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Theorem itgsubsticclem 40663
Description: lemma for itgsubsticc 40664. (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 6340 . . . 4 (𝑢 = 𝑤 → (𝐺𝑢) = (𝐺𝑤))
2 nfcv 2890 . . . 4 𝑤(𝐺𝑢)
3 itgsubsticclem.2 . . . . . 6 𝐺 = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))))
4 nfmpt1 4887 . . . . . 6 𝑢(𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))))
53, 4nfcxfr 2888 . . . . 5 𝑢𝐺
6 nfcv 2890 . . . . 5 𝑢𝑤
75, 6nffv 6347 . . . 4 𝑢(𝐺𝑤)
81, 2, 7cbvditg 23788 . . 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 40199 . . . . . . . 8 (𝜑 → (𝐾[,]𝐿) ⊆ ℝ)
1312adantr 472 . . . . . . 7 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐾[,]𝐿) ⊆ ℝ)
14 ioossicc 12423 . . . . . . . . 9 (𝐾(,)𝐿) ⊆ (𝐾[,]𝐿)
1514sseli 3728 . . . . . . . 8 (𝑢 ∈ (𝐾(,)𝐿) → 𝑢 ∈ (𝐾[,]𝐿))
1615adantl 473 . . . . . . 7 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ (𝐾[,]𝐿))
1713, 16sseldd 3733 . . . . . 6 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ ℝ)
1816iftrued 4226 . . . . . . 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 22868 . . . . . . . . . . . . 13 (𝐹 ∈ ((𝐾[,]𝐿)–cn→ℂ) → 𝐹:(𝐾[,]𝐿)⟶ℂ)
2321, 22syl 17 . . . . . . . . . . . 12 (𝜑𝐹:(𝐾[,]𝐿)⟶ℂ)
2420, 23feq1dd 39815 . . . . . . . . . . 11 (𝜑 → (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶):(𝐾[,]𝐿)⟶ℂ)
2524mptex2 6535 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝐾[,]𝐿)) → 𝐶 ∈ ℂ)
2616, 25syldan 488 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → 𝐶 ∈ ℂ)
2719fvmpt2 6441 . . . . . . . . 9 ((𝑢 ∈ (𝐾[,]𝐿) ∧ 𝐶 ∈ ℂ) → (𝐹𝑢) = 𝐶)
2816, 26, 27syl2anc 696 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐹𝑢) = 𝐶)
2928, 26eqeltrd 2827 . . . . . . 7 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐹𝑢) ∈ ℂ)
3018, 29eqeltrd 2827 . . . . . 6 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))) ∈ ℂ)
313fvmpt2 6441 . . . . . 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 696 . . . . 5 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐺𝑢) = if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))))
3332, 18, 283eqtrd 2786 . . . 4 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐺𝑢) = 𝐶)
349, 33ditgeq3d 40652 . . 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 10259 . . . . 5 -∞ ∈ ℝ*
3938a1i 11 . . . 4 (𝜑 → -∞ ∈ ℝ*)
40 pnfxr 10255 . . . . 5 +∞ ∈ ℝ*
4140a1i 11 . . . 4 (𝜑 → +∞ ∈ ℝ*)
42 ioomax 12412 . . . . . . . . 9 (-∞(,)+∞) = ℝ
4342eqcomi 2757 . . . . . . . 8 ℝ = (-∞(,)+∞)
4443a1i 11 . . . . . . 7 (𝜑 → ℝ = (-∞(,)+∞))
4512, 44sseqtrd 3770 . . . . . 6 (𝜑 → (𝐾[,]𝐿) ⊆ (-∞(,)+∞))
46 ax-resscn 10156 . . . . . . 7 ℝ ⊆ ℂ
4744, 46syl6eqssr 3785 . . . . . 6 (𝜑 → (-∞(,)+∞) ⊆ ℂ)
48 cncfss 22874 . . . . . 6 (((𝐾[,]𝐿) ⊆ (-∞(,)+∞) ∧ (-∞(,)+∞) ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) ⊆ ((𝑋[,]𝑌)–cn→(-∞(,)+∞)))
4945, 47, 48syl2anc 696 . . . . 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 3733 . . . 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 4887 . . . . . . . . . . 11 𝑢(𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)
5419, 53nfcxfr 2888 . . . . . . . . . 10 𝑢𝐹
55 eqid 2748 . . . . . . . . . 10 (topGen‘ran (,)) = (topGen‘ran (,))
56 eqid 2748 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
57 eqid 2748 . . . . . . . . . . . 12 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
5857cnfldtop 22759 . . . . . . . . . . 11 (TopOpen‘ℂfld) ∈ Top
5958a1i 11 . . . . . . . . . 10 (𝜑 → (TopOpen‘ℂfld) ∈ Top)
6012, 46syl6ss 3744 . . . . . . . . . . . . 13 (𝜑 → (𝐾[,]𝐿) ⊆ ℂ)
61 ssid 3753 . . . . . . . . . . . . 13 ℂ ⊆ ℂ
62 eqid 2748 . . . . . . . . . . . . . 14 ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) = ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿))
63 unicntop 22761 . . . . . . . . . . . . . . . . 17 ℂ = (TopOpen‘ℂfld)
6463restid 16267 . . . . . . . . . . . . . . . 16 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
