Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  itgsubsticclem Structured version   Visualization version   GIF version

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  ℂcc 10097  ℝcr 10098   · cmul 10104  +∞cpnf 10234  -∞cmnf 10235  ℝ*cxr 10236   < clt 10237   ≤ cle 10238  (,)cioo 12339  [,]cicc 12342   ↾t crest 16254  TopOpenctopn 16255  topGenctg 16271  ℂfldccnfld 19919  Topctop 20871  TopOnctopon 20888   Cn ccn 21201  –cn→ccncf 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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