itgsubsticclem - Mathbox for Glauco Siliprandi

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Theorem itgsubsticclem 46425

Description: lemma for itgsubsticc 46426. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
itgsubsticclem.1	⊢ 𝐹 = (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)
itgsubsticclem.2	⊢ 𝐺 = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))))
itgsubsticclem.3	⊢ (𝜑 → 𝑋 ∈ ℝ)
itgsubsticclem.4	⊢ (𝜑 → 𝑌 ∈ ℝ)
itgsubsticclem.5	⊢ (𝜑 → 𝑋 ≤ 𝑌)
itgsubsticclem.6	⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)))
itgsubsticclem.7	⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿¹))
itgsubsticclem.8	⊢ (𝜑 → 𝐹 ∈ ((𝐾[,]𝐿)–cn→ℂ))
itgsubsticclem.9	⊢ (𝜑 → 𝐾 ∈ ℝ)
itgsubsticclem.10	⊢ (𝜑 → 𝐿 ∈ ℝ)
itgsubsticclem.11	⊢ (𝜑 → 𝐾 ≤ 𝐿)
itgsubsticclem.12	⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
itgsubsticclem.13	⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸)
itgsubsticclem.14	⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾)
itgsubsticclem.15	⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿)

**Assertion**
Ref	Expression
itgsubsticclem	⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)

Distinct variable groups: 𝑢,𝐴 𝑢,𝐸 𝑥,𝐺 𝑢,𝐾,𝑥 𝑢,𝐿,𝑥 𝑢,𝑋,𝑥 𝑢,𝑌,𝑥 𝜑,𝑢,𝑥 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥,𝑢) 𝐶(𝑥,𝑢) 𝐸(𝑥) 𝐹(𝑥,𝑢) 𝐺(𝑢)