6558, 64ax-mp 5 . . . . . . . . . . . . . . 15 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
6665eqcomi 2757 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
6757, 62, 66cncfcn 22884 . . . . . . . . . . . . 13 (((𝐾[,]𝐿) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐾[,]𝐿)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
6860, 61, 67sylancl 697 . . . . . . . . . . . 12 (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
69 reex 10190 . . . . . . . . . . . . . . . 16 ℝ ∈ V
7069a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → ℝ ∈ V)
71 restabs 21142 . . . . . . . . . . . . . . 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 1463 . . . . . . . . . . . . . 14 (𝜑 → (((TopOpen‘ℂfld) ↾t ℝ) ↾t (𝐾[,]𝐿)) = ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)))
7357tgioo2 22778 . . . . . . . . . . . . . . . . 17 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
7473eqcomi 2757 . . . . . . . . . . . . . . . 16 ((TopOpen‘ℂfld) ↾t ℝ) = (topGen‘ran (,))
7574a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → ((TopOpen‘ℂfld) ↾t ℝ) = (topGen‘ran (,)))
7675oveq1d 6816 . . . . . . . . . . . . . 14 (𝜑 → (((TopOpen‘ℂfld) ↾t ℝ) ↾t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾t (𝐾[,]𝐿)))
7772, 76eqtr3d 2784 . . . . . . . . . . . . 13 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾t (𝐾[,]𝐿)))
7877oveq1d 6816 . . . . . . . . . . . 12 (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)) = (((topGen‘ran (,)) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
7968, 78eqtrd 2782 . . . . . . . . . . 11 (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((topGen‘ran (,)) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
8021, 79eleqtrd 2829 . . . . . . . . . 10 (𝜑𝐹 ∈ (((topGen‘ran (,)) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
8154, 55, 56, 3, 10, 11, 9, 59, 80icccncfext 40572 . . . . . . . . 9 (𝜑 → (𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)) ∧ (𝐺 ↾ (𝐾[,]𝐿)) = 𝐹))
8281simpld 477 . . . . . . . 8 (𝜑𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)))
83 uniretop 22738 . . . . . . . . 9 ℝ = (topGen‘ran (,))
84 eqid 2748 . . . . . . . . 9 ((TopOpen‘ℂfld) ↾t ran 𝐹) = ((TopOpen‘ℂfld) ↾t ran 𝐹)
8583, 84cnf 21223 . . . . . . . 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 6180 . . . . . . 7 (𝜑 → (𝐺:ℝ⟶ ((TopOpen‘ℂfld) ↾t ran 𝐹) ↔ 𝐺:(-∞(,)+∞)⟶ ((TopOpen‘ℂfld) ↾t ran 𝐹)))
8886, 87mpbid 222 . . . . . 6 (𝜑𝐺:(-∞(,)+∞)⟶ ((TopOpen‘ℂfld) ↾t ran 𝐹))
8988feqmptd 6399 . . . . 5 (𝜑𝐺 = (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺𝑤)))
90 frn 6202 . . . . . . . 8 (𝐹:(𝐾[,]𝐿)⟶ℂ → ran 𝐹 ⊆ ℂ)
9123, 90syl 17 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ ℂ)
92 cncfss 22874 . . . . . . 7 ((ran 𝐹 ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((-∞(,)+∞)–cn→ran 𝐹) ⊆ ((-∞(,)+∞)–cn→ℂ))
9391, 61, 92sylancl 697 . . . . . 6 (𝜑 → ((-∞(,)+∞)–cn→ran 𝐹) ⊆ ((-∞(,)+∞)–cn→ℂ))
9443oveq2i 6812 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t ℝ) = ((TopOpen‘ℂfld) ↾t (-∞(,)+∞))
9573, 94eqtri 2770 . . . . . . . . . 10 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t (-∞(,)+∞))
96 eqid 2748 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t ran 𝐹) = ((TopOpen‘ℂfld) ↾t ran 𝐹)
9757, 95, 96cncfcn 22884 . . . . . . . . 9 (((-∞(,)+∞) ⊆ ℂ ∧ ran 𝐹 ⊆ ℂ) → ((-∞(,)+∞)–cn→ran 𝐹) = ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)))
9847, 91, 97syl2anc 696 . . . . . . . 8 (𝜑 → ((-∞(,)+∞)–cn→ran 𝐹) = ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)))
9998eqcomd 2754 . . . . . . 7 (𝜑 → ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)) = ((-∞(,)+∞)–cn→ran 𝐹))
10082, 99eleqtrd 2829 . . . . . 6 (𝜑𝐺 ∈ ((-∞(,)+∞)–cn→ran 𝐹))
10193, 100sseldd 3733 . . . . 5 (𝜑𝐺 ∈ ((-∞(,)+∞)–cn→ℂ))
10289, 101eqeltrrd 2828 . . . 4 (𝜑 → (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺𝑤)) ∈ ((-∞(,)+∞)–cn→ℂ))
