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

Theorem itgsubsticclem 42615
 Description: lemma for itgsubsticc 42616. (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 6645 . . . 4 (𝑢 = 𝑤 → (𝐺𝑢) = (𝐺𝑤))
2 nfcv 2955 . . . 4 𝑤(𝐺𝑢)
3 itgsubsticclem.2 . . . . . 6 𝐺 = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))))
4 nfmpt1 5128 . . . . . 6 𝑢(𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))))
53, 4nfcxfr 2953 . . . . 5 𝑢𝐺
6 nfcv 2955 . . . . 5 𝑢𝑤
75, 6nffv 6655 . . . 4 𝑢(𝐺𝑤)
81, 2, 7cbvditg 24457 . . 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 12812 . . . . . . . 8 (𝜑 → (𝐾[,]𝐿) ⊆ ℝ)
1312adantr 484 . . . . . . 7 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐾[,]𝐿) ⊆ ℝ)
14 ioossicc 12811 . . . . . . . . 9 (𝐾(,)𝐿) ⊆ (𝐾[,]𝐿)
1514sseli 3911 . . . . . . . 8 (𝑢 ∈ (𝐾(,)𝐿) → 𝑢 ∈ (𝐾[,]𝐿))
1615adantl 485 . . . . . . 7 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ (𝐾[,]𝐿))
1713, 16sseldd 3916 . . . . . 6 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → 𝑢 ∈ ℝ)
1816iftrued 4433 . . . . . . 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 23498 . . . . . . . . . . . . 13 (𝐹 ∈ ((𝐾[,]𝐿)–cn→ℂ) → 𝐹:(𝐾[,]𝐿)⟶ℂ)
2321, 22syl 17 . . . . . . . . . . . 12 (𝜑𝐹:(𝐾[,]𝐿)⟶ℂ)
2420, 23feq1dd 41789 . . . . . . . . . . 11 (𝜑 → (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶):(𝐾[,]𝐿)⟶ℂ)
2524fvmptelrn 6854 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝐾[,]𝐿)) → 𝐶 ∈ ℂ)
2616, 25syldan 594 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → 𝐶 ∈ ℂ)
2719fvmpt2 6756 . . . . . . . . 9 ((𝑢 ∈ (𝐾[,]𝐿) ∧ 𝐶 ∈ ℂ) → (𝐹𝑢) = 𝐶)
2816, 26, 27syl2anc 587 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐹𝑢) = 𝐶)
2928, 26eqeltrd 2890 . . . . . . 7 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐹𝑢) ∈ ℂ)
3018, 29eqeltrd 2890 . . . . . 6 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))) ∈ ℂ)
313fvmpt2 6756 . . . . . 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 587 . . . . 5 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐺𝑢) = if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))))
3332, 18, 283eqtrd 2837 . . . 4 ((𝜑𝑢 ∈ (𝐾(,)𝐿)) → (𝐺𝑢) = 𝐶)
349, 33ditgeq3d 42604 . . 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 10687 . . . . 5 -∞ ∈ ℝ*
3938a1i 11 . . . 4 (𝜑 → -∞ ∈ ℝ*)
40 pnfxr 10684 . . . . 5 +∞ ∈ ℝ*
4140a1i 11 . . . 4 (𝜑 → +∞ ∈ ℝ*)
42 ioomax 12800 . . . . . . . . 9 (-∞(,)+∞) = ℝ
4342eqcomi 2807 . . . . . . . 8 ℝ = (-∞(,)+∞)
4443a1i 11 . . . . . . 7 (𝜑 → ℝ = (-∞(,)+∞))
4512, 44sseqtrd 3955 . . . . . 6 (𝜑 → (𝐾[,]𝐿) ⊆ (-∞(,)+∞))
46 ax-resscn 10583 . . . . . . 7 ℝ ⊆ ℂ
4744, 46eqsstrrdi 3970 . . . . . 6 (𝜑 → (-∞(,)+∞) ⊆ ℂ)
48 cncfss 23504 . . . . . 6 (((𝐾[,]𝐿) ⊆ (-∞(,)+∞) ∧ (-∞(,)+∞) ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) ⊆ ((𝑋[,]𝑌)–cn→(-∞(,)+∞)))
4945, 47, 48syl2anc 587 . . . . 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 3916 . . . 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 5128 . . . . . . . . . . 11 𝑢(𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)
5419, 53nfcxfr 2953 . . . . . . . . . 10 𝑢𝐹
55 eqid 2798 . . . . . . . . . 10 (topGen‘ran (,)) = (topGen‘ran (,))
56 eqid 2798 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
57 eqid 2798 . . . . . . . . . . . 12 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
5857cnfldtop 23389 . . . . . . . . . . 11 (TopOpen‘ℂfld) ∈ Top
5958a1i 11 . . . . . . . . . 10 (𝜑 → (TopOpen‘ℂfld) ∈ Top)
6012, 46sstrdi 3927 . . . . . . . . . . . . 13 (𝜑 → (𝐾[,]𝐿) ⊆ ℂ)
61 ssid 3937 . . . . . . . . . . . . 13 ℂ ⊆ ℂ
62 eqid 2798 . . . . . . . . . . . . . 14 ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) = ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿))
63 unicntop 23391 . . . . . . . . . . . . . . . . 17 ℂ = (TopOpen‘ℂfld)
6463restid 16699 . . . . . . . . . . . . . . . 16 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
6558, 64ax-mp 5 . . . . . . . . . . . . . . 15 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
