑧) ∈ ℝ)
236188adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝐹𝑧) + (𝐺𝑧)) ∈ ℝ)
237 simplrl 773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ+)
238237rpred 12421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ)
239 min1 12572 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
240192, 193, 239syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ ℝ+𝑑 ∈ ℝ+) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
241240ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ≤ 𝑐)
242182, 238, 227, 241leadd2dd 11244 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝑓𝑧) + 𝑐))
243227, 238readdcld 10659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓𝑧) + 𝑐) ∈ ℝ)
244 letr 10723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧)))
247246imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ (𝐹𝑧))
248162, 164addge01d 11217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ ℝ) → (0 ≤ (𝐺𝑧) ↔ (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧))))
24976, 248mpbid 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
250160, 249sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
251250adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝐹𝑧) ≤ ((𝐹𝑧) + (𝐺𝑧)))
252234, 235, 236, 247, 251letrd 10786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → ((𝑓𝑧) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
253252adantlr 711 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5085 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
255222, 254, 212syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
256255ex 413 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔𝑧) = 0) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
257256adantlr 711 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
258257adantrd 492 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓𝑧) = 0) ∧ (𝑔𝑧) = 0) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ 0 ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
259166adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → 0 ≤ ((𝐹𝑧) + (𝐺𝑧)))
260182recnd 10658 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐𝑑, 𝑐, 𝑑) ∈ ℂ)
261228, 176, 260addassd 10652 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))))
262261adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) = ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))))
263227, 237ltaddrpd 12454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) < ((𝑓𝑧) + 𝑐))
264227, 243, 263ltled 10777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐))
265 letr 10723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓𝑧) ∈ ℝ ∧ ((𝑓𝑧) + 𝑐) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧)))
266227, 243, 187, 265syl3anc 1365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) ≤ ((𝑓𝑧) + 𝑐) ∧ ((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧)) → (𝑓𝑧) ≤ (𝐹𝑧)))
267264, 266mpand 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) → (𝑓𝑧) ≤ (𝐹𝑧)))
268 le2add 11111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 607 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
271270imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → ((𝑓𝑧) + ((𝑔𝑧) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
272262, 271eqbrtrd 5085 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ≤ ((𝐹𝑧) + (𝐺𝑧)))
273259, 272, 212syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧))) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧)))
274273ex 413 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓𝑧) + 𝑐) ≤ (𝐹𝑧) ∧ ((𝑔𝑧) + 𝑑) ≤ (𝐺𝑧)) → if(((𝑓𝑧) + (𝑔𝑧)) = 0, 0, (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑))) ≤ ((𝐹𝑧) + (𝐺𝑧))))
275274ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 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 4507 . . . . . . . . . . . . . . . . . . . . . 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 4507 . . . . . . . . . . . . . . . . . . . . 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 3182 . . . . . . . . . . . . . . . . . . . 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 7181 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓𝑧) + 𝑐) ∈ V
28021, 279ifex 4518 . . . . . . . . . . . . . . . . . . . . . . . . 25 if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ∈ V
281280a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧 ∈ ℝ) → if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ∈ V)
282 eqidd 2827 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))))
28320, 281, 14, 282, 24ofrfval2 7417 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ↔ ∀𝑧 ∈ ℝ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐)) ≤ (𝐹𝑧)))
284 ovex 7181 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔𝑧) + 𝑑) ∈ V
28521, 284ifex 4518 . . . . . . . . . . . . . . . . . . . . . . . . 25 if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ∈ V
286285a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧 ∈ ℝ) → if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ∈ V)
287 eqidd 2827 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))))