103 itgsubsticclem.12 . . . 4 (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
104 fveq2 6340 . . . 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 23982 . . 3 (𝜑 → ⨜[𝐾𝐿](𝐺𝑤) d𝑤 = ⨜[𝑋𝑌]((𝐺𝐴) · 𝐵) d𝑥)
1088, 34, 1073eqtr3a 2806 . 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 479 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 = 𝐴)
11157cnfldtopon 22758 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
11235, 36iccssred 40199 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
113112, 46syl6ss 3744 . . . . . . . . . . . . . 14 (𝜑 → (𝑋[,]𝑌) ⊆ ℂ)
114 resttopon 21138 . . . . . . . . . . . . . 14 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝑋[,]𝑌) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)))
115111, 113, 114sylancr 698 . . . . . . . . . . . . 13 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)))
116 resttopon 21138 . . . . . . . . . . . . . 14 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐾[,]𝐿) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)))
117111, 60, 116sylancr 698 . . . . . . . . . . . . 13 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)))
118 eqid 2748 . . . . . . . . . . . . . . . 16 ((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) = ((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌))
11957, 118, 62cncfcn 22884 . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . 14 (𝜑 → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) = (((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿))))
12150, 120eleqtrd 2829 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ (((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿))))
122 cnf2 21226 . . . . . . . . . . . . 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 1463 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
124123adantr 472 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
125 eqid 2748 . . . . . . . . . . . 12 (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)
126125fmpt 6532 . . . . . . . . . . 11 (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿) ↔ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
127124, 126sylibr 224 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿))
128 ioossicc 12423 . . . . . . . . . . . 12 (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
129128sseli 3728 . . . . . . . . . . 11 (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
130129adantl 473 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝑥 ∈ (𝑋[,]𝑌))
131 rsp 3055 . . . . . . . . . 10 (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿) → (𝑥 ∈ (𝑋[,]𝑌) → 𝐴 ∈ (𝐾[,]𝐿)))
132127, 130, 131sylc 65 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ (𝐾[,]𝐿))
133132adantr 472 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐴 ∈ (𝐾[,]𝐿))
134110, 133eqeltrd 2827 . . . . . . 7 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 ∈ (𝐾[,]𝐿))
135134iftrued 4226 . . . . . 6 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))) = (𝐹𝑢))
136 simpll 807 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝜑)
137136, 134, 25syl2anc 696 . . . . . . 7 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 ∈ ℂ)
138134, 137, 27syl2anc 696 . . . . . 6 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → (𝐹𝑢) = 𝐶)
139 itgsubsticclem.13 . . . . . . 7 (𝑢 = 𝐴𝐶 = 𝐸)
140139adantl 473 . . . . . 6 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 = 𝐸)
141135, 138, 1403eqtrd 2786 . . . . 5 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))) = 𝐸)
14212adantr 472 . . . . . 6 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → (𝐾[,]𝐿) ⊆ ℝ)
143142, 132sseldd 3733 . . . . 5 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ ℝ)
144 elex 3340 . . . . . . . 8 (𝐴 ∈ (𝐾[,]𝐿) → 𝐴 ∈ V)
145132, 144syl 17 . . . . . . 7 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ V)
146 isset 3335 . . . . . . 7 (𝐴 ∈ V ↔ ∃𝑢 𝑢 = 𝐴)
