itgsubsticclem - Mathbox for Glauco Siliprandi

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Theorem itgsubsticclem 46803

Description: lemma for itgsubsticc 46804. (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 6878	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐺‘𝑢) = (𝐺‘𝑤))
2		nfcv 2922	. . . 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 5204	. . . . . 6 ⊢ Ⅎ𝑢(𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))))
5	3, 4	nfcxfr 2920	. . . . 5 ⊢ Ⅎ𝑢𝐺
6		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑢𝑤
7	5, 6	nffv 6888	. . . 4 ⊢ Ⅎ𝑢(𝐺‘𝑤)
8	1, 2, 7	cbvditg 26081	. . 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 13487	. . . . . . . 8 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ ℝ)
13	12	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐾[,]𝐿) ⊆ ℝ)
14		ioossicc 13486	. . . . . . . . 9 ⊢ (𝐾(,)𝐿) ⊆ (𝐾[,]𝐿)
15	14	sseli 3927	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐾(,)𝐿) → 𝑢 ∈ (𝐾[,]𝐿))
16	15	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ (𝐾[,]𝐿))
17	13, 16	sseldd 3932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ ℝ)
18	16	iftrued 4490	. . . . . . 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 25121	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ ((𝐾[,]𝐿)–cn→ℂ) → 𝐹:(𝐾[,]𝐿)⟶ℂ)
23	21, 22	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(𝐾[,]𝐿)⟶ℂ)
24	20, 23	feq1dd 6685	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶):(𝐾[,]𝐿)⟶ℂ)
25	24	fvmptelcdm 7106	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾[,]𝐿)) → 𝐶 ∈ ℂ)
26	16, 25	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝐶 ∈ ℂ)
27	19	fvmpt2 6998	. . . . . . . . 9 ⊢ ((𝑢 ∈ (𝐾[,]𝐿) ∧ 𝐶 ∈ ℂ) → (𝐹‘𝑢) = 𝐶)
28	16, 26, 27	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐹‘𝑢) = 𝐶)
29	28, 26	eqeltrd 2860	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐹‘𝑢) ∈ ℂ)
30	18, 29	eqeltrd 2860	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) ∈ ℂ)
31	3	fvmpt2 6998	. . . . . 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 596	. . . . 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 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐺‘𝑢) = 𝐶)
34	9, 33	ditgeq3d 46792	. . 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 11290	. . . . 5 ⊢ -∞ ∈ ℝ^*
39	38	a1i 11	. . . 4 ⊢ (𝜑 → -∞ ∈ ℝ^*)
40		pnfxr 11287	. . . . 5 ⊢ +∞ ∈ ℝ^*
41	40	a1i 11	. . . 4 ⊢ (𝜑 → +∞ ∈ ℝ^*)
42		ioomax 13475	. . . . . . . . 9 ⊢ (-∞(,)+∞) = ℝ
43	42	eqcomi 2769	. . . . . . . 8 ⊢ ℝ = (-∞(,)+∞)
44	43	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ = (-∞(,)+∞))
45	12, 44	sseqtrd 3967	. . . . . 6 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ (-∞(,)+∞))
46		ax-resscn 11181	. . . . . . 7 ⊢ ℝ ⊆ ℂ
47	44, 46	eqsstrrdi 3976	. . . . . 6 ⊢ (𝜑 → (-∞(,)+∞) ⊆ ℂ)
48		cncfss 25127	. . . . . 6 ⊢ (((𝐾[,]𝐿) ⊆ (-∞(,)+∞) ∧ (-∞(,)+∞) ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) ⊆ ((𝑋[,]𝑌)–cn→(-∞(,)+∞)))
49	45, 47, 48	syl2anc 596	. . . . 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 3932	. . . 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 5204	. . . . . . . . . . 11 ⊢ Ⅎ𝑢(𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)
54	19, 53	nfcxfr 2920	. . . . . . . . . 10 ⊢ Ⅎ𝑢𝐹
55		eqid 2760	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
56		eqid 2760	. . . . . . . . . 10 ⊢ ∪ (TopOpen‘ℂ_fld) = ∪ (TopOpen‘ℂ_fld)
57		eqid 2760	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
58	57	cnfldtop 25009	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) ∈ Top
59	58	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
60	12, 46	sstrdi 3943	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ ℂ)
61		ssid 3953	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
62		eqid 2760	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) = ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))
63		unicntop 25011	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
64	63	restid 17518	. . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
67	57, 62, 66	cncfcn 25138	. . . . . . . . . . . . 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 598	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
69		reex 11215	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
70	69	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℝ ∈ V)
