itgsubsticclem - Mathbox for Glauco Siliprandi

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Theorem itgsubsticclem 46013

Description: lemma for itgsubsticc 46014. (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 6817	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐺‘𝑢) = (𝐺‘𝑤))
2		nfcv 2894	. . . 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 5185	. . . . . 6 ⊢ Ⅎ𝑢(𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))))
5	3, 4	nfcxfr 2892	. . . . 5 ⊢ Ⅎ𝑢𝐺
6		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑢𝑤
7	5, 6	nffv 6827	. . . 4 ⊢ Ⅎ𝑢(𝐺‘𝑤)
8	1, 2, 7	cbvditg 25777	. . 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 13329	. . . . . . . 8 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ ℝ)
13	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐾[,]𝐿) ⊆ ℝ)
14		ioossicc 13328	. . . . . . . . 9 ⊢ (𝐾(,)𝐿) ⊆ (𝐾[,]𝐿)
15	14	sseli 3925	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐾(,)𝐿) → 𝑢 ∈ (𝐾[,]𝐿))
16	15	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ (𝐾[,]𝐿))
17	13, 16	sseldd 3930	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ ℝ)
18	16	iftrued 4478	. . . . . . 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 24808	. . . . . . . . . . . . 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 6629	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶):(𝐾[,]𝐿)⟶ℂ)
25	24	fvmptelcdm 7041	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾[,]𝐿)) → 𝐶 ∈ ℂ)
26	16, 25	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝐶 ∈ ℂ)
27	19	fvmpt2 6935	. . . . . . . . 9 ⊢ ((𝑢 ∈ (𝐾[,]𝐿) ∧ 𝐶 ∈ ℂ) → (𝐹‘𝑢) = 𝐶)
28	16, 26, 27	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐹‘𝑢) = 𝐶)
29	28, 26	eqeltrd 2831	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐹‘𝑢) ∈ ℂ)
30	18, 29	eqeltrd 2831	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) ∈ ℂ)
31	3	fvmpt2 6935	. . . . . 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 584	. . . . 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 2770	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐺‘𝑢) = 𝐶)
34	9, 33	ditgeq3d 46002	. . 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 11164	. . . . 5 ⊢ -∞ ∈ ℝ^*
39	38	a1i 11	. . . 4 ⊢ (𝜑 → -∞ ∈ ℝ^*)
40		pnfxr 11161	. . . . 5 ⊢ +∞ ∈ ℝ^*
41	40	a1i 11	. . . 4 ⊢ (𝜑 → +∞ ∈ ℝ^*)
42		ioomax 13317	. . . . . . . . 9 ⊢ (-∞(,)+∞) = ℝ
43	42	eqcomi 2740	. . . . . . . 8 ⊢ ℝ = (-∞(,)+∞)
44	43	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ = (-∞(,)+∞))
45	12, 44	sseqtrd 3966	. . . . . 6 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ (-∞(,)+∞))
46		ax-resscn 11058	. . . . . . 7 ⊢ ℝ ⊆ ℂ
47	44, 46	eqsstrrdi 3975	. . . . . 6 ⊢ (𝜑 → (-∞(,)+∞) ⊆ ℂ)
48		cncfss 24814	. . . . . 6 ⊢ (((𝐾[,]𝐿) ⊆ (-∞(,)+∞) ∧ (-∞(,)+∞) ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) ⊆ ((𝑋[,]𝑌)–cn→(-∞(,)+∞)))
49	45, 47, 48	syl2anc 584	. . . . 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 3930	. . . 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 5185	. . . . . . . . . . 11 ⊢ Ⅎ𝑢(𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)
54	19, 53	nfcxfr 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑢𝐹
55		eqid 2731	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
56		eqid 2731	. . . . . . . . . 10 ⊢ ∪ (TopOpen‘ℂ_fld) = ∪ (TopOpen‘ℂ_fld)
57		eqid 2731	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
58	57	cnfldtop 24693	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) ∈ Top
59	58	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
60	12, 46	sstrdi 3942	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ ℂ)
61		ssid 3952	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
62		eqid 2731	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) = ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))
63		unicntop 24695	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
64	63	restid 17332	. . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
67	57, 62, 66	cncfcn 24825	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
69		reex 11092	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
70	69	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℝ ∈ V)
71		restabs 23075	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐾[,]𝐿)) = ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)))
73		tgioo4 24715	. . . . . . . . . . . . . . . . 17 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
74	73	eqcomi 2740	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,))
75	74	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,)))
76	75	oveq1d 7356	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)))
77	72, 76	eqtr3d 2768	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)))
78	77	oveq1d 7356	. . . . . . . . . . . 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 2766	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
80	21, 79	eleqtrd 2833	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
81	54, 55, 56, 3, 10, 11, 9, 59, 80	icccncfext 45925	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)) ∧ (𝐺 ↾ (𝐾[,]𝐿)) = 𝐹))
82	81	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
83		uniretop 24672	. . . . . . . . 9 ⊢ ℝ = ∪ (topGen‘ran (,))
84		eqid 2731	. . . . . . . . 9 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) = ∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)
85	83, 84	cnf 23156	. . . . . . . 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 6630	. . . . . . 7 ⊢ (𝜑 → (𝐺:ℝ⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) ↔ 𝐺:(-∞(,)+∞)⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
88	86, 87	mpbid 232	. . . . . 6 ⊢ (𝜑 → 𝐺:(-∞(,)+∞)⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹))
89	88	feqmptd 6885	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺‘𝑤)))
90	23	frnd 6654	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
