itgsubsticclem - Mathbox for Glauco Siliprandi

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Theorem itgsubsticclem 45973

Description: lemma for itgsubsticc 45974. (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 6858	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐺‘𝑢) = (𝐺‘𝑤))
2		nfcv 2891	. . . 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 5206	. . . . . 6 ⊢ Ⅎ𝑢(𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))))
5	3, 4	nfcxfr 2889	. . . . 5 ⊢ Ⅎ𝑢𝐺
6		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑢𝑤
7	5, 6	nffv 6868	. . . 4 ⊢ Ⅎ𝑢(𝐺‘𝑤)
8	1, 2, 7	cbvditg 25755	. . 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 13395	. . . . . . . 8 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ ℝ)
13	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐾[,]𝐿) ⊆ ℝ)
14		ioossicc 13394	. . . . . . . . 9 ⊢ (𝐾(,)𝐿) ⊆ (𝐾[,]𝐿)
15	14	sseli 3942	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐾(,)𝐿) → 𝑢 ∈ (𝐾[,]𝐿))
16	15	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ (𝐾[,]𝐿))
17	13, 16	sseldd 3947	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ ℝ)
18	16	iftrued 4496	. . . . . . 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 24786	. . . . . . . . . . . . 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 6671	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶):(𝐾[,]𝐿)⟶ℂ)
25	24	fvmptelcdm 7085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾[,]𝐿)) → 𝐶 ∈ ℂ)
26	16, 25	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → 𝐶 ∈ ℂ)
27	19	fvmpt2 6979	. . . . . . . . 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 2828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐹‘𝑢) ∈ ℂ)
30	18, 29	eqeltrd 2828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))) ∈ ℂ)
31	3	fvmpt2 6979	. . . . . 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 2768	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐾(,)𝐿)) → (𝐺‘𝑢) = 𝐶)
34	9, 33	ditgeq3d 45962	. . 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 11231	. . . . 5 ⊢ -∞ ∈ ℝ^*
39	38	a1i 11	. . . 4 ⊢ (𝜑 → -∞ ∈ ℝ^*)
40		pnfxr 11228	. . . . 5 ⊢ +∞ ∈ ℝ^*
41	40	a1i 11	. . . 4 ⊢ (𝜑 → +∞ ∈ ℝ^*)
42		ioomax 13383	. . . . . . . . 9 ⊢ (-∞(,)+∞) = ℝ
43	42	eqcomi 2738	. . . . . . . 8 ⊢ ℝ = (-∞(,)+∞)
44	43	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ = (-∞(,)+∞))
45	12, 44	sseqtrd 3983	. . . . . 6 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ (-∞(,)+∞))
46		ax-resscn 11125	. . . . . . 7 ⊢ ℝ ⊆ ℂ
47	44, 46	eqsstrrdi 3992	. . . . . 6 ⊢ (𝜑 → (-∞(,)+∞) ⊆ ℂ)
48		cncfss 24792	. . . . . 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 3947	. . . 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 5206	. . . . . . . . . . 11 ⊢ Ⅎ𝑢(𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)
54	19, 53	nfcxfr 2889	. . . . . . . . . 10 ⊢ Ⅎ𝑢𝐹
55		eqid 2729	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
56		eqid 2729	. . . . . . . . . 10 ⊢ ∪ (TopOpen‘ℂ_fld) = ∪ (TopOpen‘ℂ_fld)
57		eqid 2729	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
58	57	cnfldtop 24671	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) ∈ Top
59	58	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
60	12, 46	sstrdi 3959	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾[,]𝐿) ⊆ ℂ)
61		ssid 3969	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
62		eqid 2729	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) = ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))
63		unicntop 24673	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
64	63	restid 17396	. . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
67	57, 62, 66	cncfcn 24803	. . . . . . . . . . . . 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 11159	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
70	69	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℝ ∈ V)
71		restabs 23052	. . . . . . . . . . . . . . 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 24693	. . . . . . . . . . . . . . . . 17 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
74	73	eqcomi 2738	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,))
75	74	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,)))
76	75	oveq1d 7402	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)))
77	72, 76	eqtr3d 2766	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)))
78	77	oveq1d 7402	. . . . . . . . . . . 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 2764	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
80	21, 79	eleqtrd 2830	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) ↾_t (𝐾[,]𝐿)) Cn (TopOpen‘ℂ_fld)))
81	54, 55, 56, 3, 10, 11, 9, 59, 80	icccncfext 45885	. . . . . . . . 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 24650	. . . . . . . . 9 ⊢ ℝ = ∪ (topGen‘ran (,))
