Theorem itg2addnc 37919

Description: Alternate proof of itg2add 25728 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 25677, 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 10357, 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	⊢ (𝜑 → 𝐹 ∈ MblFn)
itg2addnc.f2	⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
itg2addnc.f3	⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)
itg2addnc.g2	⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞))
itg2addnc.g3	⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ)

Assertion

Ref	Expression
itg2addnc	⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺)))

Proof of Theorem itg2addnc

Dummy variables 𝑡 𝑠 𝑢 𝑥 𝑦 𝑧 𝑓 𝑔 ℎ 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 773	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 = (∫1‘𝑓))
2		itg1cl 25654	. . . . . . . 8 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘𝑓) ∈ ℝ)
3	2	adantr 480	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ)
4	1, 3	eqeltrd 2837	. . . . . 6 ⊢ ((𝑓 ∈ dom ∫1 ∧ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 ∈ ℝ)
5	4	rexlimiva 3131	. . . . 5 ⊢ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → 𝑥 ∈ ℝ)
6	5	abssi 4022	. . . 4 ⊢ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ
7	6	a1i 11	. . 3 ⊢ (𝜑 → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ)
8		i1f0 25656	. . . . . 6 ⊢ (ℝ × {0}) ∈ dom ∫1
9		3nn 12236	. . . . . . . 8 ⊢ 3 ∈ ℕ
10		nnrp 12929	. . . . . . . 8 ⊢ (3 ∈ ℕ → 3 ∈ ℝ+)
11		ne0i 4295	. . . . . . . 8 ⊢ (3 ∈ ℝ+ → ℝ+ ≠ ∅)
12	9, 10, 11	mp2b 10	. . . . . . 7 ⊢ ℝ+ ≠ ∅
13		itg2addnc.f2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
14	13	ffvelcdmda 7038	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ (0[,)+∞))
15		elrege0 13382	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) ∈ (0[,)+∞) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑧)))
16	14, 15	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑧)))
17	16	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹‘𝑧))
18	17	ralrimiva 3130	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐹‘𝑧))
19		reex 11129	. . . . . . . . . . 11 ⊢ ℝ ∈ V
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ V)
21		c0ex 11138	. . . . . . . . . . 11 ⊢ 0 ∈ V
22	21	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ∈ V)
23		eqidd 2738	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
24	13	feqmptd 6910	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹‘𝑧)))
25	20, 22, 14, 23, 24	ofrfval2 7653	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐹‘𝑧)))
26	18, 25	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
27	26	ralrimivw 3134	. . . . . . 7 ⊢ (𝜑 → ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
28		r19.2z 4454	. . . . . . 7 ⊢ ((ℝ+ ≠ ∅ ∧ ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹) → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
29	12, 27, 28	sylancr 588	. . . . . 6 ⊢ (𝜑 → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
30		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑓 = (ℝ × {0}) → (∫1‘𝑓) = (∫1‘(ℝ × {0})))
31		itg10 25657	. . . . . . . . . 10 ⊢ (∫1‘(ℝ × {0})) = 0
32	30, 31	eqtr2di 2789	. . . . . . . . 9 ⊢ (𝑓 = (ℝ × {0}) → 0 = (∫1‘𝑓))
33	32	biantrud 531	. . . . . . . 8 ⊢ (𝑓 = (ℝ × {0}) → (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑓))))
34		fveq1 6841	. . . . . . . . . . . . 13 ⊢ (𝑓 = (ℝ × {0}) → (𝑓‘𝑧) = ((ℝ × {0})‘𝑧))
35	21	fvconst2 7160	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
36	34, 35	sylan9eq 2792	. . . . . . . . . . . 12 ⊢ ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) = 0)
37	36	iftrued 4489	. . . . . . . . . . 11 ⊢ ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) = 0)
38	37	mpteq2dva 5193	. . . . . . . . . 10 ⊢ (𝑓 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ 0))
39	38	breq1d 5110	. . . . . . . . 9 ⊢ (𝑓 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹))
40	39	rexbidv 3162	. . . . . . . 8 ⊢ (𝑓 = (ℝ × {0}) → (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹))
41	33, 40	bitr3d 281	. . . . . . 7 ⊢ (𝑓 = (ℝ × {0}) → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑓)) ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹))
42	41	rspcev 3578	. . . . . 6 ⊢ (((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹) → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑓)))
43	8, 29, 42	sylancr 588	. . . . 5 ⊢ (𝜑 → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑓)))
44		eqeq1 2741	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 = (∫1‘𝑓) ↔ 0 = (∫1‘𝑓)))
45	44	anbi2d 631	. . . . . . 7 ⊢ (𝑥 = 0 → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑓))))
46	45	rexbidv 3162	. . . . . 6 ⊢ (𝑥 = 0 → (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑓))))
47	21, 46	elab 3636	. . . . 5 ⊢ (0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ↔ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑓)))
48	43, 47	sylibr 234	. . . 4 ⊢ (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))})
49	48	ne0d 4296	. . 3 ⊢ (𝜑 → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ≠ ∅)
50		icossicc 13364	. . . . . . 7 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
51		fss 6686	. . . . . . 7 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
52	50, 51	mpan2 692	. . . . . 6 ⊢ (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶(0[,]+∞))
53		eqid 2737	. . . . . . 7 ⊢ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} = {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}
54	53	itg2addnclem 37916	. . . . . 6 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
55	13, 52, 54	3syl 18	. . . . 5 ⊢ (𝜑 → (∫2‘𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
56		itg2addnc.f3	. . . . 5 ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)
57	55, 56	eqeltrrd 2838	. . . 4 ⊢ (𝜑 → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ∈ ℝ)
58		ressxr 11188	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
59	6, 58	sstri 3945	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ*
60		supxrub 13251	. . . . . 6 ⊢ (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ* ∧ 𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}) → 𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
61	59, 60	mpan 691	. . . . 5 ⊢ (𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} → 𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