91		cncfss 24814	. . . . . . 7 ⊢ ((ran 𝐹 ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((-∞(,)+∞)–cn→ran 𝐹) ⊆ ((-∞(,)+∞)–cn→ℂ))
92	90, 61, 91	sylancl 586	. . . . . 6 ⊢ (𝜑 → ((-∞(,)+∞)–cn→ran 𝐹) ⊆ ((-∞(,)+∞)–cn→ℂ))
93	43	oveq2i 7352	. . . . . . . . . . 11 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t (-∞(,)+∞))
94	73, 93	eqtri 2754	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t (-∞(,)+∞))
95		eqid 2731	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) = ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)
96	57, 94, 95	cncfcn 24825	. . . . . . . . 9 ⊢ (((-∞(,)+∞) ⊆ ℂ ∧ ran 𝐹 ⊆ ℂ) → ((-∞(,)+∞)–cn→ran 𝐹) = ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
97	47, 90, 96	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((-∞(,)+∞)–cn→ran 𝐹) = ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
98	97	eqcomd 2737	. . . . . . 7 ⊢ (𝜑 → ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)) = ((-∞(,)+∞)–cn→ran 𝐹))
99	82, 98	eleqtrd 2833	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ ((-∞(,)+∞)–cn→ran 𝐹))
100	92, 99	sseldd 3930	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ ((-∞(,)+∞)–cn→ℂ))
101	89, 100	eqeltrrd 2832	. . . 4 ⊢ (𝜑 → (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺‘𝑤)) ∈ ((-∞(,)+∞)–cn→ℂ))
102		itgsubsticclem.12	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
103		fveq2 6817	. . . 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 25978	. . 3 ⊢ (𝜑 → ⨜[𝐾 → 𝐿](𝐺‘𝑤) d𝑤 = ⨜[𝑋 → 𝑌]((𝐺‘𝐴) · 𝐵) d𝑥)
107	8, 34, 106	3eqtr3a 2790	. 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 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 = 𝐴)
110	57	cnfldtopon 24692	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
111	35, 36	iccssred 13329	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
112	111, 46	sstrdi 3942	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℂ)
113		resttopon 23071	. . . . . . . . . . . . . 14 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝑋[,]𝑌) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)))
114	110, 112, 113	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)))
115		resttopon 23071	. . . . . . . . . . . . . 14 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐾[,]𝐿) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)))
116	110, 60, 115	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)))
117		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) = ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌))
118	57, 117, 62	cncfcn 24825	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) = (((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))))
120	50, 119	eleqtrd 2833	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ (((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))))
121		cnf2 23159	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
123	122	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
124		eqid 2731	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)
125	124	fmpt 7038	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿) ↔ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
126	123, 125	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿))
127		ioossicc 13328	. . . . . . . . . . . 12 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
128	127	sseli 3925	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
129	128	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝑥 ∈ (𝑋[,]𝑌))
130		rsp 3220	. . . . . . . . . 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 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐴 ∈ (𝐾[,]𝐿))
133	109, 132	eqeltrd 2831	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 ∈ (𝐾[,]𝐿))
134	133	iftrued 4478	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) = (𝐹‘𝑢))
135		simpll 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝜑)
136	135, 133, 25	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 ∈ ℂ)
137	133, 136, 27	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → (𝐹‘𝑢) = 𝐶)
138		itgsubsticclem.13	. . . . . . 7 ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸)
139	138	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 = 𝐸)
140	134, 137, 139	3eqtrd 2770	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) = 𝐸)
141	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐾[,]𝐿) ⊆ ℝ)
142	141, 131	sseldd 3930	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ ℝ)
143		elex 3457	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐾[,]𝐿) → 𝐴 ∈ V)
144	131, 143	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ V)
145		isset 3450	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∃𝑢 𝑢 = 𝐴)
146	144, 145	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ∃𝑢 𝑢 = 𝐴)
147	139, 136	eqeltrrd 2832	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐸 ∈ ℂ)
148	146, 147	exlimddv 1936	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐸 ∈ ℂ)
149	108, 140, 142, 148	fvmptd 6931	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐺‘𝐴) = 𝐸)
150	149	oveq1d 7356	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝐺‘𝐴) · 𝐵) = (𝐸 · 𝐵))
151	37, 150	ditgeq3d 46002	. 2 ⊢ (𝜑 → ⨜[𝑋 → 𝑌]((𝐺‘𝐴) · 𝐵) d𝑥 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
152	107, 151	eqtrd 2766	1 ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∃wex 1780 ∈ wcel 2111 ∀wral 3047 Vcvv 3436 ∩ cin 3896 ⊆ wss 3897 ifcif 4470 ∪ cuni 4854 class class class wbr 5086 ↦ cmpt 5167 ran crn 5612 ↾ cres 5613 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 ℂcc 10999 ℝcr 11000 · cmul 11006 +∞cpnf 11138 -∞cmnf 11139 ℝ^*cxr 11140 < clt 11141 ≤ cle 11142 (,)cioo 13240 [,]cicc 13243 ↾_t crest 17319 TopOpenctopn 17320 topGenctg 17336 ℂ_fldccnfld 21286 Topctop 22803 TopOnctopon 22820 Cn ccn 23134 –cn→ccncf 24791 𝐿¹cibl 25540 ⨜cdit 25769 D cdv 25786