84		eqid 2729	. . . . . . . . 9 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) = ∪ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)
85	83, 84	cnf 23133	. . . . . . . 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 6672	. . . . . . 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 6929	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺‘𝑤)))
90	23	frnd 6696	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
91		cncfss 24792	. . . . . . 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 7398	. . . . . . . . . . 11 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t (-∞(,)+∞))
94	73, 93	eqtri 2752	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t (-∞(,)+∞))
95		eqid 2729	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t ran 𝐹) = ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)
96	57, 94, 95	cncfcn 24803	. . . . . . . . 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 2735	. . . . . . 7 ⊢ (𝜑 → ((topGen‘ran (,)) Cn ((TopOpen‘ℂ_fld) ↾_t ran 𝐹)) = ((-∞(,)+∞)–cn→ran 𝐹))
99	82, 98	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ ((-∞(,)+∞)–cn→ran 𝐹))
100	92, 99	sseldd 3947	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ ((-∞(,)+∞)–cn→ℂ))
101	89, 100	eqeltrrd 2829	. . . 4 ⊢ (𝜑 → (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺‘𝑤)) ∈ ((-∞(,)+∞)–cn→ℂ))
102		itgsubsticclem.12	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
103		fveq2 6858	. . . 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 25956	. . 3 ⊢ (𝜑 → ⨜[𝐾 → 𝐿](𝐺‘𝑤) d𝑤 = ⨜[𝑋 → 𝑌]((𝐺‘𝐴) · 𝐵) d𝑥)
107	8, 34, 106	3eqtr3a 2788	. 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 24670	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
111	35, 36	iccssred 13395	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
112	111, 46	sstrdi 3959	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℂ)
113		resttopon 23048	. . . . . . . . . . . . . 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 23048	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) = ((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌))
118	57, 117, 62	cncfcn 24803	. . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ (((TopOpen‘ℂ_fld) ↾_t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂ_fld) ↾_t (𝐾[,]𝐿))))
121		cnf2 23136	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)
125	124	fmpt 7082	. . . . . . . . . . 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 13394	. . . . . . . . . . . 12 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
128	127	sseli 3942	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
129	128	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝑥 ∈ (𝑋[,]𝑌))
130		rsp 3225	. . . . . . . . . 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 2828	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 ∈ (𝐾[,]𝐿))
134	133	iftrued 4496	. . . . . 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 2768	. . . . 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 3947	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ ℝ)
143		elex 3468	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐾[,]𝐿) → 𝐴 ∈ V)
144	131, 143	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ V)
145		isset 3461	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∃𝑢 𝑢 = 𝐴)
146	144, 145	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ∃𝑢 𝑢 = 𝐴)
147	139, 136	eqeltrrd 2829	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐸 ∈ ℂ)
148	146, 147	exlimddv 1935	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐸 ∈ ℂ)
149	108, 140, 142, 148	fvmptd 6975	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐺‘𝐴) = 𝐸)
150	149	oveq1d 7402	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝐺‘𝐴) · 𝐵) = (𝐸 · 𝐵))
151	37, 150	ditgeq3d 45962	. 2 ⊢ (𝜑 → ⨜[𝑋 → 𝑌]((𝐺‘𝐴) · 𝐵) d𝑥 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
152	107, 151	eqtrd 2764	1 ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ∀wral 3044 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 ifcif 4488 ∪ cuni 4871 class class class wbr 5107 ↦ cmpt 5188 ran crn 5639 ↾ cres 5640 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 · cmul 11073 +∞cpnf 11205 -∞cmnf 11206 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 (,)cioo 13306 [,]cicc 13309 ↾_t crest 17383 TopOpenctopn 17384 topGenctg 17400 ℂ_fldccnfld 21264 Topctop 22780 TopOnctopon 22797 Cn ccn 23111 –cn→ccncf 24769 𝐿¹cibl 25518 ⨜cdit 25747 D cdv 25764

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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cc 10388 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-symdif 4216 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-ovol 25365 df-vol 25366 df-mbf 25520 df-itg1 25521 df-itg2 25522 df-ibl 25523 df-itg 25524 df-0p 25571 df-ditg 25748 df-limc 25767 df-dv 25768

This theorem is referenced by: itgsubsticc 45974

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