Theorem itg2addnc 37995

Description: Alternate proof of itg2add 25726 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 25675, 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 25652	. . . . . . . 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 2836	. . . . . 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 3130	. . . . 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 4008	. . . 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 25654	. . . . . 6 ⊢ (ℝ × {0}) ∈ dom ∫1
9		3nn 12260	. . . . . . . 8 ⊢ 3 ∈ ℕ
10		nnrp 12954	. . . . . . . 8 ⊢ (3 ∈ ℕ → 3 ∈ ℝ+)
11		ne0i 4281	. . . . . . . 8 ⊢ (3 ∈ ℝ+ → ℝ+ ≠ ∅)
12	9, 10, 11	mp2b 10	. . . . . . 7 ⊢ ℝ+ ≠ ∅
13		itg2addnc.f2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
14	13	ffvelcdmda 7036	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ (0[,)+∞))
15		elrege0 13407	. . . . . . . . . . . 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 3129	. . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
24	13	feqmptd 6908	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹‘𝑧)))
25	20, 22, 14, 23, 24	ofrfval2 7652	. . . . . . . . 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 3133	. . . . . . 7 ⊢ (𝜑 → ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
28		r19.2z 4439	. . . . . . 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 6840	. . . . . . . . . 10 ⊢ (𝑓 = (ℝ × {0}) → (∫1‘𝑓) = (∫1‘(ℝ × {0})))
31		itg10 25655	. . . . . . . . . 10 ⊢ (∫1‘(ℝ × {0})) = 0
32	30, 31	eqtr2di 2788	. . . . . . . . 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 6839	. . . . . . . . . . . . 13 ⊢ (𝑓 = (ℝ × {0}) → (𝑓‘𝑧) = ((ℝ × {0})‘𝑧))
35	21	fvconst2 7159	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
36	34, 35	sylan9eq 2791	. . . . . . . . . . . 12 ⊢ ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) = 0)
37	36	iftrued 4474	. . . . . . . . . . 11 ⊢ ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) = 0)
38	37	mpteq2dva 5178	. . . . . . . . . 10 ⊢ (𝑓 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ 0))
39	38	breq1d 5095	. . . . . . . . 9 ⊢ (𝑓 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹))
40	39	rexbidv 3161	. . . . . . . 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 3564	. . . . . 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 2740	. . . . . . . 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 3161	. . . . . 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 3622	. . . . 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 4282	. . 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 13389	. . . . . . 7 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
51		fss 6684	. . . . . . 7 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
52	50, 51	mpan2 692	. . . . . 6 ⊢ (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶(0[,]+∞))
53		eqid 2736	. . . . . . 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 37992	. . . . . 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 2837	. . . 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 11189	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
59	6, 58	sstri 3931	. . . . . 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 13276	. . . . . 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 3053	. . . 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 5145	. . . 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 25652	. . . . . . . 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 2836	. . . . . 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 3130	. . . . 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 4008	. . . 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 7036	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ (0[,)+∞))
74		elrege0 13407	. . . . . . . . . . . 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 3129	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐺‘𝑧))
78	72	feqmptd 6908	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℝ ↦ (𝐺‘𝑧)))
79	20, 22, 73, 23, 78	ofrfval2 7652	. . . . . . . . 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 3133	. . . . . . 7 ⊢ (𝜑 → ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺)
82		r19.2z 4439	. . . . . . 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 6840	. . . . . . . . . 10 ⊢ (𝑔 = (ℝ × {0}) → (∫1‘𝑔) = (∫1‘(ℝ × {0})))
85	84, 31	eqtr2di 2788	. . . . . . . . 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 6839	. . . . . . . . . . . . 13 ⊢ (𝑔 = (ℝ × {0}) → (𝑔‘𝑧) = ((ℝ × {0})‘𝑧))
88	87, 35	sylan9eq 2791	. . . . . . . . . . . 12 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = 0)
89	88	iftrued 4474	. . . . . . . . . . 11 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) = 0)
90	89	mpteq2dva 5178	. . . . . . . . . 10 ⊢ (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ 0))
91	90	breq1d 5095	. . . . . . . . 9 ⊢ (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺))
92	91	rexbidv 3161	. . . . . . . 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 3564	. . . . . 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 2740	. . . . . . . 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 3161	. . . . . 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 3622	. . . . 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 4282	. . 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 6684	. . . . . . 7 ⊢ ((𝐺:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐺:ℝ⟶(0[,]+∞))
103	50, 102	mpan2 692	. . . . . 6 ⊢ (𝐺:ℝ⟶(0[,)+∞) → 𝐺:ℝ⟶(0[,]+∞))
104		eqid 2736	. . . . . . 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 37992	. . . . . 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 2837	. . . 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 3931	. . . . . 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 13276	. . . . . 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 3053	. . . 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 5145	. . . 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 2736	. . 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 12124	. 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 13279	. . . . 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 2771	. . 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 13279	. . . . 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 2771	. . 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 7385	. 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 13413	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞))