Proof of Theorem itgsubsticclem Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6834	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐺‘𝑢) = (𝐺‘𝑤))
2		nfcv 2902	. . . 4 ⊢ Ⅎ𝑤(𝐺‘𝑢)
3		itgsubsticclem.2	. . . . . 6 ⊢ 𝐺 = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))))
4		nfmpt1 5178	. . . . . 6 ⊢ Ⅎ𝑢(𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))))
5	3, 4	nfcxfr 2900	. . . . 5 ⊢ Ⅎ𝑢𝐺
6		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑢𝑤
7	5, 6	nffv 6844	. . . 4 ⊢ Ⅎ𝑢(𝐺‘𝑤)
8	1, 2, 7	cbvditg 25846	. . 3 ⊢ ⨜[𝐾 → 𝐿](𝐺‘𝑢) d𝑢 = ⨜[𝐾 → 𝐿](𝐺‘𝑤) d𝑤
9		itgsubsticclem.11	. . . 4 ⊢ (𝜑 → 𝐾 ≤ 𝐿)
10		itgsubsticclem.9	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℝ)
11		itgsubsticclem.10	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ℝ)
12	10, 11	iccssred 13385	. . . . . . . 8 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ ℝ)
13	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐾[,]𝐿) ⊆ ℝ)
14		ioossicc 13384	. . . . . . . . 9 ⊢ (𝐾(,)𝐿) ⊆ (𝐾[,]𝐿)
15	14	sseli 3918	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐾(,)𝐿) → 𝑢 ∈ (𝐾[,]𝐿))
16	15	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ (𝐾[,]𝐿))
17	13, 16	sseldd 3923	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ ℝ)
18	16	iftrued 4469	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) = (𝐹‘𝑢))
19		itgsubsticclem.1	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶))
21		itgsubsticclem.8	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ ((𝐾[,]𝐿)–cn→ℂ))
22		cncff 24885	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ ((𝐾[,]𝐿)–cn→ℂ) → 𝐹:(𝐾[,]𝐿)⟶ℂ)
23	21, 22	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(𝐾[,]𝐿)⟶ℂ)
24	20, 23	feq1dd 6645	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶):(𝐾[,]𝐿)⟶ℂ)
25	24	fvmptelcdm 7061	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾[,]𝐿)) → 𝐶 ∈ ℂ)
26	16, 25	syldan 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝐶 ∈ ℂ)
27	19	fvmpt2 6954	. . . . . . . . 9 ⊢ ((𝑢 ∈ (𝐾[,]𝐿) ∧ 𝐶 ∈ ℂ) → (𝐹‘𝑢) = 𝐶)
28	16, 26, 27	syl2anc 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐹‘𝑢) = 𝐶)
29	28, 26	eqeltrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐹‘𝑢) ∈ ℂ)
30	18, 29	eqeltrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) ∈ ℂ)
31	3	fvmpt2 6954	. . . . . 6 ⊢ ((𝑢 ∈ ℝ ∧ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) ∈ ℂ) → (𝐺‘𝑢) = if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))))
32	17, 30, 31	syl2anc 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐺‘𝑢) = if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))))
33	32, 18, 28	3eqtrd 2779	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐺‘𝑢) = 𝐶)
34	9, 33	ditgeq3d 46414	. . 3 ⊢ (𝜑 → ⨜[𝐾 → 𝐿](𝐺‘𝑢) d𝑢 = ⨜[𝐾 → 𝐿]𝐶 d𝑢)
35		itgsubsticclem.3	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ)
36		itgsubsticclem.4	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ℝ)
37		itgsubsticclem.5	. . . 4 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
38		mnfxr 11200	. . . . 5 ⊢ -∞ ∈ ℝ^*
39	38	a1i 11	. . . 4 ⊢ (𝜑 → -∞ ∈ ℝ^*)
40		pnfxr 11197	. . . . 5 ⊢ +∞ ∈ ℝ^*
41	40	a1i 11	. . . 4 ⊢ (𝜑 → +∞ ∈ ℝ^*)
42		ioomax 13373	. . . . . . . . 9 ⊢ (-∞(,)+∞) = ℝ
43	42	eqcomi 2749	. . . . . . . 8 ⊢ ℝ = (-∞(,)+∞)
44	43	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ = (-∞(,)+∞))
45	12, 44	sseqtrd 3958	. . . . . 6 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ (-∞(,)+∞))
46		ax-resscn 11093	. . . . . . 7 ⊢ ℝ ⊆ ℂ
47	44, 46	eqsstrrdi 3967	. . . . . 6 ⊢ (𝜑 → (-∞(,)+∞) ⊆ ℂ)
48		cncfss 24891	. . . . . 6 ⊢ (((𝐾[,]𝐿) ⊆ (-∞(,)+∞) ∧ (-∞(,)+∞) ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) ⊆ ((𝑋[,]𝑌)–cn→(-∞(,)+∞)))
49	45, 47, 48	syl2anc 590	. . . . 5 ⊢ (𝜑 → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) ⊆ ((𝑋[,]𝑌)–cn→(-∞(,)+∞)))
50		itgsubsticclem.6	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)))
51	49, 50	sseldd 3923	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(-∞(,)+∞)))
52		itgsubsticclem.7	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿¹))
53		nfmpt1 5178	. . . . . . . . . . 11 ⊢ Ⅎ𝑢(𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)
54	19, 53	nfcxfr 2900	. . . . . . . . . 10 ⊢ Ⅎ𝑢𝐹
55		eqid 2740	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
56		eqid 2740	. . . . . . . . . 10 ⊢ ∪ (TopOpen‘ℂ_fld) = ∪ (TopOpen‘ℂ_fld)
57		eqid 2740	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
58	57	cnfldtop 24773	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) ∈ Top
59	58	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
60	12, 46	sstrdi 3934	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ ℂ)
61		ssid 3944	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
62		eqid 2740	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) = ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))
63		unicntop 24775	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
64	63	restid 17394	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
65	58, 64	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
66	65	eqcomi 2749	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