62	61	rgen 3054	. . . 4 ⊢ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
63		brralrspcev 5160	. . . 4 ⊢ ((sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ∈ ℝ ∧ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 ≤ 𝑎)
64	57, 62, 63	sylancl 587	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 ≤ 𝑎)
65		simprr 773	. . . . . . 7 ⊢ ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))) → 𝑥 = (∫1‘𝑔))
66		itg1cl 25654	. . . . . . . 8 ⊢ (𝑔 ∈ dom ∫1 → (∫1‘𝑔) ∈ ℝ)
67	66	adantr 480	. . . . . . 7 ⊢ ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))) → (∫1‘𝑔) ∈ ℝ)
68	65, 67	eqeltrd 2837	. . . . . 6 ⊢ ((𝑔 ∈ dom ∫1 ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))) → 𝑥 ∈ ℝ)
69	68	rexlimiva 3131	. . . . 5 ⊢ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔)) → 𝑥 ∈ ℝ)
70	69	abssi 4022	. . . 4 ⊢ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ
71	70	a1i 11	. . 3 ⊢ (𝜑 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ)
72		itg2addnc.g2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞))
73	72	ffvelcdmda 7038	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ (0[,)+∞))
74		elrege0 13382	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑧) ∈ (0[,)+∞) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑧)))
75	73, 74	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((𝐺‘𝑧) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑧)))
76	75	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐺‘𝑧))
77	76	ralrimiva 3130	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐺‘𝑧))
78	72	feqmptd 6910	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℝ ↦ (𝐺‘𝑧)))
79	20, 22, 73, 23, 78	ofrfval2 7653	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐺‘𝑧)))
80	77, 79	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺)
81	80	ralrimivw 3134	. . . . . . 7 ⊢ (𝜑 → ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺)
82		r19.2z 4454	. . . . . . 7 ⊢ ((ℝ+ ≠ ∅ ∧ ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺) → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺)
83	12, 81, 82	sylancr 588	. . . . . 6 ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺)
84		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑔 = (ℝ × {0}) → (∫1‘𝑔) = (∫1‘(ℝ × {0})))
85	84, 31	eqtr2di 2789	. . . . . . . . 9 ⊢ (𝑔 = (ℝ × {0}) → 0 = (∫1‘𝑔))
86	85	biantrud 531	. . . . . . . 8 ⊢ (𝑔 = (ℝ × {0}) → (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ↔ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 = (∫1‘𝑔))))
87		fveq1 6841	. . . . . . . . . . . . 13 ⊢ (𝑔 = (ℝ × {0}) → (𝑔‘𝑧) = ((ℝ × {0})‘𝑧))
88	87, 35	sylan9eq 2792	. . . . . . . . . . . 12 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = 0)
89	88	iftrued 4489	. . . . . . . . . . 11 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) = 0)
90	89	mpteq2dva 5193	. . . . . . . . . 10 ⊢ (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ 0))
91	90	breq1d 5110	. . . . . . . . 9 ⊢ (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺))
92	91	rexbidv 3162	. . . . . . . 8 ⊢ (𝑔 = (ℝ × {0}) → (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺))
93	86, 92	bitr3d 281	. . . . . . 7 ⊢ (𝑔 = (ℝ × {0}) → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 = (∫1‘𝑔)) ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺))
94	93	rspcev 3578	. . . . . 6 ⊢ (((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺) → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 = (∫1‘𝑔)))
95	8, 83, 94	sylancr 588	. . . . 5 ⊢ (𝜑 → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 = (∫1‘𝑔)))
96		eqeq1 2741	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 = (∫1‘𝑔) ↔ 0 = (∫1‘𝑔)))
97	96	anbi2d 631	. . . . . . 7 ⊢ (𝑥 = 0 → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔)) ↔ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 = (∫1‘𝑔))))
98	97	rexbidv 3162	. . . . . 6 ⊢ (𝑥 = 0 → (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 = (∫1‘𝑔))))
99	21, 98	elab 3636	. . . . 5 ⊢ (0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ↔ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 = (∫1‘𝑔)))
100	95, 99	sylibr 234	. . . 4 ⊢ (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})
101	100	ne0d 4296	. . 3 ⊢ (𝜑 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ≠ ∅)
102		fss 6686	. . . . . . 7 ⊢ ((𝐺:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐺:ℝ⟶(0[,]+∞))
103	50, 102	mpan2 692	. . . . . 6 ⊢ (𝐺:ℝ⟶(0[,)+∞) → 𝐺:ℝ⟶(0[,]+∞))
104		eqid 2737	. . . . . . 7 ⊢ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}
105	104	itg2addnclem 37916	. . . . . 6 ⊢ (𝐺:ℝ⟶(0[,]+∞) → (∫2‘𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
106	72, 103, 105	3syl 18	. . . . 5 ⊢ (𝜑 → (∫2‘𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
107		itg2addnc.g3	. . . . 5 ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ)
108	106, 107	eqeltrrd 2838	. . . 4 ⊢ (𝜑 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ∈ ℝ)
109	70, 58	sstri 3945	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ*
110		supxrub 13251	. . . . . 6 ⊢ (({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ* ∧ 𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) → 𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
111	109, 110	mpan 691	. . . . 5 ⊢ (𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} → 𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
112	111	rgen 3054	. . . 4 ⊢ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )
113		brralrspcev 5160	. . . 4 ⊢ ((sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ∈ ℝ ∧ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 ≤ 𝑎)
114	108, 112, 113	sylancl 587	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 ≤ 𝑎)
115		eqid 2737	. . 3 ⊢ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} = {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}
116	7, 49, 64, 71, 101, 114, 115	supadd 12122	. 2 ⊢ (𝜑 → (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ, < )) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < ))
117		supxrre 13254	. . . . 5 ⊢ (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ ∧ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ≠ ∅ ∧ ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 ≤ 𝑎) → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ, < ))
118	7, 49, 64, 117	syl3anc 1374	. . . 4 ⊢ (𝜑 → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ, < ))
119	55, 118	eqtrd 2772	. . 3 ⊢ (𝜑 → (∫2‘𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ, < ))