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cc 10321 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-symdif 4198 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-ofr 7606 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-dju 9789 df-card 9827 df-acn 9830 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ioo 13244 df-ioc 13245 df-ico 13246 df-icc 13247 df-fz 13403 df-fzo 13550 df-fl 13691 df-mod 13769 df-seq 13904 df-exp 13964 df-hash 14233 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-limsup 15373 df-clim 15390 df-rlim 15391 df-sum 15589 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-hom 17180 df-cco 17181 df-rest 17321 df-topn 17322 df-0g 17340 df-gsum 17341 df-topgen 17342 df-pt 17343 df-prds 17346 df-xrs 17401 df-qtop 17406 df-imas 17407 df-xps 17409 df-mre 17483 df-mrc 17484 df-acs 17486 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19224 df-cmn 19689 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-cld 22929 df-ntr 22930 df-cls 22931 df-nei 23008 df-lp 23046 df-perf 23047 df-cn 23137 df-cnp 23138 df-haus 23225 df-cmp 23297 df-tx 23472 df-hmeo 23665 df-fil 23756 df-fm 23848 df-flim 23849 df-flf 23850 df-xms 24230 df-ms 24231 df-tms 24232 df-cncf 24793 df-ovol 25387 df-vol 25388 df-mbf 25542 df-itg1 25543 df-itg2 25544 df-ibl 25545 df-itg 25546 df-0p 25593 df-ditg 25770 df-limc 25789 df-dv 25790

This theorem is referenced by: itgsubsticc 46014

W3C validator