itgsubsticclem - Mathbox for Glauco Siliprandi

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Theorem itgsubsticclem 44206

Description: lemma for itgsubsticc 44207. (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 6842	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐺‘𝑢) = (𝐺‘𝑤))
2		nfcv 2907	. . . 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 5213	. . . . . 6 ⊢ Ⅎ𝑢(𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))))
5	3, 4	nfcxfr 2905	. . . . 5 ⊢ Ⅎ𝑢𝐺
6		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑢𝑤
7	5, 6	nffv 6852	. . . 4 ⊢ Ⅎ𝑢(𝐺‘𝑤)
8	1, 2, 7	cbvditg 25218	. . 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 13351	. . . . . . . 8 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ ℝ)
13	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐾[,]𝐿) ⊆ ℝ)
14		ioossicc 13350	. . . . . . . . 9 ⊢ (𝐾(,)𝐿) ⊆ (𝐾[,]𝐿)
15	14	sseli 3940	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐾(,)𝐿) → 𝑢 ∈ (𝐾[,]𝐿))
16	15	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ (𝐾[,]𝐿))
17	13, 16	sseldd 3945	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ ℝ)
18	16	iftrued 4494	. . . . . . 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 24256	. . . . . . . . . . . . 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 43374	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶):(𝐾[,]𝐿)⟶ℂ)
25	24	fvmptelcdm 7061	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾[,]𝐿)) → 𝐶 ∈ ℂ)
26	16, 25	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝐶 ∈ ℂ)
27	19	fvmpt2 6959	. . . . . . . . 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 2838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐹‘𝑢) ∈ ℂ)
30	18, 29	eqeltrd 2838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) ∈ ℂ)
31	3	fvmpt2 6959	. . . . . 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 2780	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐺‘𝑢) = 𝐶)
34	9, 33	ditgeq3d 44195	. . 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 11212	. . . . 5 ⊢ -∞ ∈ ℝ^*
39	38	a1i 11	. . . 4 ⊢ (𝜑 → -∞ ∈ ℝ^*)
40		pnfxr 11209	. . . . 5 ⊢ +∞ ∈ ℝ^*
41	40	a1i 11	. . . 4 ⊢ (𝜑 → +∞ ∈ ℝ^*)
42		ioomax 13339	. . . . . . . . 9 ⊢ (-∞(,)+∞) = ℝ
43	42	eqcomi 2745	. . . . . . . 8 ⊢ ℝ = (-∞(,)+∞)
44	43	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ = (-∞(,)+∞))
45	12, 44	sseqtrd 3984	. . . . . 6 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ (-∞(,)+∞))
46		ax-resscn 11108	. . . . . . 7 ⊢ ℝ ⊆ ℂ
47	44, 46	eqsstrrdi 3999	. . . . . 6 ⊢ (𝜑 → (-∞(,)+∞) ⊆ ℂ)
48		cncfss 24262	. . . . . 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 3945	. . . 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 5213	. . . . . . . . . . 11 ⊢ Ⅎ𝑢(𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)
54	19, 53	nfcxfr 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑢𝐹
55		eqid 2736	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
56		eqid 2736	. . . . . . . . . 10 ⊢ ∪ (TopOpen‘ℂ_fld) = ∪ (TopOpen‘ℂ_fld)
57		eqid 2736	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
58	57	cnfldtop 24147	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) ∈ Top
59	58	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
60	12, 46	sstrdi 3956	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ ℂ)
61		ssid 3966	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
62		eqid 2736	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) = ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))
63		unicntop 24149	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
64	63	restid 17315	. . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
67	57, 62, 66	cncfcn 24273	. . . . . . . . . . . . 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 11142	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
70	69	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℝ ∈ V)
71		restabs 22516	. . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐾[,]𝐿)) = ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)))
73	57	tgioo2 24166	. . . . . . . . . . . . . . . . 17 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
74	73	eqcomi 2745	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,))
75	74	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,)))
76	75	oveq1d 7372	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)))
77	72, 76	eqtr3d 2778	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)))
78	77	oveq1d 7372	. . . . . . . . . . . 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 2776	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
80	21, 79	eleqtrd 2840	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
81	54, 55, 56, 3, 10, 11, 9, 59, 80	icccncfext 44118	. . . . . . . . 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 24126	. . . . . . . . 9 ⊢ ℝ = ∪ (topGen‘ran (,))
84		eqid 2736	. . . . . . . . 9 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) = ∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)
