Theorem itg2addnc 34950

Description: Alternate proof of itg2add 24363 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 24312, 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 9860, 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 771	. . . . . . 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 24289	. . . . . . . 8 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘𝑓) ∈ ℝ)
3	2	adantr 483	. . . . . . 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 2916	. . . . . 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 3284	. . . . 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 4049	. . . 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 24291	. . . . . 6 ⊢ (ℝ × {0}) ∈ dom ∫1
9		3nn 11719	. . . . . . . 8 ⊢ 3 ∈ ℕ
10		nnrp 12403	. . . . . . . 8 ⊢ (3 ∈ ℕ → 3 ∈ ℝ+)
11		ne0i 4303	. . . . . . . 8 ⊢ (3 ∈ ℝ+ → ℝ+ ≠ ∅)
12	9, 10, 11	mp2b 10	. . . . . . 7 ⊢ ℝ+ ≠ ∅
13		itg2addnc.f2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
14	13	ffvelrnda 6854	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ (0[,)+∞))
15		elrege0 12845	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) ∈ (0[,)+∞) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑧)))
16	14, 15	sylib 220	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑧)))
17	16	simprd 498	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹‘𝑧))
18	17	ralrimiva 3185	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐹‘𝑧))
19		reex 10631	. . . . . . . . . . 11 ⊢ ℝ ∈ V
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ V)
21		c0ex 10638	. . . . . . . . . . 11 ⊢ 0 ∈ V
22	21	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ∈ V)
23		eqidd 2825	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
24	13	feqmptd 6736	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹‘𝑧)))
25	20, 22, 14, 23, 24	ofrfval2 7430	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐹‘𝑧)))
26	18, 25	mpbird 259	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
27	26	ralrimivw 3186	. . . . . . 7 ⊢ (𝜑 → ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
28		r19.2z 4443	. . . . . . 7 ⊢ ((ℝ+ ≠ ∅ ∧ ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹) → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
29	12, 27, 28	sylancr 589	. . . . . 6 ⊢ (𝜑 → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹)
30		fveq2 6673	. . . . . . . . . 10 ⊢ (𝑓 = (ℝ × {0}) → (∫1‘𝑓) = (∫1‘(ℝ × {0})))
31		itg10 24292	. . . . . . . . . 10 ⊢ (∫1‘(ℝ × {0})) = 0
32	30, 31	syl6req 2876	. . . . . . . . 9 ⊢ (𝑓 = (ℝ × {0}) → 0 = (∫1‘𝑓))
33	32	biantrud 534	. . . . . . . 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 6672	. . . . . . . . . . . . 13 ⊢ (𝑓 = (ℝ × {0}) → (𝑓‘𝑧) = ((ℝ × {0})‘𝑧))
35	21	fvconst2 6969	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
36	34, 35	sylan9eq 2879	. . . . . . . . . . . 12 ⊢ ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) = 0)
37	36	iftrued 4478	. . . . . . . . . . 11 ⊢ ((𝑓 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) = 0)
38	37	mpteq2dva 5164	. . . . . . . . . 10 ⊢ (𝑓 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ 0))
39	38	breq1d 5079	. . . . . . . . 9 ⊢ (𝑓 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐹))
40	39	rexbidv 3300	. . . . . . . 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 283	. . . . . . 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 3626	. . . . . 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 589	. . . . 5 ⊢ (𝜑 → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 = (∫1‘𝑓)))
44		eqeq1 2828	. . . . . . . 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 3300	. . . . . 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 3670	. . . . 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 236	. . . 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 4304	. . 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 12827	. . . . . . 7 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
51		fss 6530	. . . . . . 7 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
52	50, 51	mpan2 689	. . . . . 6 ⊢ (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶(0[,]+∞))
53		eqid 2824	. . . . . . 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 34947	. . . . . 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 2917	. . . 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 10688	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
59	6, 58	sstri 3979	. . . . . 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 12720	. . . . . 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 688	. . . . 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 3151	. . . 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 5129	. . . 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 588	. . 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 771	. . . . . . 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 24289	. . . . . . . 8 ⊢ (𝑔 ∈ dom ∫1 → (∫1‘𝑔) ∈ ℝ)
67	66	adantr 483	. . . . . . 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 2916	. . . . . 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 3284	. . . . 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 4049	. . . 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	ffvelrnda 6854	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ (0[,)+∞))
74		elrege0 12845	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑧) ∈ (0[,)+∞) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑧)))
75	73, 74	sylib 220	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((𝐺‘𝑧) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑧)))
76	75	simprd 498	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐺‘𝑧))
77	76	ralrimiva 3185	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐺‘𝑧))
78	72	feqmptd 6736	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℝ ↦ (𝐺‘𝑧)))
79	20, 22, 73, 23, 78	ofrfval2 7430	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐺‘𝑧)))