28820, 286, 73, 287, 78ofrfval2 7417 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺 ↔ ∀𝑧 ∈ ℝ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑)) ≤ (𝐺𝑧)))
289283, 288anbi12d 630 . . . . . . . . . . . . . . . . . . . . . 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 3175 . . . . . . . . . . . . . . . . . . . . . 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 290 . . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . . 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 7181 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓𝑧) + (𝑔𝑧)) + if(𝑐𝑑, 𝑐, 𝑑)) ∈ V
29521, 294ifex 4518 . . . . . . . . . . . . . . . . . . . . . 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 7183 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑧) + (𝐺𝑧)) ∈ V)
298225ffnd 6512 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 ∈ dom ∫1𝑓 Fn ℝ)
299298adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓 Fn ℝ)
300299ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑓 Fn ℝ)
301173ffnd 6512 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔 ∈ dom ∫1𝑔 Fn ℝ)
302301adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑔 Fn ℝ)
303302ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → 𝑔 Fn ℝ)
304 eqidd 2827 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓𝑧) = (𝑓𝑧))
305 eqidd 2827 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔𝑧) = (𝑔𝑧))
306300, 303, 293, 293, 127, 304, 305ofval 7408 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓f + 𝑔)‘𝑧) = ((𝑓𝑧) + (𝑔𝑧)))
307306eqeq1d 2828 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓f + 𝑔)‘𝑧) = 0 ↔ ((𝑓𝑧) + (𝑔𝑧)) = 0))
308306oveq1d 7163 . . . . . . . . . . . . . . . . . . . . . . 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 4495 . . . . . . . . . . . . . . . . . . . . . 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 5158 . . . . . . . . . . . . . . . . . . . . 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 6512 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐹 Fn ℝ)
31272ffnd 6512 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐺 Fn ℝ)
313 eqidd 2827 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
314 eqidd 2827 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
315311, 312, 20, 20, 127, 313, 314offval 7406 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐹f + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹𝑧) + (𝐺𝑧))))
316315ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (𝐹f + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹𝑧) + (𝐺𝑧))))
317293, 296, 297, 310, 316ofrfval2 7417 . . . . . . . . . . . . . . . . . . . 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 295 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺) → (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹f + 𝐺)))
319318imp 407 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺)) → (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹f + 𝐺))
320 oveq2 7156 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = if(𝑐𝑑, 𝑐, 𝑑) → (((𝑓f + 𝑔)‘𝑧) + 𝑦) = (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))
321320ifeq2d 4489 . . . . . . . . . . . . . . . . . . . . 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 5159 . . . . . . . . . . . . . . . . . . . 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 5073 . . . . . . . . . . . . . . . . . . 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 3627 . . . . . . . . . . . . . . . . . 18 ((if(𝑐𝑑, 𝑐, 𝑑) ∈ ℝ+ ∧ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + if(𝑐𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹f + 𝐺)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺))
325142, 319, 324syl2anc 584 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺))
326325ex 413 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+𝑑 ∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺)))
327326rexlimdvva 3299 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∃𝑐 ∈ ℝ+𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺)))
328140, 327syl5 34 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺)))
329328a1dd 50 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺))))
330329imp31 418 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺))
331 oveq12 7157 . . . . . . . . . . . . . . 15 ((𝑡 = (∫1𝑓) ∧ 𝑢 = (∫1𝑔)) → (𝑡 + 𝑢) = ((∫1𝑓) + (∫1𝑔)))
332331ad2ant2l 742 . . . . . . . . . . . . . 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 24217 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∫1‘(𝑓f + 𝑔)) = ((∫1𝑓) + (∫1𝑔)))
334333eqcomd 2832 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((∫1𝑓) + (∫1𝑔)) = (∫1‘(𝑓f + 𝑔)))
335334adantl 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((∫1𝑓) + (∫1𝑔)) = (∫1‘(𝑓f + 𝑔)))
336332, 335sylan9eqr 2883 . . . . . . . . . . . . 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 2846 . . . . . . . . . . . . . 14 ((𝑠 = (𝑡 + 𝑢) ∧ (𝑡 + 𝑢) = (∫1‘(𝑓f + 𝑔))) → 𝑠 = (∫1‘(𝑓f + 𝑔)))
338337ancoms 459 . . . . . . . . . . . . 13 (((𝑡 + 𝑢) = (∫1‘(𝑓f + 𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓f + 𝑔)))
339336, 338sylan 580 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓f + 𝑔)))
340 fveq1 6666 . . . . . . . . . . . . . . . . . . 19 ( = (𝑓f + 𝑔) → (𝑧) = ((𝑓f + 𝑔)‘𝑧))
341340eqeq1d 2828 . . . . . . . . . . . . . . . . . 18 ( = (𝑓f + 𝑔) → ((𝑧) = 0 ↔ ((𝑓f + 𝑔)‘𝑧) = 0))
342340oveq1d 7163 . . . . . . . . . . . . . . . . . 18 ( = (𝑓f + 𝑔) → ((𝑧) + 𝑦) = (((𝑓f + 𝑔)‘𝑧) + 𝑦))
343341, 342ifbieq2d 4495 . . . . . . . . . . . . . . . . 17 ( = (𝑓f + 𝑔) → if((𝑧) = 0, 0, ((𝑧) + 𝑦)) = if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦)))
344343mpteq2dv 5159 . . . . . . . . . . . . . . . 16 ( = (𝑓f + 𝑔) → (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(((𝑓f + 𝑔)‘𝑧) = 0, 0, (((𝑓f + 𝑔)‘𝑧) + 𝑦))))
345344breq1d 5073 . . . . . . . . . . . . . . 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 3302 . . . . . . . . . . . . . 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 6667 . . . . . . . . . . . . . . 15 ( = (𝑓f + 𝑔) → (∫1) = (∫1‘(𝑓f + 𝑔)))
348347eqeq2d 2837 . . . . . . . . . . . . . 14 ( = (𝑓f + 𝑔) → (𝑠 = (∫1) ↔ 𝑠 = (∫1‘(𝑓f + 𝑔))))
349346, 348anbi12d 630 . . . . . . . . . . . . 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 3627 . . . . . . . . . . . 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 834 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1)))
352351exp31 420 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1)))))
353352rexlimdvva 3299 . . . . . . . . 9 (𝜑 → (∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1)))))
354353impd 411 . . . . . . . 8 (𝜑 → ((∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1))))
355354exlimdvv 1928 . . . . . . 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 213 . . . . . 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 2830 . . . . . . . . . 10 (𝑥 = 𝑡 → (𝑥 = (∫1𝑓) ↔ 𝑡 = (∫1𝑓)))
358357anbi2d 628 . . . . . . . . 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 3302 . . . . . . . 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 3690 . . . . . . 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 2830 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → (𝑥 = (∫1𝑔) ↔ 𝑢 = (∫1𝑔)))
362361anbi2d 628 . . . . . . . . . . . 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 3302 . . . . . . . . . . 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 3690 . . . . . . . . . 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 622 . . . . . . . . 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 1947 . . . . . . . . 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 3373 . . . . . . . . . . . 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 623 . . . . . . . . . . 11 ((∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) ↔ ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
369 anass 469 . . . . . . . . . . 11 (((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑡 = (∫1𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑢 = (∫1𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))))
370368, 369bitr2i 277 . . . . . . . . . 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 1841 . . . . . . . . 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 300 . . . . . . . 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 1841 . . . . . . 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 276 . . . . . 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 290 . . . . 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 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)})
377376supeq1d 8899 . . 3 (𝜑 → sup({𝑠 ∣ ∃ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑧) = 0, 0, ((𝑧) + 𝑦))) ∘r ≤ (𝐹f + 𝐺) ∧ 𝑠 = (∫1))}, ℝ*, < ) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, < ))
378 simpr 485 . . . . . . . . 9 (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (𝑡 + 𝑢))
3796sseli 3967 . . . . . . . . . . 11 (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} → 𝑡 ∈ ℝ)
380379ad2antrr 722 . . . . . . . . . 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 3967 . . . . . . . . . . 11 (𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))} → 𝑢 ∈ ℝ)
382381ad2antlr 723 . . . . . . . . . 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 10659 . . . . . . . . 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 2918 . . . . . . . 8 (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 ∈ ℝ)
385384ex 413 . . . . . . 7 ((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) → (𝑠 = (𝑡 + 𝑢) → 𝑠 ∈ ℝ))
386385rexlimivv 3297 . . . . . 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 4050 . . . . 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 2835 . . . . . . . 8 0 = (0 + 0)
390 rspceov 7195 . . . . . . . 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 1443 . . . . . . 7 ((0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}) → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢))
39248, 100, 391syl2anc 584 . . . . . 6 (𝜑 → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢))
393 eqeq1 2830 . . . . . . . 8 (𝑠 = 0 → (𝑠 = (𝑡 + 𝑢) ↔ 0 = (𝑡 + 𝑢)))