120		supxrre 13254	. . . . 5 ⊢ (({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ ∧ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ≠ ∅ ∧ ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 ≤ 𝑎) → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ, < ))
121	71, 101, 114, 120	syl3anc 1374	. . . 4 ⊢ (𝜑 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ, < ))
122	106, 121	eqtrd 2772	. . 3 ⊢ (𝜑 → (∫2‘𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ, < ))
123	119, 122	oveq12d 7386	. 2 ⊢ (𝜑 → ((∫2‘𝐹) + (∫2‘𝐺)) = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ, < )))
124		ge0addcl 13388	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞))
125	50, 124	sselid 3933	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,]+∞))
126	125	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,]+∞))
127		inidm 4181	. . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ
128	126, 13, 72, 20, 20, 127	off 7650	. . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶(0[,]+∞))
129		eqid 2737	. . . . 5 ⊢ {𝑠 ∣ ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))} = {𝑠 ∣ ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))}
130	129	itg2addnclem 37916	. . . 4 ⊢ ((𝐹 ∘f + 𝐺):ℝ⟶(0[,]+∞) → (∫2‘(𝐹 ∘f + 𝐺)) = sup({𝑠 ∣ ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))}, ℝ*, < ))
131	128, 130	syl 17	. . 3 ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = sup({𝑠 ∣ ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))}, ℝ*, < ))
132		itg2addnc.f1	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ MblFn)
133	132, 13, 56, 72, 107	itg2addnclem3 37918	. . . . . . 7 ⊢ (𝜑 → (∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)) → ∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))))
134		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 𝑓 ∈ dom ∫1)
135		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 𝑔 ∈ dom ∫1)
136	134, 135	i1fadd 25664	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑓 ∘f + 𝑔) ∈ dom ∫1)
137	136	ad3antlr 732	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → (𝑓 ∘f + 𝑔) ∈ dom ∫1)
138		reeanv 3210	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑐 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺))
139	138	biimpri 228	. . . . . . . . . . . . . . . 16 ⊢ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) → ∃𝑐 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺))
140	139	ad2ant2r 748	. . . . . . . . . . . . . . 15 ⊢ (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) → ∃𝑐 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺))
141		ifcl 4527	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ+)
142	141	ad2antlr 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺)) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ+)
143		breq1 5103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) → (0 ≤ (𝐹‘𝑧) ↔ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧)))
144	143	anbi1d 632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) → ((0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) ↔ (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))))
145	144	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) → (((0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) ↔ ((if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))))
146		breq1 5103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓‘𝑧) + 𝑐) = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) → (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ↔ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧)))
147	146	anbi1d 632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓‘𝑧) + 𝑐) = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) ↔ (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))))
148	147	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓‘𝑧) + 𝑐) = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) → (((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) ↔ ((if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))))
149		breq1 5103	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → (0 ≤ (𝐺‘𝑧) ↔ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))
150	149	anbi2d 631	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → ((0 ≤ (𝐹‘𝑧) ∧ 0 ≤ (𝐺‘𝑧)) ↔ (0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))))
151	150	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → (((0 ≤ (𝐹‘𝑧) ∧ 0 ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) ↔ ((0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))))
152		breq1 5103	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑔‘𝑧) + 𝑑) = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → (((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧) ↔ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))
153	152	anbi2d 631	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑔‘𝑧) + 𝑑) = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → ((0 ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) ↔ (0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))))
154	153	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑔‘𝑧) + 𝑑) = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → (((0 ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) ↔ ((0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))))
155		oveq12 7377	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓‘𝑧) = 0 ∧ (𝑔‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = (0 + 0))
156		00id 11320	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0 + 0) = 0
157	155, 156	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓‘𝑧) = 0 ∧ (𝑔‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = 0)
158	157	iftrued 4489	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓‘𝑧) = 0 ∧ (𝑔‘𝑧) = 0) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) = 0)
159	158	adantll 715	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) ∧ (𝑔‘𝑧) = 0) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) = 0)
160		simpll 767	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝜑)
161	15	simplbi 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹‘𝑧) ∈ (0[,)+∞) → (𝐹‘𝑧) ∈ ℝ)
162	14, 161	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ ℝ)
163	74	simplbi 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺‘𝑧) ∈ (0[,)+∞) → (𝐺‘𝑧) ∈ ℝ)
164	73, 163	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ℝ)