80	77, 79	mpbird 259	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺)
81	80	ralrimivw 3186	. . . . . . 7 ⊢ (𝜑 → ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺)
82		r19.2z 4443	. . . . . . 7 ⊢ ((ℝ+ ≠ ∅ ∧ ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺) → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺)
83	12, 81, 82	sylancr 589	. . . . . 6 ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺)
84		fveq2 6673	. . . . . . . . . 10 ⊢ (𝑔 = (ℝ × {0}) → (∫1‘𝑔) = (∫1‘(ℝ × {0})))
85	84, 31	syl6req 2876	. . . . . . . . 9 ⊢ (𝑔 = (ℝ × {0}) → 0 = (∫1‘𝑔))
86	85	biantrud 534	. . . . . . . 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 6672	. . . . . . . . . . . . 13 ⊢ (𝑔 = (ℝ × {0}) → (𝑔‘𝑧) = ((ℝ × {0})‘𝑧))
88	87, 35	sylan9eq 2879	. . . . . . . . . . . 12 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = 0)
89	88	iftrued 4478	. . . . . . . . . . 11 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) = 0)
90	89	mpteq2dva 5164	. . . . . . . . . 10 ⊢ (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ 0))
91	90	breq1d 5079	. . . . . . . . 9 ⊢ (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ↔ (𝑧 ∈ ℝ ↦ 0) ∘r ≤ 𝐺))
92	91	rexbidv 3300	. . . . . . . 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 283	. . . . . . 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 3626	. . . . . 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 589	. . . . 5 ⊢ (𝜑 → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 = (∫1‘𝑔)))
96		eqeq1 2828	. . . . . . . 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 3300	. . . . . 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 3670	. . . . 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 236	. . . 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 4304	. . 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 6530	. . . . . . 7 ⊢ ((𝐺:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐺:ℝ⟶(0[,]+∞))
103	50, 102	mpan2 689	. . . . . 6 ⊢ (𝐺:ℝ⟶(0[,)+∞) → 𝐺:ℝ⟶(0[,]+∞))
104		eqid 2824	. . . . . . 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 34947	. . . . . 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 2917	. . . 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 3979	. . . . . 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 12720	. . . . . 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 688	. . . . 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 3151	. . . 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 5129	. . . 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 588	. . 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 2824	. . 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 11612	. 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 12723	. . . . 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 1367	. . . 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 2859	. . 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 12723	. . . . 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 1367	. . . 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 2859	. . 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 7177	. 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 12851	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞))
125	50, 124	sseldi 3968	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,]+∞))
126	125	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,]+∞))
127		inidm 4198	. . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ
128	126, 13, 72, 20, 20, 127	off 7427	. . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶(0[,]+∞))
129		eqid 2824	. . . . 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 34947	. . . 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 34949	. . . . . . 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 485	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 𝑓 ∈ dom ∫1)
135		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 𝑔 ∈ dom ∫1)
136	134, 135	i1fadd 24299	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑓 ∘f + 𝑔) ∈ dom ∫1)
137	136	ad3antlr 729	. . . . . . . . . . . 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 3370	. . . . . . . . . . . . . . . . 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 230	. . . . . . . . . . . . . . . 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 745	. . . . . . . . . . . . . . 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 4514	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ+)
142	141	ad2antlr 725	. . . . . . . . . . . . . . . . . 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 5072	. . . . . . . . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . . . . . . 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 5072	. . . . . . . . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . . . . . . 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 5072	. . . . . . . . . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . . . . . . . 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 5072	. . . . . . . . . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . . . . . . . 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 7168	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓‘𝑧) = 0 ∧ (𝑔‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = (0 + 0))
156		00id 10818	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0 + 0) = 0
157	155, 156	syl6eq 2875	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓‘𝑧) = 0 ∧ (𝑔‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = 0)
158	157	iftrued 4478	. . . . . . . . . . . . . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝜑)
161	15	simplbi 500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹‘𝑧) ∈ (0[,)+∞) → (𝐹‘𝑧) ∈ ℝ)
162	14, 161	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ ℝ)
163	74	simplbi 500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11219	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
