Theorem itg2addnc 37681

Description: Alternate proof of itg2add 25794 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 25743, 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 10475, 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 25720	. . . . . . . 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 2841	. . . . . 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 3147	. . . . 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 4070	. . . 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 25722	. . . . . 6 ⊢ (ℝ × {0}) ∈ dom ∫1
9		3nn 12345	. . . . . . . 8 ⊢ 3 ∈ ℕ
10		nnrp 13046	. . . . . . . 8 ⊢ (3 ∈ ℕ → 3 ∈ ℝ+)
11		ne0i 4341	. . . . . . . 8 ⊢ (3 ∈ ℝ+ → ℝ+ ≠ ∅)
12	9, 10, 11	mp2b 10	. . . . . . 7 ⊢ ℝ+ ≠ ∅
13		itg2addnc.f2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
14	13	ffvelcdmda 7104	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ (0[,)+∞))
15		elrege0 13494	. . . . . . . . . . . 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 3146	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐹‘𝑧))
19		reex 11246	. . . . . . . . . . 11 ⊢ ℝ ∈ V
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ V)
21		c0ex 11255	. . . . . . . . . . 11 ⊢ 0 ∈ V
22	21	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ∈ V)
23		eqidd 2738	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
24	13	feqmptd 6977	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹‘𝑧)))
25	20, 22, 14, 23, 24	ofrfval2 7718	. . . . . . . . 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 3150	. . . . . . 7 ⊢ (𝜑 → ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
28		r19.2z 4495	. . . . . . 7 ⊢ ((ℝ+ ≠ ∅ ∧ ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹) → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
29	12, 27, 28	sylancr 587	. . . . . 6 ⊢ (𝜑 → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
30		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑓 = (ℝ × {0}) → (∫1‘𝑓) = (∫1‘(ℝ × {0})))
31		itg10 25723	. . . . . . . . . 10 ⊢ (∫1‘(ℝ × {0})) = 0
32	30, 31	eqtr2di 2794	. . . . . . . . 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 6905	. . . . . . . . . . . . 13 ⊢ (𝑓 = (ℝ × {0}) → (𝑓‘𝑧) = ((ℝ × {0})‘𝑧))
35	21	fvconst2 7224	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
36	34, 35	sylan9eq 2797	. . . . . . . . . . . 12 ⊢ ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) = 0)
37	36	iftrued 4533	. . . . . . . . . . 11 ⊢ ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) = 0)
38	37	mpteq2dva 5242	. . . . . . . . . 10 ⊢ (𝑓 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ 0))
39	38	breq1d 5153	. . . . . . . . 9 ⊢ (𝑓 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹))
40	39	rexbidv 3179	. . . . . . . 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 3622	. . . . . 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 587	. . . . 5 ⊢ (𝜑 → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑓)))
44		eqeq1 2741	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 = (∫1‘𝑓) ↔ 0 = (∫1‘𝑓)))
45	44	anbi2d 630	. . . . . . 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 3179	. . . . . 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 3679	. . . . 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 4342	. . 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 13476	. . . . . . 7 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
51		fss 6752	. . . . . . 7 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
52	50, 51	mpan2 691	. . . . . 6 ⊢ (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶(0[,]+∞))
53		eqid 2737	. . . . . . 7 ⊢ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} = {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}
54	53	itg2addnclem 37678	. . . . . 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 2842	. . . 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 11305	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
59	6, 58	sstri 3993	. . . . . 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 13366	. . . . . 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 690	. . . . 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 3063	. . . 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 5203	. . . 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 586	. . 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 25720	. . . . . . . 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 2841	. . . . . 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 3147	. . . . 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 4070	. . . 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 7104	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ (0[,)+∞))
74		elrege0 13494	. . . . . . . . . . . 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 3146	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐺‘𝑧))
78	72	feqmptd 6977	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℝ ↦ (𝐺‘𝑧)))
79	20, 22, 73, 23, 78	ofrfval2 7718	. . . . . . . . 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 3150	. . . . . . 7 ⊢ (𝜑 → ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺)
82		r19.2z 4495	. . . . . . 7 ⊢ ((ℝ+ ≠ ∅ ∧ ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺) → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺)
83	12, 81, 82	sylancr 587	. . . . . 6 ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺)
84		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑔 = (ℝ × {0}) → (∫1‘𝑔) = (∫1‘(ℝ × {0})))
85	84, 31	eqtr2di 2794	. . . . . . . . 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 6905	. . . . . . . . . . . . 13 ⊢ (𝑔 = (ℝ × {0}) → (𝑔‘𝑧) = ((ℝ × {0})‘𝑧))
88	87, 35	sylan9eq 2797	. . . . . . . . . . . 12 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = 0)
89	88	iftrued 4533	. . . . . . . . . . 11 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) = 0)
90	89	mpteq2dva 5242	. . . . . . . . . 10 ⊢ (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ 0))
91	90	breq1d 5153	. . . . . . . . 9 ⊢ (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺))
92	91	rexbidv 3179	. . . . . . . 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 3622	. . . . . 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 587	. . . . 5 ⊢ (𝜑 → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 = (∫1‘𝑔)))