147145, 146sylib 208 . . . . . 6 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → ∃𝑢 𝑢 = 𝐴)
148140, 137eqeltrrd 2828 . . . . . 6 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐸 ∈ ℂ)
149147, 148exlimddv 2000 . . . . 5 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐸 ∈ ℂ)
150109, 141, 143, 149fvmptd 6438 . . . 4 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → (𝐺𝐴) = 𝐸)
151150oveq1d 6816 . . 3 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → ((𝐺𝐴) · 𝐵) = (𝐸 · 𝐵))
15237, 151ditgeq3d 40652 . 2 (𝜑 → ⨜[𝑋𝑌]((𝐺𝐴) · 𝐵) d𝑥 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
153108, 152eqtrd 2782 1 (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wex 1841  wcel 2127  wral 3038  Vcvv 3328  cin 3702  wss 3703  ifcif 4218   cuni 4576   class class class wbr 4792  cmpt 4869  ran crn 5255  cres 5256  wf 6033  cfv 6037  (class class class)co 6801  cc 10097  cr 10098   · cmul 10104  +∞cpnf 10234  -∞cmnf 10235  *cxr 10236   < clt 10237  cle 10238  (,)cioo 12339  [,]cicc 12342  t crest 16254  TopOpenctopn 16255  topGenctg 16271  fldccnfld 19919  Topctop 20871  TopOnctopon 20888   Cn ccn 21201  cnccncf 22851  𝐿1cibl 23556  cdit 23780   D cdv 23797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-inf2 8699  ax-cc 9420  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176  ax-pre-sup 10177  ax-addf 10178  ax-mulf 10179
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-fal 1626  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-iin 4663  df-disj 4761  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-se 5214  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-isom 6046  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-of 7050  df-ofr 7051  df-om 7219  df-1st 7321  df-2nd 7322  df-supp 7452  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-omul 7722  df-er 7899  df-map 8013  df-pm 8014  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8429  df-fi 8470  df-sup 8501  df-inf 8502  df-oi 8568  df-card 8926  df-acn 8929  df-cda 9153  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-div 10848  df-nn 11184  df-2 11242  df-3 11243  df-4 11244  df-5 11245  df-6 11246  df-7 11247  df-8 11248  df-9 11249  df-n0 11456  df-z 11541  df-dec 11657  df-uz 11851  df-q 11953  df-rp 11997  df-xneg 12110  df-xadd 12111  df-xmul 12112  df-ioo 12343  df-ioc 12344  df-ico 12345  df-icc 12346  df-fz 12491  df-fzo 12631  df-fl 12758  df-mod 12834  df-seq 12967  df-exp 13026  df-hash 13283  df-cj 14009  df-re 14010  df-im 14011  df-sqrt 14145  df-abs 14146  df-limsup 14372  df-clim 14389  df-rlim 14390  df-sum 14587  df-struct 16032  df-ndx 16033  df-slot 16034  df-base 16036  df-sets 16037  df-ress 16038  df-plusg 16127  df-mulr 16128  df-starv 16129  df-sca 16130  df-vsca 16131  df-ip 16132  df-tset 16133  df-ple 16134  df-ds 16137  df-unif 16138  df-hom 16139  df-cco 16140  df-rest 16256  df-topn 16257  df-0g 16275  df-gsum 16276  df-topgen 16277  df-pt 16278  df-prds 16281  df-xrs 16335  df-qtop 16340  df-imas 16341  df-xps 16343  df-mre 16419  df-mrc 16420  df-acs 16422  df-mgm 17414  df-sgrp 17456  df-mnd 17467  df-submnd 17508  df-mulg 17713  df-cntz 17921  df-cmn 18366  df-psmet 19911  df-xmet 19912  df-met 19913  df-bl 19914  df-mopn 19915  df-fbas 19916  df-fg 19917  df-cnfld 19920  df-top 20872  df-topon 20889  df-topsp 20910  df-bases 20923  df-cld 20996  df-ntr 20997  df-cls 20998  df-nei 21075  df-lp 21113  df-perf 21114  df-cn 21204  df-cnp 21205  df-haus 21292  df-cmp 21363  df-tx 21538  df-hmeo 21731  df-fil 21822  df-fm 21914  df-flim 21915  df-flf 21916  df-xms 22297  df-ms 22298  df-tms 22299  df-cncf 22853  df-ovol 23404  df-vol 23405  df-mbf 23558  df-itg1 23559  df-itg2 23560  df-ibl 23561  df-itg 23562  df-0p 23607  df-ditg 23781  df-limc 23800  df-dv 23801
This theorem is referenced by:  itgsubsticc  40664
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