85	83, 84	cnf 22597	. . . . . . . 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 6654	. . . . . . 7 ⊢ (𝜑 → (𝐺:ℝ⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) ↔ 𝐺:(-∞(,)+∞)⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
88	86, 87	mpbid 231	. . . . . 6 ⊢ (𝜑 → 𝐺:(-∞(,)+∞)⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹))
89	88	feqmptd 6910	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺‘𝑤)))
90	23	frnd 6676	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
91		cncfss 24262	. . . . . . 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 7368	. . . . . . . . . . 11 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t (-∞(,)+∞))
94	73, 93	eqtri 2764	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t (-∞(,)+∞))
95		eqid 2736	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) = ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)
96	57, 94, 95	cncfcn 24273	. . . . . . . . 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 2742	. . . . . . 7 ⊢ (𝜑 → ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)) = ((-∞(,)+∞)–cn→ran 𝐹))
99	82, 98	eleqtrd 2840	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ ((-∞(,)+∞)–cn→ran 𝐹))
100	92, 99	sseldd 3945	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ ((-∞(,)+∞)–cn→ℂ))
101	89, 100	eqeltrrd 2839	. . . 4 ⊢ (𝜑 → (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺‘𝑤)) ∈ ((-∞(,)+∞)–cn→ℂ))
102		itgsubsticclem.12	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
103		fveq2 6842	. . . 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 25413	. . 3 ⊢ (𝜑 → ⨜[𝐾 → 𝐿](𝐺‘𝑤) d𝑤 = ⨜[𝑋 → 𝑌]((𝐺‘𝐴) · 𝐵) d𝑥)
107	8, 34, 106	3eqtr3a 2800	. 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 24146	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
111	35, 36	iccssred 13351	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
112	111, 46	sstrdi 3956	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℂ)
113		resttopon 22512	. . . . . . . . . . . . . 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 22512	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) = ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌))
118	57, 117, 62	cncfcn 24273	. . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ (((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))))
121		cnf2 22600	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 2736	. . . . . . . . . . . 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 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿))
127		ioossicc 13350	. . . . . . . . . . . 12 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
128	127	sseli 3940	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
129	128	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝑥 ∈ (𝑋[,]𝑌))
130		rsp 3230	. . . . . . . . . 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 2838	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 ∈ (𝐾[,]𝐿))
134	133	iftrued 4494	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) = (𝐹‘𝑢))
135		simpll 765	. . . . . . . 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 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 = 𝐸)
140	134, 137, 139	3eqtrd 2780	. . . . 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 3945	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ ℝ)
143		elex 3463	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐾[,]𝐿) → 𝐴 ∈ V)
144	131, 143	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ V)
145		isset 3458	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∃𝑢 𝑢 = 𝐴)
146	144, 145	sylib 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ∃𝑢 𝑢 = 𝐴)
147	139, 136	eqeltrrd 2839	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐸 ∈ ℂ)
148	146, 147	exlimddv 1938	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐸 ∈ ℂ)
149	108, 140, 142, 148	fvmptd 6955	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐺‘𝐴) = 𝐸)
150	149	oveq1d 7372	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝐺‘𝐴) · 𝐵) = (𝐸 · 𝐵))
151	37, 150	ditgeq3d 44195	. 2 ⊢ (𝜑 → ⨜[𝑋 → 𝑌]((𝐺‘𝐴) · 𝐵) d𝑥 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
152	107, 151	eqtrd 2776	1 ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3064 Vcvv 3445 ∩ cin 3909 ⊆ wss 3910 ifcif 4486 ∪ cuni 4865 class class class wbr 5105 ↦ cmpt 5188 ran crn 5634 ↾ cres 5635 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 · cmul 11056 +∞cpnf 11186 -∞cmnf 11187 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 (,)cioo 13264 [,]cicc 13267 ↾_t crest 17302 TopOpenctopn 17303 topGenctg 17319 ℂ_fldccnfld 20796 Topctop 22242 TopOnctopon 22259 Cn ccn 22575 –cn→ccncf 24239 𝐿¹cibl 24981 ⨜cdit 25210 D cdv 25227

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-symdif 4202 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-ovol 24828 df-vol 24829 df-mbf 24983 df-itg1 24984 df-itg2 24985 df-ibl 24986 df-itg 24987 df-0p 25034 df-ditg 25211 df-limc 25230 df-dv 25231

This theorem is referenced by: itgsubsticc 44207

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