96		eqeq1 2741	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 = (∫1‘𝑔) ↔ 0 = (∫1‘𝑔)))
97	96	anbi2d 630	. . . . . . 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 3179	. . . . . 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 3679	. . . . 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 4342	. . 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 6752	. . . . . . 7 ⊢ ((𝐺:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐺:ℝ⟶(0[,]+∞))
103	50, 102	mpan2 691	. . . . . 6 ⊢ (𝐺:ℝ⟶(0[,)+∞) → 𝐺:ℝ⟶(0[,]+∞))
104		eqid 2737	. . . . . . 7 ⊢ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}
105	104	itg2addnclem 37678	. . . . . 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 2842	. . . 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 3993	. . . . . 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 13366	. . . . . 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 690	. . . . 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 3063	. . . 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 5203	. . . 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 586	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 ≤ 𝑎)
115		eqid 2737	. . 3 ⊢ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} = {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}
116	7, 49, 64, 71, 101, 114, 115	supadd 12236	. 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 13369	. . . . 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 1373	. . . 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 2777	. . 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 13369	. . . . 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 1373	. . . 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 2777	. . 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 7449	. 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 13500	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞))
125	50, 124	sselid 3981	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,]+∞))
126	125	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,]+∞))
127		inidm 4227	. . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ
128	126, 13, 72, 20, 20, 127	off 7715	. . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶(0[,]+∞))
129		eqid 2737	. . . . 5 ⊢ {𝑠 ∣ ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))} = {𝑠 ∣ ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))}
130	129	itg2addnclem 37678	. . . 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 37680	. . . . . . 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 25730	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑓 ∘f + 𝑔) ∈ dom ∫1)
137	136	ad3antlr 731	. . . . . . . . . . . 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 3229	. . . . . . . . . . . . . . . . 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 747	. . . . . . . . . . . . . . 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 4571	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ+)
142	141	ad2antlr 727	. . . . . . . . . . . . . . . . . 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 5146	. . . . . . . . . . . . . . . . . . . . . . . 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 631	. . . . . . . . . . . . . . . . . . . . . . 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 5146	. . . . . . . . . . . . . . . . . . . . . . . 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 631	. . . . . . . . . . . . . . . . . . . . . . 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 5146	. . . . . . . . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . . . . . . . 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 5146	. . . . . . . . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . . . . . . . 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 7440	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓‘𝑧) = 0 ∧ (𝑔‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = (0 + 0))
156		00id 11436	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0 + 0) = 0
157	155, 156	eqtrdi 2793	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓‘𝑧) = 0 ∧ (𝑔‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = 0)
158	157	iftrued 4533	. . . . . . . . . . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹‘𝑧) ∈ (0[,)+∞) → (𝐹‘𝑧) ∈ ℝ)
162	14, 161	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ ℝ)
163	74	simplbi 497	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11839	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
166	160, 165	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
167	166	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 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 5165	. . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 25711	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑔 ∈ dom ∫1 → 𝑔:ℝ⟶ℝ)
174	173	ffvelcdmda 7104	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) ∈ ℝ)
175	172, 174	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) ∈ ℝ)
176	175	recnd 11289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) ∈ ℂ)
177	176	addlidd 11462	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (0 + (𝑔‘𝑧)) = (𝑔‘𝑧))
178	171, 177	sylan9eqr 2799	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ)
182	181	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ)
183	175, 182	readdcld 11290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ ℝ)
188	187, 185	readdcld 11290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ)
192		rpre 13043	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑐 ∈ ℝ+ → 𝑐 ∈ ℝ)
193		rpre 13043	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑑 ∈ ℝ+ → 𝑑 ∈ ℝ)
194		min2 13232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑑)
195	192, 193, 194	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑑)
196	195	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11878	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) + 𝑑) ∈ ℝ)
199		letr 11355	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11852	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5165	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5146	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5146	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4565	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4564	. . . . . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 25711	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑓 ∈ dom ∫1 → 𝑓:ℝ⟶ℝ)