166	160, 165	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
167	166	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . 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 5091	. . . . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7166	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓‘𝑧) = 0 → ((𝑓‘𝑧) + (𝑔‘𝑧)) = (0 + (𝑔‘𝑧)))
172		simplrr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑔 ∈ dom ∫1)
173		i1ff 24280	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑔 ∈ dom ∫1 → 𝑔:ℝ⟶ℝ)
174	173	ffvelrnda 6854	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) ∈ ℝ)
175	172, 174	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) ∈ ℝ)
176	175	recnd 10672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) ∈ ℂ)
177	176	addid2d 10844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (0 + (𝑔‘𝑧)) = (𝑔‘𝑧))
178	171, 177	sylan9eqr 2881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7174	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ)
182	181	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ)
183	175, 182	readdcld 10673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ℝ)
186	185	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ ℝ)
188	187, 185	readdcld 10673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) + (𝐺‘𝑧)) ∈ ℝ)
189	188	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ+)
191	190	rpred 12434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ)
192		rpre 12400	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑐 ∈ ℝ+ → 𝑐 ∈ ℝ)
193		rpre 12400	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑑 ∈ ℝ+ → 𝑑 ∈ ℝ)
194		min2 12586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑑)
195	192, 193, 194	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑑)
196	195	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) + 𝑑) ∈ ℝ)
199		letr 10737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (0 ≤ (𝐹‘𝑧) ↔ (𝐺‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
204	17, 203	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
205	160, 204	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
206	205	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10800	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5091	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5072	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5072	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4508	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . 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 4507	. . . . . . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑔‘𝑧) = 0 → ((𝑓‘𝑧) + (𝑔‘𝑧)) = ((𝑓‘𝑧) + 0))
224		simplrl 775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑓 ∈ dom ∫1)
225		i1ff 24280	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑓 ∈ dom ∫1 → 𝑓:ℝ⟶ℝ)
226	225	ffvelrnda 6854	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
227	224, 226	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
228	227	recnd 10672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℂ)
229	228	addid1d 10843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 0) = (𝑓‘𝑧))
230	223, 229	sylan9eqr 2881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7174	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ+)
238	237	rpred 12434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ)
239		min1 12585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑐)
240	192, 193, 239	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑐)
241	240	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 𝑐) ∈ ℝ)
244		letr 10737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (0 ≤ (𝐺‘𝑧) ↔ (𝐹‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
249	76, 248	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
250	160, 249	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))
251	250	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10800	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5091	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . 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 10672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10666	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 12467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) < ((𝑓‘𝑧) + 𝑐))
264	227, 243, 263	ltled 10791	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑐))
265		letr 10737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧)) → ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
270	267, 201, 269	syl2and 609	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))
271	270	imp 409	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5091	. . . . . . . . . . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . . 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 4507	. . . . . . . . . . . . . . . . . . . . . 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 4507	. . . . . . . . . . . . . . . . . . . . 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 3180	. . . . . . . . . . . . . . . . . . . 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 7192	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓‘𝑧) + 𝑐) ∈ V
280	21, 279	ifex 4518	. . . . . . . . . . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . . . . . . . . . . . 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 7192	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑔‘𝑧) + 𝑑) ∈ V
285	21, 284	ifex 4518	. . . . . . . . . . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . . . . . . . . . . . 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 3173	. . . . . . . . . . . . . . . . . . . . . 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	syl6bbr 291	. . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . 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 7192	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ V