125	50, 124	sselid 3919	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,]+∞))
126	125	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,]+∞))
127		inidm 4167	. . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ
128	126, 13, 72, 20, 20, 127	off 7649	. . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶(0[,]+∞))
129		eqid 2736	. . . . 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 37992	. . . 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 37994	. . . . . . 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 25662	. . . . . . . . . . . . 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 3209	. . . . . . . . . . . . . . . . 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 4512	. . . . . . . . . . . . . . . . . . 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 5088	. . . . . . . . . . . . . . . . . . . . . . . 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 5088	. . . . . . . . . . . . . . . . . . . . . . . 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 5088	. . . . . . . . . . . . . . . . . . . . . . . . 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 5088	. . . . . . . . . . . . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓‘𝑧) = 0 ∧ (𝑔‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = (0 + 0))
156		00id 11321	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0 + 0) = 0
157	155, 156	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓‘𝑧) = 0 ∧ (𝑔‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = 0)
158	157	iftrued 4474	. . . . . . . . . . . . . . . . . . . . . . . . . 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 11726	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5107	. . . . . . . . . . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 25643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑔 ∈ dom ∫1 → 𝑔:ℝ⟶ℝ)
174	173	ffvelcdmda 7036	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) ∈ ℂ)
177	176	addlidd 11347	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (0 + (𝑔‘𝑧)) = (𝑔‘𝑧))
178	171, 177	sylan9eqr 2793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11174	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11174	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ)
192		rpre 12951	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑐 ∈ ℝ+ → 𝑐 ∈ ℝ)
193		rpre 12951	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑑 ∈ ℝ+ → 𝑑 ∈ ℝ)
194		min2 13142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11174	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) + 𝑑) ∈ ℝ)
199		letr 11240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11303	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5088	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5088	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4506	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4505	. . . . . . . . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 25643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑓 ∈ dom ∫1 → 𝑓:ℝ⟶ℝ)
226	225	ffvelcdmda 7036	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℂ)
229	228	addridd 11346	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 0) = (𝑓‘𝑧))
230	223, 229	sylan9eqr 2793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11174	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ)
239		min1 13141	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11174	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 𝑐) ∈ ℝ)
244		letr 11240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11303	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 11173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11167	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13019	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) < ((𝑓‘𝑧) + 𝑐))
264	227, 243, 263	ltled 11294	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑐))
265		letr 11240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5107	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4505	. . . . . . . . . . . . . . . . . . . . . 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 4505	. . . . . . . . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . . . . . . . . . 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 7400	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓‘𝑧) + 𝑐) ∈ V
280	21, 279	ifex 4517	. . . . . . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . 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 7652	. . . . . . . . . . . . . . . . . . . . . . 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 7400	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑔‘𝑧) + 𝑑) ∈ V
285	21, 284	ifex 4517	. . . . . . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . 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 7652	. . . . . . . . . . . . . . . . . . . . . . 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 3097	. . . . . . . . . . . . . . . . . . . . . 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 7400	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ V
295	21, 294	ifex 4517	. . . . . . . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) + (𝐺‘𝑧)) ∈ V)
298	225	ffnd 6669	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) = (𝑓‘𝑧))
305		eqidd 2737	. . . . . . . . . . . . . . . . . . . . . . . . 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 7642	. . . . . . . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . . . . . . . . . . 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 4493	. . . . . . . . . . . . . . . . . . . . . 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 5178	. . . . . . . . . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐹 Fn ℝ)
312	72	ffnd 6669	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐺 Fn ℝ)
313		eqidd 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
314		eqidd 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