71		restabs 23390	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐾[,]𝐿)) = ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)))
73		tgioo4 25031	. . . . . . . . . . . . . . . . 17 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
74	73	eqcomi 2769	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,))
75	74	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,)))
76	75	oveq1d 7428	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)))
77	72, 76	eqtr3d 2797	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)))
78	77	oveq1d 7428	. . . . . . . . . . . 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 2795	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
80	21, 79	eleqtrd 2862	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
81	54, 55, 56, 3, 10, 11, 9, 59, 80	icccncfext 46715	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)) ∧ (𝐺 ↾ (𝐾[,]𝐿)) = 𝐹))
82	81	simpld 500	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
83		uniretop 24988	. . . . . . . . 9 ⊢ ℝ = ∪ (topGen‘ran (,))
84		eqid 2760	. . . . . . . . 9 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) = ∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)
85	83, 84	cnf 23471	. . . . . . . 8 ⊢ (𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)) → 𝐺:ℝ⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹))
86	82, 85	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℝ⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹))
87	44	feq2d 6686	. . . . . . 7 ⊢ (𝜑 → (𝐺:ℝ⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) ↔ 𝐺:(-∞(,)+∞)⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
88	86, 87	mpbid 235	. . . . . 6 ⊢ (𝜑 → 𝐺:(-∞(,)+∞)⟶∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹))
89	88	feqmptd 6946	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺‘𝑤)))
90	23	frnd 6711	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
91		cncfss 25127	. . . . . . 7 ⊢ ((ran 𝐹 ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((-∞(,)+∞)–cn→ran 𝐹) ⊆ ((-∞(,)+∞)–cn→ℂ))
92	90, 61, 91	sylancl 598	. . . . . 6 ⊢ (𝜑 → ((-∞(,)+∞)–cn→ran 𝐹) ⊆ ((-∞(,)+∞)–cn→ℂ))
93	43	oveq2i 7424	. . . . . . . . . . 11 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t (-∞(,)+∞))
94	73, 93	eqtri 2783	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t (-∞(,)+∞))
95		eqid 2760	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) = ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)
96	57, 94, 95	cncfcn 25138	. . . . . . . . 9 ⊢ (((-∞(,)+∞) ⊆ ℂ ∧ ran 𝐹 ⊆ ℂ) → ((-∞(,)+∞)–cn→ran 𝐹) = ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
97	47, 90, 96	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → ((-∞(,)+∞)–cn→ran 𝐹) = ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)))
98	97	eqcomd 2766	. . . . . . 7 ⊢ (𝜑 → ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)) = ((-∞(,)+∞)–cn→ran 𝐹))
99	82, 98	eleqtrd 2862	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ ((-∞(,)+∞)–cn→ran 𝐹))
100	92, 99	sseldd 3932	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ ((-∞(,)+∞)–cn→ℂ))
101	89, 100	eqeltrrd 2861	. . . 4 ⊢ (𝜑 → (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺‘𝑤)) ∈ ((-∞(,)+∞)–cn→ℂ))
102		itgsubsticclem.12	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
103		fveq2 6878	. . . 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 26276	. . 3 ⊢ (𝜑 → ⨜[𝐾 → 𝐿](𝐺‘𝑤) d𝑤 = ⨜[𝑋 → 𝑌]((𝐺‘𝐴) · 𝐵) d𝑥)
107	8, 34, 106	3eqtr3a 2819	. 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 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 = 𝐴)
110	57	cnfldtopon 25008	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
111	35, 36	iccssred 13487	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
112	111, 46	sstrdi 3943	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℂ)
113		resttopon 23386	. . . . . . . . . . . . . 14 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝑋[,]𝑌) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)))
114	110, 112, 113	sylancr 599	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)))
115		resttopon 23386	. . . . . . . . . . . . . 14 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐾[,]𝐿) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)))
116	110, 60, 115	sylancr 599	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)))