226	225	ffvelcdmda 7104	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
227	224, 226	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
228	227	recnd 11289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℂ)
229	228	addridd 11461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 0) = (𝑓‘𝑧))
230	223, 229	sylan9eqr 2799	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ)
239		min1 13231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑐)
240	192, 193, 239	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑐)
241	240	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11878	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 𝑐) ∈ ℝ)
244		letr 11355	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5165	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . . . . . . 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 11289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11283	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13110	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) < ((𝑓‘𝑧) + 𝑐))
264	227, 243, 263	ltled 11409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑐))
265		letr 11355	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5165	. . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . 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 4564	. . . . . . . . . . . . . . . . . . . . . 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 4564	. . . . . . . . . . . . . . . . . . . . 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 3167	. . . . . . . . . . . . . . . . . . . 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 7464	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓‘𝑧) + 𝑐) ∈ V
280	21, 279	ifex 4576	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ∈ V
281	280	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ∈ V)
282		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))))
283	20, 281, 14, 282, 24	ofrfval2 7718	. . . . . . . . . . . . . . . . . . . . . . 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 7464	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑔‘𝑧) + 𝑑) ∈ V
285	21, 284	ifex 4576	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ∈ V
286	285	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ∈ V)
287		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))))
288	20, 286, 73, 287, 78	ofrfval2 7718	. . . . . . . . . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . . . . . . . . . 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 3111	. . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . 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 7464	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ V
295	21, 294	ifex 4576	. . . . . . . . . . . . . . . . . . . . . 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 7466	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) + (𝐺‘𝑧)) ∈ V)
298	225	ffnd 6737	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 Fn ℝ)
299	298	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 𝑓 Fn ℝ)
300	299	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑓 Fn ℝ)
301	173	ffnd 6737	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑔 ∈ dom ∫1 → 𝑔 Fn ℝ)
302	301	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 𝑔 Fn ℝ)
303	302	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑔 Fn ℝ)
304		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) = (𝑓‘𝑧))
305		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = (𝑔‘𝑧))
306	300, 303, 293, 293, 127, 304, 305	ofval 7708	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓 ∘f + 𝑔)‘𝑧) = ((𝑓‘𝑧) + (𝑔‘𝑧)))
307	306	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓 ∘f + 𝑔)‘𝑧) = 0 ↔ ((𝑓‘𝑧) + (𝑔‘𝑧)) = 0))
308	306	oveq1d 7446	. . . . . . . . . . . . . . . . . . . . . . 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 4552	. . . . . . . . . . . . . . . . . . . . . 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 5242	. . . . . . . . . . . . . . . . . . . . 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 6737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐹 Fn ℝ)
312	72	ffnd 6737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐺 Fn ℝ)
313		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
314		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
315	311, 312, 20, 20, 127, 313, 314	offval 7706	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧) + (𝐺‘𝑧))))
316	315	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 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 7718	. . . . . . . . . . . . . . . . . . . 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 7439	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = if(𝑐 ≤ 𝑑, 𝑐, 𝑑) → (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦) = (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))
321	320	ifeq2d 4546	. . . . . . . . . . . . . . . . . . . . 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 5244	. . . . . . . . . . . . . . . . . . . 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 5153	. . . . . . . . . . . . . . . . . . 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 3622	. . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . 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 3213	. . . . . . . . . . . . . . 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 7440	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 = (∫1‘𝑓) ∧ 𝑢 = (∫1‘𝑔)) → (𝑡 + 𝑢) = ((∫1‘𝑓) + (∫1‘𝑔)))
332	331	ad2ant2l 746	. . . . . . . . . . . . . 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 25736	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (∫1‘(𝑓 ∘f + 𝑔)) = ((∫1‘𝑓) + (∫1‘𝑔)))
334	333	eqcomd 2743	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ((∫1‘𝑓) + (∫1‘𝑔)) = (∫1‘(𝑓 ∘f + 𝑔)))
335	334	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) → ((∫1‘𝑓) + (∫1‘𝑔)) = (∫1‘(𝑓 ∘f + 𝑔)))
336	332, 335	sylan9eqr 2799	. . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . 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 580	. . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → (ℎ‘𝑧) = ((𝑓 ∘f + 𝑔)‘𝑧))
341	340	eqeq1d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((ℎ‘𝑧) = 0 ↔ ((𝑓 ∘f + 𝑔)‘𝑧) = 0))