315	311, 312, 20, 20, 127, 313, 314	offval 7640	. . . . . . . . . . . . . . . . . . . . . 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 7652	. . . . . . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = if(𝑐 ≤ 𝑑, 𝑐, 𝑑) → (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦) = (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))
321	320	ifeq2d 4487	. . . . . . . . . . . . . . . . . . . . 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 5179	. . . . . . . . . . . . . . . . . . . 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 5095	. . . . . . . . . . . . . . . . . . 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 3564	. . . . . . . . . . . . . . . . . 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 3194	. . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . 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 25668	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (∫1‘(𝑓 ∘f + 𝑔)) = ((∫1‘𝑓) + (∫1‘𝑔)))
334	333	eqcomd 2742	. . . . . . . . . . . . . . 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 2793	. . . . . . . . . . . . 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 2756	. . . . . . . . . . . . . 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 6839	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → (ℎ‘𝑧) = ((𝑓 ∘f + 𝑔)‘𝑧))
341	340	eqeq1d 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((ℎ‘𝑧) = 0 ↔ ((𝑓 ∘f + 𝑔)‘𝑧) = 0))
342	340	oveq1d 7382	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((ℎ‘𝑧) + 𝑦) = (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))
343	341, 342	ifbieq2d 4493	. . . . . . . . . . . . . . . . 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 5179	. . . . . . . . . . . . . . . 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 5095	. . . . . . . . . . . . . . 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 3161	. . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → (∫1‘ℎ) = (∫1‘(𝑓 ∘f + 𝑔)))
348	347	eqeq2d 2747	. . . . . . . . . . . . . 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 3564	. . . . . . . . . . . 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 3194	. . . . . . . . 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 2740	. . . . . . . . . 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 3161	. . . . . . . 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 3641	. . . . . . 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 2740	. . . . . . . . . . . . 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 3161	. . . . . . . . . . 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 3641	. . . . . . . . . 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 3209	. . . . . . . . . . . 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 2802	. . . 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 9359	. . 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 3917	. . . . . . . . . . 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 3917	. . . . . . . . . . 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 11174	. . . . . . . . 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 2836	. . . . . . . 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 3179	. . . . . 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 4008	. . . . 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 2745	. . . . . . . 8 ⊢ 0 = (0 + 0)
390		rspceov 7416	. . . . . . . 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 2740	. . . . . . . 8 ⊢ (𝑠 = 0 → (𝑠 = (𝑡 + 𝑢) ↔ 0 = (𝑡 + 𝑢)))
394	393	2rexbidv 3202	. . . . . . 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 3548	. . . . . 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 4325	. . . . 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 11174	. . . . 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 13276	. . . . . . . . . . . . 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 13276	. . . . . . . . . . . . 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 11769	. . . . . . . . . 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 5107	. . . . . . . 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 3194	. . . . . 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 5089	. . . . . . . 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 3160	. . . . . . 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 2740	. . . . . . . . 9 ⊢ (𝑠 = 𝑏 → (𝑠 = (𝑡 + 𝑢) ↔ 𝑏 = (𝑡 + 𝑢)))
420	419	2rexbidv 3202	. . . . . . . 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 3639	. . . . . . 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 3564	. . . . 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 13279	. . . 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 2775	. 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 2782	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 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  ifcif 4466  {csn 4567   class class class wbr 5085   ↦ cmpt 5166   × cxp 5629  dom cdm 5631   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629   ∘_r cofr 7630  supcsup 9353  ℝcr 11037  0cc0 11038   + caddc 11041  +∞cpnf 11176  ℝ^*cxr 11178   < clt 11179   ≤ cle 11180  ℕcn 12174  3c3 12237  ℝ⁺crp 12942  [,)cico 13300  [,]cicc 13301  MblFncmbf 25581  ∫₁citg1 25582  ∫₂citg2 25583

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-disj 5053  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-ofr 7632  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fi 9324  df-sup 9355  df-inf 9356  df-oi 9425  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-q 12899  df-rp 12943  df-xneg 13063  df-xadd 13064  df-xmul 13065  df-ioo 13302  df-ico 13304  df-icc 13305  df-fz 13462  df-fzo 13609  df-fl 13751  df-seq 13964  df-exp 14024  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-clim 15450  df-sum 15649  df-rest 17385  df-topgen 17406  df-psmet 21344  df-xmet 21345  df-met 21346  df-bl 21347  df-mopn 21348  df-top 22859  df-topon 22876  df-bases 22911  df-cmp 23352  df-ovol 25431  df-vol 25432  df-mbf 25586  df-itg1 25587  df-itg2 25588

This theorem is referenced by:  ibladdnclem  37997  itgaddnclem1  37999  iblabsnclem  38004  iblabsnc  38005  iblmulc2nc  38006  ftc1anclem4  38017  ftc1anclem5  38018  ftc1anclem6  38019  ftc1anclem8  38021