117		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) = ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌))
118	57, 117, 62	cncfcn 25138	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) = (((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))))
120	50, 119	eleqtrd 2862	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ (((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))))
121		cnf2 23474	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
123	122	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
124		eqid 2760	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)
125	124	fmpt 7103	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿) ↔ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
126	123, 125	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿))
127		ioossicc 13486	. . . . . . . . . . . 12 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
128	127	sseli 3927	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
129	128	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝑥 ∈ (𝑋[,]𝑌))
130		rsp 3250	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿) → (𝑥 ∈ (𝑋[,]𝑌) → 𝐴 ∈ (𝐾[,]𝐿)))
131	126, 129, 130	sylc 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ (𝐾[,]𝐿))
132	131	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐴 ∈ (𝐾[,]𝐿))
133	109, 132	eqeltrd 2860	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 ∈ (𝐾[,]𝐿))
134	133	iftrued 4490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) = (𝐹‘𝑢))
135		simpll 779	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝜑)
136	135, 133, 25	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 ∈ ℂ)
137	133, 136, 27	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → (𝐹‘𝑢) = 𝐶)
138		itgsubsticclem.13	. . . . . . 7 ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸)
139	138	adantl 487	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 = 𝐸)
140	134, 137, 139	3eqtrd 2799	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) = 𝐸)
141	12	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐾[,]𝐿) ⊆ ℝ)
142	141, 131	sseldd 3932	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ ℝ)
143		elex 3471	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐾[,]𝐿) → 𝐴 ∈ V)
144	131, 143	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ V)
145		isset 3464	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∃𝑢 𝑢 = 𝐴)
146	144, 145	sylib 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ∃𝑢 𝑢 = 𝐴)
147	139, 136	eqeltrrd 2861	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐸 ∈ ℂ)
148	146, 147	exlimddv 1968	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐸 ∈ ℂ)
149	108, 140, 142, 148	fvmptd 6994	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐺‘𝐴) = 𝐸)
150	149	oveq1d 7428	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝐺‘𝐴) · 𝐵) = (𝐸 · 𝐵))
151	37, 150	ditgeq3d 46792	. 2 ⊢ (𝜑 → ⨜[𝑋 → 𝑌]((𝐺‘𝐴) · 𝐵) d𝑥 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
152	107, 151	eqtrd 2795	1 ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ∀wral 3076 Vcvv 3450 ∩ cin 3898 ⊆ wss 3899 ifcif 4482 ∪ cuni 4867 class class class wbr 5103 ↦ cmpt 5186 ran crn 5656 ↾ cres 5657 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 · cmul 11129 +∞cpnf 11264 -∞cmnf 11265 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 (,)cioo 13398 [,]cicc 13401 ↾_t crest 17505 TopOpenctopn 17506 topGenctg 17522 ℂ_fldccnfld 21585 Topctop 23118 TopOnctopon 23135 Cn ccn 23449 –cn→ccncf 25104 𝐿¹cibl 25845 ⨜cdit 26073 D cdv 26090

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cc 10437 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-ofr 7679 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-omul 8460 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-acn 9947 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-sum 15774 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-lp 23361 df-perf 23362 df-cn 23452 df-cnp 23453 df-haus 23540 df-cmp 23612 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-ovol 25692 df-vol 25693 df-mbf 25847 df-itg1 25848 df-itg2 25849 df-ibl 25850 df-itg 25851 df-0p 25898 df-ditg 26074 df-limc 26093 df-dv 26094

This theorem is used by: itgsubsticc 46804

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