67	57, 62, 66	cncfcn 24902	. . . . . . . . . . . . 13 ⊢ (((𝐾[,]𝐿) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐾[,]𝐿)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
68	60, 61, 67	sylancl 592	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
69		reex 11127	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
70	69	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℝ ∈ V)
71		restabs 23155	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐾[,]𝐿) ⊆ ℝ ∧ ℝ ∈ V) → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐾[,]𝐿)) = ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)))
72	59, 12, 70, 71	syl3anc 1379	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐾[,]𝐿)) = ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)))
73		tgioo4 24795	. . . . . . . . . . . . . . . . 17 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
74	73	eqcomi 2749	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,))
75	74	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,)))
76	75	oveq1d 7378	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)))
77	72, 76	eqtr3d 2777	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)))
78	77	oveq1d 7378	. . . . . . . . . . . 12 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)) = (((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
79	68, 78	eqtrd 2775	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
80	21, 79	eleqtrd 2842	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
81	54, 55, 56, 3, 10, 11, 9, 59, 80	icccncfext 46337	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)) ∧ (𝐺 ↾ (𝐾[,]𝐿)) = 𝐹))
82	81	simpld 495	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
83		uniretop 24752	. . . . . . . . 9 ⊢ ℝ = ∪ (topGen‘ran (,))
84		eqid 2740	. . . . . . . . 9 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) = ∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)
85	83, 84	cnf 23236	. . . . . . . 8 ⊢ (𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)) → 𝐺:ℝ⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹))
86	82, 85	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℝ⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹))
87	44	feq2d 6646	. . . . . . 7 ⊢ (𝜑 → (𝐺:ℝ⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) ↔ 𝐺:(-∞(,)+∞)⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
88	86, 87	mpbid 233	. . . . . 6 ⊢ (𝜑 → 𝐺:(-∞(,)+∞)⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹))
89	88	feqmptd 6902	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺‘𝑤)))
90	23	frnd 6670	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
91		cncfss 24891	. . . . . . 7 ⊢ ((ran 𝐹 ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((-∞(,)+∞)–cn→ran 𝐹) ⊆ ((-∞(,)+∞)–cn→ℂ))
92	90, 61, 91	sylancl 592	. . . . . 6 ⊢ (𝜑 → ((-∞(,)+∞)–cn→ran 𝐹) ⊆ ((-∞(,)+∞)–cn→ℂ))
93	43	oveq2i 7374	. . . . . . . . . . 11 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t (-∞(,)+∞))
94	73, 93	eqtri 2763	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t (-∞(,)+∞))
95		eqid 2740	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) = ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)
96	57, 94, 95	cncfcn 24902	. . . . . . . . 9 ⊢ (((-∞(,)+∞) ⊆ ℂ ∧ ran 𝐹 ⊆ ℂ) → ((-∞(,)+∞)–cn→ran 𝐹) = ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
97	47, 90, 96	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → ((-∞(,)+∞)–cn→ran 𝐹) = ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
98	97	eqcomd 2746	. . . . . . 7 ⊢ (𝜑 → ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)) = ((-∞(,)+∞)–cn→ran 𝐹))
99	82, 98	eleqtrd 2842	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ ((-∞(,)+∞)–cn→ran 𝐹))
100	92, 99	sseldd 3923	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ ((-∞(,)+∞)–cn→ℂ))
101	89, 100	eqeltrrd 2841	. . . 4 ⊢ (𝜑 → (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺‘𝑤)) ∈ ((-∞(,)+∞)–cn→ℂ))
102		itgsubsticclem.12	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
103		fveq2 6834	. . . 4 ⊢ (𝑤 = 𝐴 → (𝐺‘𝑤) = (𝐺‘𝐴))
104		itgsubsticclem.14	. . . 4 ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾)
105		itgsubsticclem.15	. . . 4 ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿)
106	35, 36, 37, 39, 41, 51, 52, 101, 102, 103, 104, 105	itgsubst 26041	. . 3 ⊢ (𝜑 → ⨜[𝐾 → 𝐿](𝐺‘𝑤) d𝑤 = ⨜[𝑋 → 𝑌]((𝐺‘𝐴) · 𝐵) d𝑥)
107	8, 34, 106	3eqtr3a 2799	. 2 ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌]((𝐺‘𝐴) · 𝐵) d𝑥)
108	3	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐺 = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿)))))
109		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 = 𝐴)
110	57	cnfldtopon 24772	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
111	35, 36	iccssred 13385	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
112	111, 46	sstrdi 3934	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℂ)
113		resttopon 23151	. . . . . . . . . . . . . 14 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝑋[,]𝑌) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)))
114	110, 112, 113	sylancr 593	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)))
115		resttopon 23151	. . . . . . . . . . . . . 14 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐾[,]𝐿) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)))