342	340	oveq1d 7446	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((ℎ‘𝑧) + 𝑦) = (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))
343	341, 342	ifbieq2d 4552	. . . . . . . . . . . . . . . . 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 5244	. . . . . . . . . . . . . . . 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 5153	. . . . . . . . . . . . . . 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 3179	. . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → (∫1‘ℎ) = (∫1‘(𝑓 ∘f + 𝑔)))
348	347	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → (𝑠 = (∫1‘ℎ) ↔ 𝑠 = (∫1‘(𝑓 ∘f + 𝑔))))
349	346, 348	anbi12d 632	. . . . . . . . . . . . 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 3622	. . . . . . . . . . . 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 3213	. . . . . . . . 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 1934	. . . . . . 7 ⊢ (𝜑 → (∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))))
356	133, 355	impbid 212	. . . . . 6 ⊢ (𝜑 → (∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)) ↔ ∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))))
357		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (𝑥 = (∫1‘𝑓) ↔ 𝑡 = (∫1‘𝑓)))
358	357	anbi2d 630	. . . . . . . . 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 3179	. . . . . . . 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 3700	. . . . . . 7 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)))
361		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑥 = (∫1‘𝑔) ↔ 𝑢 = (∫1‘𝑔)))
362	361	anbi2d 630	. . . . . . . . . . . 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 3179	. . . . . . . . . . 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 3700	. . . . . . . . . 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 623	. . . . . . . . 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 1953	. . . . . . . . 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 3229	. . . . . . . . . . . 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 624	. . . . . . . . . . 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 1848	. . . . . . . . 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 1848	. . . . . . 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 2808	. . . 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 9486	. . 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 3979	. . . . . . . . . . 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 726	. . . . . . . . . 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 3979	. . . . . . . . . . 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 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)) → 𝑢 ∈ ℝ)
383	380, 382	readdcld 11290	. . . . . . . . 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 2841	. . . . . . . 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 3201	. . . . . 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 4070	. . . . 5 ⊢ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ
388	387	a1i 11	. . . 4 ⊢ (𝜑 → {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ)
389	156	eqcomi 2746	. . . . . . . 8 ⊢ 0 = (0 + 0)
390		rspceov 7480	. . . . . . . 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 1452	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}0 = (𝑡 + 𝑢))
393		eqeq1 2741	. . . . . . . 8 ⊢ (𝑠 = 0 → (𝑠 = (𝑡 + 𝑢) ↔ 0 = (𝑡 + 𝑢)))
394	393	2rexbidv 3222	. . . . . . 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 3606	. . . . . 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 4385	. . . . 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 11290	. . . . 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 728	. . . . . . . . . . 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 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))})) → 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 13366	. . . . . . . . . . . . 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 690	. . . . . . . . . . . 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 728	. . . . . . . . . . 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 13366	. . . . . . . . . . . . 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 690	. . . . . . . . . . . 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 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))})) → 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 11882	. . . . . . . . . 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 5165	. . . . . . . 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 3213	. . . . . 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 1927	. . . . 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 5147	. . . . . . . 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 3178	. . . . . . 7 ⊢ (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )) → (∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ 𝑎 ↔ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) + sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))))
419		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑠 = 𝑏 → (𝑠 = (𝑡 + 𝑢) ↔ 𝑏 = (𝑡 + 𝑢)))
420	419	2rexbidv 3222	. . . . . . . 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 3697	. . . . . . 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 3622	. . . . 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 584	. . . 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 13369	. . . 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 1373	. . 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 2781	. 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 2788	1 ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  ∅c0 4333  ifcif 4525  {csn 4626   class class class wbr 5143   ↦ cmpt 5225   × cxp 5683  dom cdm 5685   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∘_f cof 7695   ∘_r cofr 7696  supcsup 9480  ℝcr 11154  0cc0 11155   + caddc 11158  +∞cpnf 11292  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296  ℕcn 12266  3c3 12322  ℝ⁺crp 13034  [,)cico 13389  [,]cicc 13390  MblFncmbf 25649  ∫₁citg1 25650  ∫₂citg2 25651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500  df-mbf 25654  df-itg1 25655  df-itg2 25656

This theorem is referenced by:  ibladdnclem  37683  itgaddnclem1  37685  iblabsnclem  37690  iblabsnc  37691  iblmulc2nc  37692  ftc1anclem4  37703  ftc1anclem5  37704  ftc1anclem6  37705  ftc1anclem8  37707