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Theorem itgsubsticclem 38867
Description: lemma for itgsubsticc 38868. (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), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))))
itgsubsticclem.3 (𝜑𝑋 ∈ ℝ)
itgsubsticclem.4 (𝜑𝑌 ∈ ℝ)
itgsubsticclem.5 (𝜑𝑋𝑌)
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 (𝜑𝐹 ∈ ((𝐾[,]𝐿)–cn→ℂ))
itgsubsticclem.9 (𝜑𝐾 ∈ ℝ)
itgsubsticclem.10 (𝜑𝐿 ∈ ℝ)
itgsubsticclem.11 (𝜑𝐾𝐿)
itgsubsticclem.12 (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
itgsubsticclem.13 (𝑢 = 𝐴𝐶 = 𝐸)
itgsubsticclem.14 (𝑥 = 𝑋𝐴 = 𝐾)
itgsubsticclem.15 (𝑥 = 𝑌𝐴 = 𝐿)
Assertion
Ref Expression
itgsubsticclem (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](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.
StepHypRef Expression
1 fveq2 6103 . . . 4 (𝑢 = 𝑤 → (𝐺𝑢) = (𝐺𝑤))
2 nfcv 2751 . . . 4 𝑤(𝐺𝑢)
3 itgsubsticclem.2 . . . . . 6 𝐺 = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))))
4 nfmpt1 4675 . . . . . 6 𝑢(𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))))
53, 4nfcxfr 2749 . . . . 5 𝑢𝐺
6 nfcv 2751 . . . . 5 𝑢𝑤
75, 6nffv 6110 . . . 4 𝑢(𝐺𝑤)
81, 2, 7cbvditg 23424 . . 3 ⨜[𝐾𝐿](𝐺𝑢) d𝑢 = ⨜[𝐾𝐿](𝐺𝑤) d𝑤
9 itgsubsticclem.11 . . . 4 (𝜑𝐾𝐿)
10 itgsubsticclem.9 . . . . . . . . 9 (𝜑𝐾 ∈ ℝ)
11 itgsubsticclem.10 . . . . . . . . 9 (𝜑𝐿 ∈ ℝ)
1210, 11iccssred 38574 . . . . . . . 8 (𝜑 → (𝐾[,]𝐿) ⊆ ℝ)
1312adantr 480 . . . . . . 7 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐾[,]𝐿) ⊆ ℝ)
14 ioossicc 12130 . . . . . . . . 9 (𝐾(,)𝐿) ⊆ (𝐾[,]𝐿)
1514sseli 3564 . . . . . . . 8 (𝑢 ∈ (𝐾(,)𝐿) → 𝑢 ∈ (𝐾[,]𝐿))
1615adantl 481 . . . . . . 7 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ (𝐾[,]𝐿))
1713, 16sseldd 3569 . . . . . 6 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ ℝ)
1816iftrued 4044 . . . . . . 7 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))) = (𝐹𝑢))
19 itgsubsticclem.1 . . . . . . . . . . . . 13 𝐹 = (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)
2019a1i 11 . . . . . . . . . . . 12 (𝜑𝐹 = (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶))
21 itgsubsticclem.8 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ ((𝐾[,]𝐿)–cn→ℂ))
22 cncff 22504 . . . . . . . . . . . . 13 (𝐹 ∈ ((𝐾[,]𝐿)–cn→ℂ) → 𝐹:(𝐾[,]𝐿)⟶ℂ)
2321, 22syl 17 . . . . . . . . . . . 12 (𝜑𝐹:(𝐾[,]𝐿)⟶ℂ)
2420, 23feq1dd 38341 . . . . . . . . . . 11 (𝜑 → (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶):(𝐾[,]𝐿)⟶ℂ)
2524mptex2 38344 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝐾[,]𝐿)) → 𝐶 ∈ ℂ)
2616, 25syldan 486 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → 𝐶 ∈ ℂ)
2719fvmpt2 6200 . . . . . . . . 9 ((𝑢 ∈ (𝐾[,]𝐿) ∧ 𝐶 ∈ ℂ) → (𝐹𝑢) = 𝐶)
2816, 26, 27syl2anc 691 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐹𝑢) = 𝐶)
2928, 26eqeltrd 2688 . . . . . . 7 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐹𝑢) ∈ ℂ)
3018, 29eqeltrd 2688 . . . . . 6 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))) ∈ ℂ)
313fvmpt2 6200 . . . . . 6 ((𝑢 ∈ ℝ ∧ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))) ∈ ℂ) → (𝐺𝑢) = if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))))
3217, 30, 31syl2anc 691 . . . . 5 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐺𝑢) = if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))))
3332, 18, 283eqtrd 2648 . . . 4 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐺𝑢) = 𝐶)
349, 33ditgeq3d 38856 . . 3 (𝜑 → ⨜[𝐾𝐿](𝐺𝑢) d𝑢 = ⨜[𝐾𝐿]𝐶 d𝑢)
35 itgsubsticclem.3 . . . 4 (𝜑𝑋 ∈ ℝ)
36 itgsubsticclem.4 . . . 4 (𝜑𝑌 ∈ ℝ)
37 itgsubsticclem.5 . . . 4 (𝜑𝑋𝑌)
38 mnfxr 9975 . . . . 5 -∞ ∈ ℝ*
3938a1i 11 . . . 4 (𝜑 → -∞ ∈ ℝ*)
40 pnfxr 9971 . . . . 5 +∞ ∈ ℝ*
4140a1i 11 . . . 4 (𝜑 → +∞ ∈ ℝ*)
42 ioomax 12119 . . . . . . . . 9 (-∞(,)+∞) = ℝ
4342eqcomi 2619 . . . . . . . 8 ℝ = (-∞(,)+∞)
4443a1i 11 . . . . . . 7 (𝜑 → ℝ = (-∞(,)+∞))
4512, 44sseqtrd 3604 . . . . . 6 (𝜑 → (𝐾[,]𝐿) ⊆ (-∞(,)+∞))