295	21, 294	ifex 4518	. . . . . . . . . . . . . . . . . . . . . 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 7194	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) + (𝐺‘𝑧)) ∈ V)
298	225	ffnd 6518	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 Fn ℝ)
299	298	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 𝑓 Fn ℝ)
300	299	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑓 Fn ℝ)
301	173	ffnd 6518	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑔 ∈ dom ∫1 → 𝑔 Fn ℝ)
302	301	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 𝑔 Fn ℝ)
303	302	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑔 Fn ℝ)
304		eqidd 2825	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) = (𝑓‘𝑧))
305		eqidd 2825	. . . . . . . . . . . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . . . . . . . . . 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 7174	. . . . . . . . . . . . . . . . . . . . . . 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 4495	. . . . . . . . . . . . . . . . . . . . . 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 5164	. . . . . . . . . . . . . . . . . . . . 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 6518	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐹 Fn ℝ)
312	72	ffnd 6518	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐺 Fn ℝ)
313		eqidd 2825	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
314		eqidd 2825	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
315	311, 312, 20, 20, 127, 313, 314	offval 7419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧) + (𝐺‘𝑧))))
316	315	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . . . . . . . . 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 296	. . . . . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . . . . 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 7167	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = if(𝑐 ≤ 𝑑, 𝑐, 𝑑) → (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦) = (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))
321	320	ifeq2d 4489	. . . . . . . . . . . . . . . . . . . . 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 5165	. . . . . . . . . . . . . . . . . . . 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 5079	. . . . . . . . . . . . . . . . . . 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 3626	. . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . 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 3297	. . . . . . . . . . . . . . 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 420	. . . . . . . . . . . 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 7168	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 = (∫1‘𝑓) ∧ 𝑢 = (∫1‘𝑔)) → (𝑡 + 𝑢) = ((∫1‘𝑓) + (∫1‘𝑔)))
332	331	ad2ant2l 744	. . . . . . . . . . . . . 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 24305	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (∫1‘(𝑓 ∘f + 𝑔)) = ((∫1‘𝑓) + (∫1‘𝑔)))
334	333	eqcomd 2830	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ((∫1‘𝑓) + (∫1‘𝑔)) = (∫1‘(𝑓 ∘f + 𝑔)))
335	334	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) → ((∫1‘𝑓) + (∫1‘𝑔)) = (∫1‘(𝑓 ∘f + 𝑔)))
336	332, 335	sylan9eqr 2881	. . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . 14 ⊢ ((𝑠 = (𝑡 + 𝑢) ∧ (𝑡 + 𝑢) = (∫1‘(𝑓 ∘f + 𝑔))) → 𝑠 = (∫1‘(𝑓 ∘f + 𝑔)))
338	337	ancoms 461	. . . . . . . . . . . . 13 ⊢ (((𝑡 + 𝑢) = (∫1‘(𝑓 ∘f + 𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓 ∘f + 𝑔)))
339	336, 338	sylan 582	. . . . . . . . . . . 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 6672	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → (ℎ‘𝑧) = ((𝑓 ∘f + 𝑔)‘𝑧))
341	340	eqeq1d 2826	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((ℎ‘𝑧) = 0 ↔ ((𝑓 ∘f + 𝑔)‘𝑧) = 0))
342	340	oveq1d 7174	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((ℎ‘𝑧) + 𝑦) = (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))
343	341, 342	ifbieq2d 4495	. . . . . . . . . . . . . . . . 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 5165	. . . . . . . . . . . . . . . 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 5079	. . . . . . . . . . . . . . 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 3300	. . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑓 ∘f + 𝑔) → (∫1‘ℎ) = (∫1‘(𝑓 ∘f + 𝑔)))
348	347	eqeq2d 2835	. . . . . . . . . . . . . 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 3626	. . . . . . . . . . . 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 834	. . . . . . . . . . 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 422	. . . . . . . . . 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 3297	. . . . . . . . 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 413	. . . . . . . 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 214	. . . . . 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 2828	. . . . . . . . . 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 3300	. . . . . . . 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 3689	. . . . . . 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 2828	. . . . . . . . . . . . 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 3300	. . . . . . . . . . 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 3689	. . . . . . . . . 10 ⊢ (∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)))
365	364	anbi2i 624	. . . . . . . . 9 ⊢ ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))))
366		19.42v 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 3370	. . . . . . . . . . . 12 ⊢ (∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))))
368	367	anbi1i 625	. . . . . . . . . . 11 ⊢ ((∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) ↔ ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))