165	162, 164, 17, 76	addge0d 11725	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
166	160, 165	sylan 581	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
167	166	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) ∧ (𝑔‘𝑧) = 0) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
168	159, 167	eqbrtrd 5122	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) ∧ (𝑔‘𝑧) = 0) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
169	168	a1d 25	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) ∧ (𝑔‘𝑧) = 0) → ((0 ≤ (𝐹‘𝑧) ∧ 0 ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
170	166	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
171		oveq1 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓‘𝑧) = 0 → ((𝑓‘𝑧) + (𝑔‘𝑧)) = (0 + (𝑔‘𝑧)))
172		simplrr 778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑔 ∈ dom ∫1)
173		i1ff 25645	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑔 ∈ dom ∫1 → 𝑔:ℝ⟶ℝ)
174	173	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) ∈ ℝ)
175	172, 174	sylan 581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) ∈ ℝ)
176	175	recnd 11172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) ∈ ℂ)
177	176	addlidd 11346	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (0 + (𝑔‘𝑧)) = (𝑔‘𝑧))
178	171, 177	sylan9eqr 2794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = (𝑔‘𝑧))
179	178	oveq1d 7383	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))
180	179	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))
181	141	rpred 12961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ)
182	181	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ)
183	175, 182	readdcld 11173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ)
184	183	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ)
185	160, 164	sylan 581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ℝ)
186	185	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → (𝐺‘𝑧) ∈ ℝ)
187	160, 162	sylan 581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ ℝ)
188	187, 185	readdcld 11173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) + (𝐺‘𝑧)) ∈ ℝ)
189	188	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝐹‘𝑧) + (𝐺‘𝑧)) ∈ ℝ)
190		simplrr 778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ+)
191	190	rpred 12961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ)
192		rpre 12926	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑐 ∈ ℝ+ → 𝑐 ∈ ℝ)
193		rpre 12926	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑑 ∈ ℝ+ → 𝑑 ∈ ℝ)
194		min2 13117	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑑)
195	192, 193, 194	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑑)
196	195	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑑)
197	182, 191, 175, 196	leadd2dd 11764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝑔‘𝑧) + 𝑑))
198	175, 191	readdcld 11173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) + 𝑑) ∈ ℝ)
199		letr 11239	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ ∧ ((𝑔‘𝑧) + 𝑑) ∈ ℝ ∧ (𝐺‘𝑧) ∈ ℝ) → ((((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝑔‘𝑧) + 𝑑) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧)))
200	183, 198, 185, 199	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝑔‘𝑧) + 𝑑) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧)))
201	197, 200	mpand 696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧)))
202	201	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧))
203	164, 162	addge02d 11738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (0 ≤ (𝐹‘𝑧) ↔ (𝐺‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
204	17, 203	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
205	160, 204	sylan 581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
206	205	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → (𝐺‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
207	184, 186, 189, 202, 206	letrd 11302	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
208	207	adantlr 716	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
209	180, 208	eqbrtrd 5122	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
210		breq1 5103	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0 = if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) → (0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)) ↔ if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
211		breq1 5103	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) → ((((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)) ↔ if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
212	210, 211	ifboth 4521	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)) ∧ (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
213	170, 209, 212	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
214	213	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) → (((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
215	214	adantld 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) → ((0 ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
216	215	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) ∧ ¬ (𝑔‘𝑧) = 0) → ((0 ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
217	151, 154, 169, 216	ifbothda 4520	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) → ((0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
218	149	anbi2d 631	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ 0 ≤ (𝐺‘𝑧)) ↔ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))))
219	218	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → (((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ 0 ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) ↔ ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))))
220	152	anbi2d 631	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑔‘𝑧) + 𝑑) = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) ↔ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))))
221	220	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑔‘𝑧) + 𝑑) = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → (((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) ↔ ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))))
222	166	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔‘𝑧) = 0) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
223		oveq2 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑔‘𝑧) = 0 → ((𝑓‘𝑧) + (𝑔‘𝑧)) = ((𝑓‘𝑧) + 0))