46 ax-resscn 9872 . . . . . . 7 ℝ ⊆ ℂ
4744, 46syl6eqssr 3619 . . . . . 6 (𝜑 → (-∞(,)+∞) ⊆ ℂ)
48 cncfss 22510 . . . . . 6 (((𝐾[,]𝐿) ⊆ (-∞(,)+∞) ∧ (-∞(,)+∞) ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) ⊆ ((𝑋[,]𝑌)–cn→(-∞(,)+∞)))
4945, 47, 48syl2anc 691 . . . . 5 (𝜑 → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) ⊆ ((𝑋[,]𝑌)–cn→(-∞(,)+∞)))
50 itgsubsticclem.6 . . . . 5 (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)))
5149, 50sseldd 3569 . . . 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 4675 . . . . . . . . . . 11 𝑢(𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)
5419, 53nfcxfr 2749 . . . . . . . . . 10 𝑢𝐹
55 eqid 2610 . . . . . . . . . 10 (topGen‘ran (,)) = (topGen‘ran (,))
56 eqid 2610 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
57 eqid 2610 . . . . . . . . . . . 12 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
5857cnfldtop 22397 . . . . . . . . . . 11 (TopOpen‘ℂfld) ∈ Top
5958a1i 11 . . . . . . . . . 10 (𝜑 → (TopOpen‘ℂfld) ∈ Top)
6012, 46syl6ss 3580 . . . . . . . . . . . . 13 (𝜑 → (𝐾[,]𝐿) ⊆ ℂ)
61 ssid 3587 . . . . . . . . . . . . 13 ℂ ⊆ ℂ
62 eqid 2610 . . . . . . . . . . . . . 14 ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) = ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿))
63 unicntop 38230 . . . . . . . . . . . . . . . . 17 ℂ = (TopOpen‘ℂfld)
6463restid 15917 . . . . . . . . . . . . . . . 16 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
6558, 64ax-mp 5 . . . . . . . . . . . . . . 15 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
6665eqcomi 2619 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
6757, 62, 66cncfcn 22520 . . . . . . . . . . . . 13 (((𝐾[,]𝐿) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐾[,]𝐿)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
6860, 61, 67sylancl 693 . . . . . . . . . . . 12 (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
69 reex 9906 . . . . . . . . . . . . . . . 16 ℝ ∈ V
7069a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → ℝ ∈ V)
71 restabs 20779 . . . . . . . . . . . . . . 15 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐾[,]𝐿) ⊆ ℝ ∧ ℝ ∈ V) → (((TopOpen‘ℂfld) ↾t ℝ) ↾t (𝐾[,]𝐿)) = ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)))
7259, 12, 70, 71syl3anc 1318 . . . . . . . . . . . . . 14 (𝜑 → (((TopOpen‘ℂfld) ↾t ℝ) ↾t (𝐾[,]𝐿)) = ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)))
7357tgioo2 22414 . . . . . . . . . . . . . . . . 17 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
7473eqcomi 2619 . . . . . . . . . . . . . . . 16 ((TopOpen‘ℂfld) ↾t ℝ) = (topGen‘ran (,))
7574a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → ((TopOpen‘ℂfld) ↾t ℝ) = (topGen‘ran (,)))
7675oveq1d 6564 . . . . . . . . . . . . . 14 (𝜑 → (((TopOpen‘ℂfld) ↾t ℝ) ↾t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾t (𝐾[,]𝐿)))
7772, 76eqtr3d 2646 . . . . . . . . . . . . 13 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾t (𝐾[,]𝐿)))
7877oveq1d 6564 . . . . . . . . . . . 12 (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)) = (((topGen‘ran (,)) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
7968, 78eqtrd 2644 . . . . . . . . . . 11 (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((topGen‘ran (,)) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
8021, 79eleqtrd 2690 . . . . . . . . . 10 (𝜑𝐹 ∈ (((topGen‘ran (,)) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
8154, 55, 56, 3, 10, 11, 9, 59, 80icccncfext 38773 . . . . . . . . 9 (𝜑 → (𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)) ∧ (𝐺 ↾ (𝐾[,]𝐿)) = 𝐹))
8281simpld 474 . . . . . . . 8 (𝜑𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)))
83 uniretop 22376 . . . . . . . . 9 ℝ = (topGen‘ran (,))
84 eqid 2610 . . . . . . . . 9 ((TopOpen‘ℂfld) ↾t ran 𝐹) = ((TopOpen‘ℂfld) ↾t ran 𝐹)