369		anass 471	. . . . . . . . . . 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 278	. . . . . . . . . 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 1847	. . . . . . . . 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 301	. . . . . . . 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 1847	. . . . . . 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 277	. . . . . 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	syl6bbr 291	. . . . 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 2888	. . . 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 8913	. . 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 487	. . . . . . . . 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 3966	. . . . . . . . . . 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 724	. . . . . . . . . 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 3966	. . . . . . . . . . 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 725	. . . . . . . . . 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 10673	. . . . . . . . 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 2916	. . . . . . . 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 415	. . . . . . 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 3295	. . . . . 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 4049	. . . . 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 2833	. . . . . . . 8 ⊢ 0 = (0 + 0)
390		rspceov 7206	. . . . . . . 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 1446	. . . . . . 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 586	. . . . . 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 2828	. . . . . . . 8 ⊢ (𝑠 = 0 → (𝑠 = (𝑡 + 𝑢) ↔ 0 = (𝑡 + 𝑢)))
394	393	2rexbidv 3303	. . . . . . 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 3610	. . . . . 6 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}0 = (𝑡 + 𝑢) → ∃𝑠∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢))
396	392, 395	syl 17	. . . . 5 ⊢ (𝜑 → ∃𝑠∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢))
397		abn0 4339	. . . . 5 ⊢ ({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅ ↔ ∃𝑠∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢))
398	396, 397	sylibr 236	. . . 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 10673	. . . . 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 487	. . . . . . . . 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 726	. . . . . . . . . . 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 727	. . . . . . . . . . 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 483	. . . . . . . . . . 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 483	. . . . . . . . . . 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 12720	. . . . . . . . . . . . 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 688	. . . . . . . . . . . 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 726	. . . . . . . . . . 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 12720	. . . . . . . . . . . . 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 688	. . . . . . . . . . . 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 727	. . . . . . . . . . 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 11262	. . . . . . . . . 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 483	. . . . . . . . 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 5091	. . . . . . . 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 415	. . . . . . 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 3297	. . . . . 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 5073	. . . . . . . 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 3200	. . . . . . 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 2828	. . . . . . . . 9 ⊢ (𝑠 = 𝑏 → (𝑠 = (𝑡 + 𝑢) ↔ 𝑏 = (𝑡 + 𝑢)))
420	419	2rexbidv 3303	. . . . . . . 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 3687	. . . . . . 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	syl6bb 289	. . . . . 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 3626	. . . . 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 586	. . . 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 12723	. . . 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 1367	. . 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 2863	. 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 2870	1 ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534   = wceq 1536  ∃wex 1779   ∈ wcel 2113  {cab 2802   ≠ wne 3019  ∀wral 3141  ∃wrex 3142  Vcvv 3497   ⊆ wss 3939  ∅c0 4294  ifcif 4470  {csn 4570   class class class wbr 5069   ↦ cmpt 5149   × cxp 5556  dom cdm 5558   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159   ∘_f cof 7410   ∘_r cofr 7411  supcsup 8907  ℝcr 10539  0cc0 10540   + caddc 10543  +∞cpnf 10675  ℝ^*cxr 10677   < clt 10678   ≤ cle 10679  ℕcn 11641  3c3 11696  ℝ⁺crp 12392  [,)cico 12743  [,]cicc 12744  MblFncmbf 24218  ∫₁citg1 24219  ∫₂citg2 24220

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-inf2 9107  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618  ax-addf 10619

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-disj 5035  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412  df-ofr 7413  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fi 8878  df-sup 8909  df-inf 8910  df-oi 8977  df-dju 9333  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-nn 11642  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-ioo 12745  df-ico 12747  df-icc 12748  df-fz 12896  df-fzo 13037  df-fl 13165  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14461  df-re 14462  df-im 14463  df-sqrt 14597  df-abs 14598  df-clim 14848  df-sum 15046  df-rest 16699  df-topgen 16720  df-psmet 20540  df-xmet 20541  df-met 20542  df-bl 20543  df-mopn 20544  df-top 21505  df-topon 21522  df-bases 21557  df-cmp 21998  df-ovol 24068  df-vol 24069  df-mbf 24223  df-itg1 24224  df-itg2 24225

This theorem is referenced by:  ibladdnclem  34952  itgaddnclem1  34954  iblabsnclem  34959  iblabsnc  34960  iblmulc2nc  34961  ftc1anclem4  34974  ftc1anclem5  34975  ftc1anclem6  34976  ftc1anclem8  34978