224		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑓 ∈ dom ∫1)
225		i1ff 25645	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑓 ∈ dom ∫1 → 𝑓:ℝ⟶ℝ)
226	225	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
227	224, 226	sylan 581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
228	227	recnd 11172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℂ)
229	228	addridd 11345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 0) = (𝑓‘𝑧))
230	223, 229	sylan9eqr 2794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = (𝑓‘𝑧))
231	230	oveq1d 7383	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔‘𝑧) = 0) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))
232	231	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔‘𝑧) = 0) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))
233	227, 182	readdcld 11173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ)
234	233	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ)
235	187	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → (𝐹‘𝑧) ∈ ℝ)
236	188	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝐹‘𝑧) + (𝐺‘𝑧)) ∈ ℝ)
237		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ+)
238	237	rpred 12961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ)
239		min1 13116	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑐)
240	192, 193, 239	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑐)
241	240	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑐)
242	182, 238, 227, 241	leadd2dd 11764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝑓‘𝑧) + 𝑐))
243	227, 238	readdcld 11173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 𝑐) ∈ ℝ)
244		letr 11239	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ ∧ ((𝑓‘𝑧) + 𝑐) ∈ ℝ ∧ (𝐹‘𝑧) ∈ ℝ) → ((((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝑓‘𝑧) + 𝑐) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐹‘𝑧)))
245	233, 243, 187, 244	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝑓‘𝑧) + 𝑐) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐹‘𝑧)))
246	242, 245	mpand 696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐹‘𝑧)))
247	246	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐹‘𝑧))
248	162, 164	addge01d 11737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (0 ≤ (𝐺‘𝑧) ↔ (𝐹‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
249	76, 248	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
250	160, 249	sylan 581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
251	250	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → (𝐹‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
252	234, 235, 236, 247, 251	letrd 11302	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
253	252	adantlr 716	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔‘𝑧) = 0) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
254	232, 253	eqbrtrd 5122	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔‘𝑧) = 0) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
255	222, 254, 212	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔‘𝑧) = 0) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
256	255	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (𝑔‘𝑧) = 0) → (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
257	256	adantlr 716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓‘𝑧) = 0) ∧ (𝑔‘𝑧) = 0) → (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
258	257	adantrd 491	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓‘𝑧) = 0) ∧ (𝑔‘𝑧) = 0) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ 0 ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
259	166	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧))) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
260	182	recnd 11172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℂ)
261	228, 176, 260	addassd 11166	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))))
262	261	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧))) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))))
263	227, 237	ltaddrpd 12994	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) < ((𝑓‘𝑧) + 𝑐))
264	227, 243, 263	ltled 11293	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑐))
265		letr 11239	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓‘𝑧) ∈ ℝ ∧ ((𝑓‘𝑧) + 𝑐) ∈ ℝ ∧ (𝐹‘𝑧) ∈ ℝ) → (((𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑐) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
266	227, 243, 187, 265	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑐) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
267	264, 266	mpand 696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) → (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
268		le2add 11631	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑓‘𝑧) ∈ ℝ ∧ ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ) ∧ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) ∈ ℝ)) → (((𝑓‘𝑧) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧)) → ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
269	227, 183, 187, 185, 268	syl22anc 839	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧)) → ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
270	267, 201, 269	syl2and 609	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
271	270	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧))) → ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
272	262, 271	eqbrtrd 5122	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧))) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
273	259, 272, 212	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧))) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
274	273	ex 412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
275	274	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓‘𝑧) = 0) ∧ ¬ (𝑔‘𝑧) = 0) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
276	219, 221, 258, 275	ifbothda 4520	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓‘𝑧) = 0) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
277	145, 148, 217, 276	ifbothda 4520	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
278	277	ralimdva 3150	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → (∀𝑧 ∈ ℝ (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → ∀𝑧 ∈ ℝ if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