3943932rexbidv 3305 . . . . . . 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 3611 . . . . . 6 (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}0 = (𝑡 + 𝑢) → ∃𝑠𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢))
396392, 395syl 17 . . . . 5 (𝜑 → ∃𝑠𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢))
397 abn0 4340 . . . . 5 ({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅ ↔ ∃𝑠𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢))
398396, 397sylibr 235 . . . 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 10659 . . . . 5 (𝜑 → (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) ∈ ℝ)
400 simpr 485 . . . . . . . . 9 (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → 𝑏 = (𝑡 + 𝑢))
401379ad2antrl 724 . . . . . . . . . . 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 725 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → 𝑢 ∈ ℝ)
40357adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) ∈ ℝ)
404108adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ) ∈ ℝ)
405 supxrub 12707 . . . . . . . . . . . . 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 686 . . . . . . . . . . . 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 724 . . . . . . . . . . 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 12707 . . . . . . . . . . . . 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 686 . . . . . . . . . . . 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 725 . . . . . . . . . . 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 11248 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → (𝑡 + 𝑢) ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))
412411adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → (𝑡 + 𝑢) ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))
413400, 412eqbrtrd 5085 . . . . . . . 8 (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))
414413ex 413 . . . . . . 7 ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))})) → (𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < ))))
415414rexlimdvva 3299 . . . . . 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 1921 . . . . 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 5067 . . . . . . . 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 3202 . . . . . . 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 2830 . . . . . . . . 9 (𝑠 = 𝑏 → (𝑠 = (𝑡 + 𝑢) ↔ 𝑏 = (𝑡 + 𝑢)))
4204192rexbidv 3305 . . . . . . . 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 3688 . . . . . . 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 288 . . . . . 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 3627 . . . . 5 (((sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )) ∈ ℝ ∧ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}, ℝ*, < )))) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏𝑎)
424399, 416, 423syl2anc 584 . . . 4 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓𝑧) = 0, 0, ((𝑓𝑧) + 𝑐))) ∘r𝐹𝑥 = (∫1𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔𝑧) = 0, 0, ((𝑔𝑧) + 𝑑))) ∘r𝐺𝑥 = (∫1𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏𝑎)
425 supxrre 12710 . . . 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 1365 . . 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 2865 . 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 2872 1 (𝜑 → (∫2‘(𝐹f + 𝐺)) = ((∫2𝐹) + (∫2𝐺)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1528   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {cab 2804   ≠ wne 3021  ∀wral 3143  ∃wrex 3144  Vcvv 3500   ⊆ wss 3940  ∅c0 4295  ifcif 4470  {csn 4564   class class class wbr 5063   ↦ cmpt 5143   × cxp 5552  dom cdm 5554   Fn wfn 6347  ⟶wf 6348  ‘cfv 6352  (class class class)co 7148   ∘f cof 7397   ∘r cofr 7398  supcsup 8893  ℝcr 10525  0cc0 10526   + caddc 10529  +∞cpnf 10661  ℝ*cxr 10663   < clt 10664   ≤ cle 10665  ℕcn 11627  3c3 11682  ℝ+crp 12379  [,)cico 12730  [,]cicc 12731  MblFncmbf 24130  ∫1citg1 24131  ∫2citg2 24132 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-disj 5029  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7399  df-ofr 7400  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-2o 8094  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-fi 8864  df-sup 8895  df-inf 8896  df-oi 8963  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-q 12338  df-rp 12380  df-xneg 12497  df-xadd 12498  df-xmul 12499  df-ioo 12732  df-ico 12734  df-icc 12735  df-fz 12883  df-fzo 13024  df-fl 13152  df-seq 13360  df-exp 13420  df-hash 13681  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-clim 14835  df-sum 15033  df-rest 16686  df-topgen 16707  df-psmet 20453  df-xmet 20454  df-met 20455  df-bl 20456  df-mopn 20457  df-top 21418  df-topon 21435  df-bases 21470  df-cmp 21911  df-ovol 23980  df-vol 23981  df-mbf 24135  df-itg1 24136  df-itg2 24137 This theorem is referenced by:  ibladdnclem  34815  itgaddnclem1  34817  iblabsnclem  34822  iblabsnc  34823  iblmulc2nc  34824  ftc1anclem4  34837  ftc1anclem5  34838  ftc1anclem6  34839  ftc1anclem8  34841
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