8583, 84cnf 20860 . . . . . . . 8 (𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)) → 𝐺:ℝ⟶ ((TopOpen‘ℂfld) ↾t ran 𝐹))
8682, 85syl 17 . . . . . . 7 (𝜑𝐺:ℝ⟶ ((TopOpen‘ℂfld) ↾t ran 𝐹))
8744feq2d 5944 . . . . . . 7 (𝜑 → (𝐺:ℝ⟶ ((TopOpen‘ℂfld) ↾t ran 𝐹) ↔ 𝐺:(-∞(,)+∞)⟶ ((TopOpen‘ℂfld) ↾t ran 𝐹)))
8886, 87mpbid 221 . . . . . 6 (𝜑𝐺:(-∞(,)+∞)⟶ ((TopOpen‘ℂfld) ↾t ran 𝐹))
8988feqmptd 6159 . . . . 5 (𝜑𝐺 = (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺𝑤)))
90 frn 5966 . . . . . . . 8 (𝐹:(𝐾[,]𝐿)⟶ℂ → ran 𝐹 ⊆ ℂ)
9123, 90syl 17 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ ℂ)
92 cncfss 22510 . . . . . . 7 ((ran 𝐹 ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((-∞(,)+∞)–cn→ran 𝐹) ⊆ ((-∞(,)+∞)–cn→ℂ))
9391, 61, 92sylancl 693 . . . . . 6 (𝜑 → ((-∞(,)+∞)–cn→ran 𝐹) ⊆ ((-∞(,)+∞)–cn→ℂ))
9443oveq2i 6560 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t ℝ) = ((TopOpen‘ℂfld) ↾t (-∞(,)+∞))
9573, 94eqtri 2632 . . . . . . . . . 10 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t (-∞(,)+∞))
96 eqid 2610 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t ran 𝐹) = ((TopOpen‘ℂfld) ↾t ran 𝐹)
9757, 95, 96cncfcn 22520 . . . . . . . . 9 (((-∞(,)+∞) ⊆ ℂ ∧ ran 𝐹 ⊆ ℂ) → ((-∞(,)+∞)–cn→ran 𝐹) = ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)))
9847, 91, 97syl2anc 691 . . . . . . . 8 (𝜑 → ((-∞(,)+∞)–cn→ran 𝐹) = ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)))
9998eqcomd 2616 . . . . . . 7 (𝜑 → ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)) = ((-∞(,)+∞)–cn→ran 𝐹))
10082, 99eleqtrd 2690 . . . . . 6 (𝜑𝐺 ∈ ((-∞(,)+∞)–cn→ran 𝐹))
10193, 100sseldd 3569 . . . . 5 (𝜑𝐺 ∈ ((-∞(,)+∞)–cn→ℂ))
10289, 101eqeltrrd 2689 . . . 4 (𝜑 → (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺𝑤)) ∈ ((-∞(,)+∞)–cn→ℂ))
103 itgsubsticclem.12 . . . 4 (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
104 fveq2 6103 . . . 4 (𝑤 = 𝐴 → (𝐺𝑤) = (𝐺𝐴))
105 itgsubsticclem.14 . . . 4 (𝑥 = 𝑋𝐴 = 𝐾)
106 itgsubsticclem.15 . . . 4 (𝑥 = 𝑌𝐴 = 𝐿)
10735, 36, 37, 39, 41, 51, 52, 102, 103, 104, 105, 106itgsubst 23616 . . 3 (𝜑 → ⨜[𝐾𝐿](𝐺𝑤) d𝑤 = ⨜[𝑋𝑌]((𝐺𝐴) · 𝐵) d𝑥)
1088, 34, 1073eqtr3a 2668 . 2 (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌]((𝐺𝐴) · 𝐵) d𝑥)
1093a1i 11 . . . . 5 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐺 = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿)))))
110 simpr 476 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 = 𝐴)
11157cnfldtopon 22396 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
11235, 36iccssred 38574 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
113112, 46syl6ss 3580 . . . . . . . . . . . . . 14 (𝜑 → (𝑋[,]𝑌) ⊆ ℂ)
114 resttopon 20775 . . . . . . . . . . . . . 14 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝑋[,]𝑌) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)))
115111, 113, 114sylancr 694 . . . . . . . . . . . . 13 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)))
116 resttopon 20775 . . . . . . . . . . . . . 14 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐾[,]𝐿) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)))
117111, 60, 116sylancr 694 . . . . . . . . . . . . 13 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)))
118 eqid 2610 . . . . . . . . . . . . . . . 16 ((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) = ((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌))
11957, 118, 62cncfcn 22520 . . . . . . . . . . . . . . 15 (((𝑋[,]𝑌) ⊆ ℂ ∧ (𝐾[,]𝐿) ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) = (((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿))))
120113, 60, 119syl2anc 691 . . . . . . . . . . . . . 14 (𝜑 → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) = (((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿))))
12150, 120eleqtrd 2690 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ (((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿))))