279		ovex 7401	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓‘𝑧) + 𝑐) ∈ V
280	21, 279	ifex 4532	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ∈ V
281	280	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ∈ V)
282		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))))
283	20, 281, 14, 282, 24	ofrfval2 7653	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧)))
284		ovex 7401	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑔‘𝑧) + 𝑑) ∈ V
285	21, 284	ifex 4532	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ∈ V
286	285	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ∈ V)
287		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))))
288	20, 286, 73, 287, 78	ofrfval2 7653	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ↔ ∀𝑧 ∈ ℝ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))
289	283, 288	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) ↔ (∀𝑧 ∈ ℝ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ ∀𝑧 ∈ ℝ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))))
290		r19.26 3098	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑧 ∈ ℝ (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) ↔ (∀𝑧 ∈ ℝ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ ∀𝑧 ∈ ℝ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))
291	289, 290	bitr4di 289	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) ↔ ∀𝑧 ∈ ℝ (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))))
292	291	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) ↔ ∀𝑧 ∈ ℝ (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))))
293	19	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → ℝ ∈ V)
294		ovex 7401	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ V
295	21, 294	ifex 4532	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ∈ V
296	295	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ∈ V)
297		ovexd 7403	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) + (𝐺‘𝑧)) ∈ V)
298	225	ffnd 6671	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 Fn ℝ)
299	298	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 𝑓 Fn ℝ)
300	299	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑓 Fn ℝ)
301	173	ffnd 6671	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑔 ∈ dom ∫1 → 𝑔 Fn ℝ)
302	301	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 𝑔 Fn ℝ)
303	302	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑔 Fn ℝ)
304		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) = (𝑓‘𝑧))
305		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = (𝑔‘𝑧))
306	300, 303, 293, 293, 127, 304, 305	ofval 7643	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓 ∘f + 𝑔)‘𝑧) = ((𝑓‘𝑧) + (𝑔‘𝑧)))
307	306	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓 ∘f + 𝑔)‘𝑧) = 0 ↔ ((𝑓‘𝑧) + (𝑔‘𝑧)) = 0))
308	306	oveq1d 7383	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))
309	307, 308	ifbieq2d 4508	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) = if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))))
310	309	mpteq2dva 5193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) = (𝑧 ∈ ℝ ↦ if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))))
311	13	ffnd 6671	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐹 Fn ℝ)
312	72	ffnd 6671	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐺 Fn ℝ)
313		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
314		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
315	311, 312, 20, 20, 127, 313, 314	offval 7641	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧) + (𝐺‘𝑧))))
316	315	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → (𝐹 ∘f + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧) + (𝐺‘𝑧))))
317	293, 296, 297, 310, 316	ofrfval2 7653	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → ((𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹 ∘f + 𝐺) ↔ ∀𝑧 ∈ ℝ if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
318	278, 292, 317	3imtr4d 294	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) → (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹 ∘f + 𝐺)))
319	318	imp 406	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺)) → (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹 ∘f + 𝐺))
320		oveq2 7376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = if(𝑐 ≤ 𝑑, 𝑐, 𝑑) → (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦) = (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))
321	320	ifeq2d 4502	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = if(𝑐 ≤ 𝑑, 𝑐, 𝑑) → if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦)) = if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))))
322	321	mpteq2dv 5194	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = if(𝑐 ≤ 𝑑, 𝑐, 𝑑) → (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))))
323	322	breq1d 5110	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = if(𝑐 ≤ 𝑑, 𝑐, 𝑑) → ((𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ↔ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹 ∘f + 𝐺)))
324	323	rspcev 3578	. . . . . . . . . . . . . . . . . 18 ⊢ ((if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ+ ∧ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹 ∘f + 𝐺)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺))
325	142, 319, 324	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺))
326	325	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺)))
327	326	rexlimdvva 3195	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) → (∃𝑐 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺)))
328	140, 327	syl5 34	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) → (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺)))
329	328	a1dd 50	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) → (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺))))
330	329	imp31 417	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺))
331		oveq12 7377	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 = (∫1‘𝑓) ∧ 𝑢 = (∫1‘𝑔)) → (𝑡 + 𝑢) = ((∫1‘𝑓) + (∫1‘𝑔)))
332	331	ad2ant2l 747	. . . . . . . . . . . . . 14 ⊢ (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) → (𝑡 + 𝑢) = ((∫1‘𝑓) + (∫1‘𝑔)))
333	134, 135	itg1add 25670	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (∫1‘(𝑓 ∘f + 𝑔)) = ((∫1‘𝑓) + (∫1‘𝑔)))
334	333	eqcomd 2743	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ((∫1‘𝑓) + (∫1‘𝑔)) = (∫1‘(𝑓 ∘f + 𝑔)))