116	110, 60, 115	sylancr 593	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)))
117		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) = ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌))
118	57, 117, 62	cncfcn 24902	. . . . . . . . . . . . . . 15 ⊢ (((𝑋[,]𝑌) ⊆ ℂ ∧ (𝐾[,]𝐿) ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) = (((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))))
119	112, 60, 118	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) = (((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))))
120	50, 119	eleqtrd 2842	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ (((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))))
121		cnf2 23239	. . . . . . . . . . . . 13 ⊢ ((((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)) ∧ ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)) ∧ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ (((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
122	114, 116, 120, 121	syl3anc 1379	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
123	122	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
124		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)
125	124	fmpt 7058	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿) ↔ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
126	123, 125	sylibr 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿))
127		ioossicc 13384	. . . . . . . . . . . 12 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
128	127	sseli 3918	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
129	128	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝑥 ∈ (𝑋[,]𝑌))
130		rsp 3228	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿) → (𝑥 ∈ (𝑋[,]𝑌) → 𝐴 ∈ (𝐾[,]𝐿)))
131	126, 129, 130	sylc 65	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ (𝐾[,]𝐿))
132	131	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐴 ∈ (𝐾[,]𝐿))
133	109, 132	eqeltrd 2840	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 ∈ (𝐾[,]𝐿))
134	133	iftrued 4469	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) = (𝐹‘𝑢))
135		simpll 772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝜑)
136	135, 133, 25	syl2anc 590	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 ∈ ℂ)
137	133, 136, 27	syl2anc 590	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → (𝐹‘𝑢) = 𝐶)
138		itgsubsticclem.13	. . . . . . 7 ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸)
139	138	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 = 𝐸)
140	134, 137, 139	3eqtrd 2779	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) = 𝐸)
141	12	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐾[,]𝐿) ⊆ ℝ)
142	141, 131	sseldd 3923	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ ℝ)
143		elex 3453	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐾[,]𝐿) → 𝐴 ∈ V)
144	131, 143	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ V)
145		isset 3446	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∃𝑢 𝑢 = 𝐴)
146	144, 145	sylib 219	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ∃𝑢 𝑢 = 𝐴)
147	139, 136	eqeltrrd 2841	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐸 ∈ ℂ)
148	146, 147	exlimddv 1942	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐸 ∈ ℂ)
149	108, 140, 142, 148	fvmptd 6950	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐺‘𝐴) = 𝐸)
150	149	oveq1d 7378	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝐺‘𝐴) · 𝐵) = (𝐸 · 𝐵))
151	37, 150	ditgeq3d 46414	. 2 ⊢ (𝜑 → ⨜[𝑋 → 𝑌]((𝐺‘𝐴) · 𝐵) d𝑥 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
152	107, 151	eqtrd 2775	1 ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∃wex 1786 ∈ wcel 2119 ∀wral 3054 Vcvv 3432 ∩ cin 3889 ⊆ wss 3890 ifcif 4461 ∪ cuni 4845 class class class wbr 5079 ↦ cmpt 5160 ran crn 5626 ↾ cres 5627 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 ℝcr 11035 · cmul 11041 +∞cpnf 11174 -∞cmnf 11175 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 (,)cioo 13296 [,]cicc 13299 ↾_t crest 17381 TopOpenctopn 17382 topGenctg 17398 ℂ_fldccnfld 21354 Topctop 22883 TopOnctopon 22900 Cn ccn 23214 –cn→ccncf 24868 𝐿¹cibl 25609 ⨜cdit 25838 D cdv 25855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cc 10355 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-symdif 4188 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-disj 5047 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-ofr 7628 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-oadd 8406 df-omul 8407 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-dju 9823 df-card 9861 df-acn 9864 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ioc 13301 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-mod 13827 df-seq 13962 df-exp 14022 df-hash 14291 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-cmp 23377 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870 df-ovol 25456 df-vol 25457 df-mbf 25611 df-itg1 25612 df-itg2 25613 df-ibl 25614 df-itg 25615 df-0p 25662 df-ditg 25839 df-limc 25858 df-dv 25859

This theorem is referenced by: itgsubsticc 46426

W3C validator