122 cnf2 20863 . . . . . . . . . . . . 13 ((((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)) ∧ ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)) ∧ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ (((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
123115, 117, 121, 122syl3anc 1318 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
124123adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
125 eqid 2610 . . . . . . . . . . . 12 (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)
126125fmpt 6289 . . . . . . . . . . 11 (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿) ↔ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
127124, 126sylibr 223 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿))
128 ioossicc 12130 . . . . . . . . . . . 12 (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
129128sseli 3564 . . . . . . . . . . 11 (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
130129adantl 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝑥 ∈ (𝑋[,]𝑌))
131 rsp 2913 . . . . . . . . . 10 (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿) → (𝑥 ∈ (𝑋[,]𝑌) → 𝐴 ∈ (𝐾[,]𝐿)))
132127, 130, 131sylc 63 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ (𝐾[,]𝐿))
133132adantr 480 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐴 ∈ (𝐾[,]𝐿))
134110, 133eqeltrd 2688 . . . . . . 7 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 ∈ (𝐾[,]𝐿))
135134iftrued 4044 . . . . . 6 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))) = (𝐹𝑢))
136 simpll 786 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝜑)
137136, 134, 25syl2anc 691 . . . . . . 7 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 ∈ ℂ)
138134, 137, 27syl2anc 691 . . . . . 6 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → (𝐹𝑢) = 𝐶)
139 itgsubsticclem.13 . . . . . . 7 (𝑢 = 𝐴𝐶 = 𝐸)
140139adantl 481 . . . . . 6 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 = 𝐸)
141135, 138, 1403eqtrd 2648 . . . . 5 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))) = 𝐸)
14212adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → (𝐾[,]𝐿) ⊆ ℝ)
143142, 132sseldd 3569 . . . . 5 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ ℝ)
144 elex 3185 . . . . . . . 8 (𝐴 ∈ (𝐾[,]𝐿) → 𝐴 ∈ V)
145132, 144syl 17 . . . . . . 7 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ V)
146 isset 3180 . . . . . . 7 (𝐴 ∈ V ↔ ∃𝑢 𝑢 = 𝐴)
147145, 146sylib 207 . . . . . 6 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → ∃𝑢 𝑢 = 𝐴)
148140, 137eqeltrrd 2689 . . . . . 6 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐸 ∈ ℂ)
149147, 148exlimddv 1850 . . . . 5 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐸 ∈ ℂ)
150109, 141, 143, 149fvmptd 6197 . . . 4 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → (𝐺𝐴) = 𝐸)
151150oveq1d 6564 . . 3 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → ((𝐺𝐴) · 𝐵) = (𝐸 · 𝐵))
15237, 151ditgeq3d 38856 . 2 (𝜑 → ⨜[𝑋𝑌]((𝐺𝐴) · 𝐵) d𝑥 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
153108, 152eqtrd 2644 1 (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wex 1695  wcel 1977  wral 2896  Vcvv 3173  cin 3539  wss 3540  ifcif 4036   cuni 4372   class class class wbr 4583  cmpt 4643  ran crn 5039  cres 5040  wf 5800  cfv 5804  (class class class)co 6549  cc 9813  cr 9814   · cmul 9820  +∞cpnf 9950  -∞cmnf 9951  *cxr 9952   < clt 9953  cle 9954  (,)cioo 12046  [,]cicc 12049  t crest 15904  TopOpenctopn 15905  topGenctg 15921  fldccnfld 19567  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  cnccncf 22487  𝐿1cibl 23192  cdit 23416   D cdv 23433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195  df-itg2 23196  df-ibl 23197  df-itg 23198  df-0p 23243  df-ditg 23417  df-limc 23436  df-dv 23437
This theorem is referenced by:  itgsubsticc  38868
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