335	334	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) → ((∫1‘𝑓) + (∫1‘𝑔)) = (∫1‘(𝑓 ∘f + 𝑔)))
336	332, 335	sylan9eqr 2794	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) → (𝑡 + 𝑢) = (∫1‘(𝑓 ∘f + 𝑔)))
337		eqtr 2757	. . . . . . . . . . . . . 14 ⊢ ((𝑠 = (𝑡 + 𝑢) ∧ (𝑡 + 𝑢) = (∫1‘(𝑓 ∘f + 𝑔))) → 𝑠 = (∫1‘(𝑓 ∘f + 𝑔)))
338	337	ancoms 458	. . . . . . . . . . . . 13 ⊢ (((𝑡 + 𝑢) = (∫1‘(𝑓 ∘f + 𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓 ∘f + 𝑔)))
339	336, 338	sylan 581	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓 ∘f + 𝑔)))
340		fveq1 6841	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → (ℎ‘𝑧) = ((𝑓 ∘f + 𝑔)‘𝑧))
341	340	eqeq1d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((ℎ‘𝑧) = 0 ↔ ((𝑓 ∘f + 𝑔)‘𝑧) = 0))
342	340	oveq1d 7383	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((ℎ‘𝑧) + 𝑦) = (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))
343	341, 342	ifbieq2d 4508	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦)) = if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦)))
344	343	mpteq2dv 5194	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))))
345	344	breq1d 5110	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ↔ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺)))
346	345	rexbidv 3162	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺)))
347		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → (∫1‘ℎ) = (∫1‘(𝑓 ∘f + 𝑔)))
348	347	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → (𝑠 = (∫1‘ℎ) ↔ 𝑠 = (∫1‘(𝑓 ∘f + 𝑔))))
349	346, 348	anbi12d 633	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘(𝑓 ∘f + 𝑔)))))
350	349	rspcev 3578	. . . . . . . . . . . 12 ⊢ (((𝑓 ∘f + 𝑔) ∈ dom ∫1 ∧ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘(𝑓 ∘f + 𝑔)))) → ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)))
351	137, 330, 339, 350	syl12anc 837	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)))
352	351	exp31 419	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) → (((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)))))
353	352	rexlimdvva 3195	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)))))
354	353	impd 410	. . . . . . . 8 ⊢ (𝜑 → ((∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))))
355	354	exlimdvv 1936	. . . . . . 7 ⊢ (𝜑 → (∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))))
356	133, 355	impbid 212	. . . . . 6 ⊢ (𝜑 → (∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)) ↔ ∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))))
357		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (𝑥 = (∫1‘𝑓) ↔ 𝑡 = (∫1‘𝑓)))
358	357	anbi2d 631	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ (∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓))))
359	358	rexbidv 3162	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓))))
360	359	rexab 3655	. . . . . . 7 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)))
361		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑥 = (∫1‘𝑔) ↔ 𝑢 = (∫1‘𝑔)))
362	361	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔)) ↔ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))))
363	362	rexbidv 3162	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))))
364	363	rexab 3655	. . . . . . . . . 10 ⊢ (∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)))
365	364	anbi2i 624	. . . . . . . . 9 ⊢ ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))))
366		19.42v 1955	. . . . . . . . 9 ⊢ (∃𝑢(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))))
367		reeanv 3210	. . . . . . . . . . . 12 ⊢ (∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))))
368	367	anbi1i 625	. . . . . . . . . . 11 ⊢ ((∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) ↔ ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
369		anass 468	. . . . . . . . . . 11 ⊢ (((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))))
370	368, 369	bitr2i 276	. . . . . . . . . 10 ⊢ ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) ↔ (∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
371	370	exbii 1850	. . . . . . . . 9 ⊢ (∃𝑢(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) ↔ ∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
372	365, 366, 371	3bitr2i 299	. . . . . . . 8 ⊢ ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ ∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
373	372	exbii 1850	. . . . . . 7 ⊢ (∃𝑡(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ ∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
374	360, 373	bitri 275	. . . . . 6 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
375	356, 374	bitr4di 289	. . . . 5 ⊢ (𝜑 → (∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)) ↔ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)))
376	375	abbidv 2803	. . . 4 ⊢ (𝜑 → {𝑠 ∣ ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))} = {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)})
377	376	supeq1d 9361	. . 3 ⊢ (𝜑 → sup({𝑠 ∣ ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))}, ℝ*, < ) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, < ))
378		simpr 484	. . . . . . . . 9 ⊢ (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (𝑡 + 𝑢))
379	6	sseli 3931	. . . . . . . . . . 11 ⊢ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} → 𝑡 ∈ ℝ)
380	379	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑡 ∈ ℝ)
381	70	sseli 3931	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} → 𝑢 ∈ ℝ)
382	381	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑢 ∈ ℝ)
383	380, 382	readdcld 11173	. . . . . . . . 9 ⊢ (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → (𝑡 + 𝑢) ∈ ℝ)
384	378, 383	eqeltrd 2837	. . . . . . . 8 ⊢ (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 ∈ ℝ)
385	384	ex 412	. . . . . . 7 ⊢ ((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) → (𝑠 = (𝑡 + 𝑢) → 𝑠 ∈ ℝ))
386	385	rexlimivv 3180	. . . . . 6 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) → 𝑠 ∈ ℝ)
387	386	abssi 4022	. . . . 5 ⊢ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ
388	387	a1i 11	. . . 4 ⊢ (𝜑 → {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ)
389	156	eqcomi 2746	. . . . . . . 8 ⊢ 0 = (0 + 0)
390		rspceov 7417	. . . . . . . 8 ⊢ ((0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ∧ 0 = (0 + 0)) → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}0 = (𝑡 + 𝑢))
391	389, 390	mp3an3 1453	. . . . . . 7 ⊢ ((0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}0 = (𝑡 + 𝑢))
392	48, 100, 391	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}0 = (𝑡 + 𝑢))
393		eqeq1 2741	. . . . . . . 8 ⊢ (𝑠 = 0 → (𝑠 = (𝑡 + 𝑢) ↔ 0 = (𝑡 + 𝑢)))
394	393	2rexbidv 3203	. . . . . . 7 ⊢ (𝑠 = 0 → (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}0 = (𝑡 + 𝑢)))
395	21, 394	spcev 3562	. . . . . 6 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}0 = (𝑡 + 𝑢) → ∃𝑠∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢))
396	392, 395	syl 17	. . . . 5 ⊢ (𝜑 → ∃𝑠∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢))
397		abn0 4339	. . . . 5 ⊢ ({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅ ↔ ∃𝑠∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢))
398	396, 397	sylibr 234	. . . 4 ⊢ (𝜑 → {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅)
399	57, 108	readdcld 11173	. . . . 5 ⊢ (𝜑 → (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )) ∈ ℝ)
400		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → 𝑏 = (𝑡 + 𝑢))
401	379	ad2antrl 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → 𝑡 ∈ ℝ)
402	381	ad2antll 730	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → 𝑢 ∈ ℝ)
403	57	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ∈ ℝ)
404	108	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) ∈ ℝ)
405		supxrub 13251	. . . . . . . . . . . . 13 ⊢ (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ* ∧ 𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}) → 𝑡 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
406	59, 405	mpan 691	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} → 𝑡 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
407	406	ad2antrl 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → 𝑡 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
408		supxrub 13251	. . . . . . . . . . . . 13 ⊢ (({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ* ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) → 𝑢 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
409	109, 408	mpan 691	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} → 𝑢 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
410	409	ad2antll 730	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → 𝑢 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
411	401, 402, 403, 404, 407, 410	le2addd 11768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → (𝑡 + 𝑢) ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )))
412	411	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → (𝑡 + 𝑢) ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )))
413	400, 412	eqbrtrd 5122	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )))
414	413	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → (𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))))
415	414	rexlimdvva 3195	. . . . . 6 ⊢ (𝜑 → (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))))
416	415	alrimiv 1929	. . . . 5 ⊢ (𝜑 → ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))))
417		breq2 5104	. . . . . . . 8 ⊢ (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )) → (𝑏 ≤ 𝑎 ↔ 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))))
418	417	ralbidv 3161	. . . . . . 7 ⊢ (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )) → (∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ 𝑎 ↔ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))))
419		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑠 = 𝑏 → (𝑠 = (𝑡 + 𝑢) ↔ 𝑏 = (𝑡 + 𝑢)))
420	419	2rexbidv 3203	. . . . . . . 8 ⊢ (𝑠 = 𝑏 → (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 = (𝑡 + 𝑢)))
421	420	ralab 3653	. . . . . . 7 ⊢ (∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )) ↔ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))))
422	418, 421	bitrdi 287	. . . . . 6 ⊢ (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )) → (∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ 𝑎 ↔ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )))))
423	422	rspcev 3578	. . . . 5 ⊢ (((sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )) ∈ ℝ ∧ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )))) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ 𝑎)
424	399, 416, 423	syl2anc 585	. . . 4 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ 𝑎)
425		supxrre 13254	. . . 4 ⊢ (({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ ∧ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅ ∧ ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ 𝑎) → sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, < ) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < ))
426	388, 398, 424, 425	syl3anc 1374	. . 3 ⊢ (𝜑 → sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, < ) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < ))
427	131, 377, 426	3eqtrd 2776	. 2 ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < ))
428	116, 123, 427	3eqtr4rd 2783	1 ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  ifcif 4481  {csn 4582   class class class wbr 5100   ↦ cmpt 5181   × cxp 5630  dom cdm 5632   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ∘_f cof 7630   ∘_r cofr 7631  supcsup 9355  ℝcr 11037  0cc0 11038   + caddc 11041  +∞cpnf 11175  ℝ^*cxr 11177   < clt 11178   ≤ cle 11179  ℕcn 12157  3c3 12213  ℝ⁺crp 12917  [,)cico 13275  [,]cicc 13276  MblFncmbf 25583  ∫₁citg1 25584  ∫₂citg2 25585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-addf 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-disj 5068  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-ofr 7633  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fi 9326  df-sup 9357  df-inf 9358  df-oi 9427  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-z 12501  df-uz 12764  df-q 12874  df-rp 12918  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ioo 13277  df-ico 13279  df-icc 13280  df-fz 13436  df-fzo 13583  df-fl 13724  df-seq 13937  df-exp 13997  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-clim 15423  df-sum 15622  df-rest 17354  df-topgen 17375  df-psmet 21313  df-xmet 21314  df-met 21315  df-bl 21316  df-mopn 21317  df-top 22850  df-topon 22867  df-bases 22902  df-cmp 23343  df-ovol 25433  df-vol 25434  df-mbf 25588  df-itg1 25589  df-itg2 25590

This theorem is referenced by:  ibladdnclem  37921  itgaddnclem1  37923  iblabsnclem  37928  iblabsnc  37929  iblmulc2nc  37930  ftc1anclem4  37941  ftc1anclem5  37942  ftc1anclem6  37943  ftc1anclem8  37945