6665eqcomi 2807 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
6757, 62, 66cncfcn 23515 . . . . . . . . . . . . 13 (((𝐾[,]𝐿) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐾[,]𝐿)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
6860, 61, 67sylancl 589 . . . . . . . . . . . 12 (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
69 reex 10617 . . . . . . . . . . . . . . . 16 ℝ ∈ V
7069a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → ℝ ∈ V)
71 restabs 21770 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 14 (𝜑 → (((TopOpen‘ℂfld) ↾t ℝ) ↾t (𝐾[,]𝐿)) = ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)))
7357tgioo2 23408 . . . . . . . . . . . . . . . . 17 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
7473eqcomi 2807 . . . . . . . . . . . . . . . 16 ((TopOpen‘ℂfld) ↾t ℝ) = (topGen‘ran (,))
7574a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → ((TopOpen‘ℂfld) ↾t ℝ) = (topGen‘ran (,)))
7675oveq1d 7150 . . . . . . . . . . . . . 14 (𝜑 → (((TopOpen‘ℂfld) ↾t ℝ) ↾t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾t (𝐾[,]𝐿)))
7772, 76eqtr3d 2835 . . . . . . . . . . . . 13 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) = ((topGen‘ran (,)) ↾t (𝐾[,]𝐿)))
7877oveq1d 7150 . . . . . . . . . . . 12 (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)) = (((topGen‘ran (,)) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
7968, 78eqtrd 2833 . . . . . . . . . . 11 (𝜑 → ((𝐾[,]𝐿)–cn→ℂ) = (((topGen‘ran (,)) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
8021, 79eleqtrd 2892 . . . . . . . . . 10 (𝜑𝐹 ∈ (((topGen‘ran (,)) ↾t (𝐾[,]𝐿)) Cn (TopOpen‘ℂfld)))
8154, 55, 56, 3, 10, 11, 9, 59, 80icccncfext 42527 . . . . . . . . 9 (𝜑 → (𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)) ∧ (𝐺 ↾ (𝐾[,]𝐿)) = 𝐹))
8281simpld 498 . . . . . . . 8 (𝜑𝐺 ∈ ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)))
83 uniretop 23368 . . . . . . . . 9 ℝ = (topGen‘ran (,))
84 eqid 2798 . . . . . . . . 9 ((TopOpen‘ℂfld) ↾t ran 𝐹) = ((TopOpen‘ℂfld) ↾t ran 𝐹)
8583, 84cnf 21851 . . . . . . . 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 6473 . . . . . . 7 (𝜑 → (𝐺:ℝ⟶ ((TopOpen‘ℂfld) ↾t ran 𝐹) ↔ 𝐺:(-∞(,)+∞)⟶ ((TopOpen‘ℂfld) ↾t ran 𝐹)))
8886, 87mpbid 235 . . . . . 6 (𝜑𝐺:(-∞(,)+∞)⟶ ((TopOpen‘ℂfld) ↾t ran 𝐹))
8988feqmptd 6708 . . . . 5 (𝜑𝐺 = (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺𝑤)))
9023frnd 6494 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ ℂ)
91 cncfss 23504 . . . . . . 7 ((ran 𝐹 ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((-∞(,)+∞)–cn→ran 𝐹) ⊆ ((-∞(,)+∞)–cn→ℂ))
9290, 61, 91sylancl 589 . . . . . 6 (𝜑 → ((-∞(,)+∞)–cn→ran 𝐹) ⊆ ((-∞(,)+∞)–cn→ℂ))
9343oveq2i 7146 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t ℝ) = ((TopOpen‘ℂfld) ↾t (-∞(,)+∞))
9473, 93eqtri 2821 . . . . . . . . . 10 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t (-∞(,)+∞))
95 eqid 2798 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t ran 𝐹) = ((TopOpen‘ℂfld) ↾t ran 𝐹)
9657, 94, 95cncfcn 23515 . . . . . . . . 9 (((-∞(,)+∞) ⊆ ℂ ∧ ran 𝐹 ⊆ ℂ) → ((-∞(,)+∞)–cn→ran 𝐹) = ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)))
9747, 90, 96syl2anc 587 . . . . . . . 8 (𝜑 → ((-∞(,)+∞)–cn→ran 𝐹) = ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)))
9897eqcomd 2804 . . . . . . 7 (𝜑 → ((topGen‘ran (,)) Cn ((TopOpen‘ℂfld) ↾t ran 𝐹)) = ((-∞(,)+∞)–cn→ran 𝐹))
9982, 98eleqtrd 2892 . . . . . 6 (𝜑𝐺 ∈ ((-∞(,)+∞)–cn→ran 𝐹))
10092, 99sseldd 3916 . . . . 5 (𝜑𝐺 ∈ ((-∞(,)+∞)–cn→ℂ))
10189, 100eqeltrrd 2891 . . . 4 (𝜑 → (𝑤 ∈ (-∞(,)+∞) ↦ (𝐺𝑤)) ∈ ((-∞(,)+∞)–cn→ℂ))
102 itgsubsticclem.12 . . . 4 (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
103 fveq2 6645 . . . 4 (𝑤 = 𝐴 → (𝐺𝑤) = (𝐺𝐴))
104 itgsubsticclem.14 . . . 4 (𝑥 = 𝑋𝐴 = 𝐾)
105 itgsubsticclem.15 . . . 4 (𝑥 = 𝑌𝐴 = 𝐿)
10635, 36, 37, 39, 41, 51, 52, 101, 102, 103, 104, 105itgsubst 24652 . . 3 (𝜑 → ⨜[𝐾𝐿](𝐺𝑤) d𝑤 = ⨜[𝑋𝑌]((𝐺𝐴) · 𝐵) d𝑥)
1078, 34, 1063eqtr3a 2857 . 2 (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌]((𝐺𝐴) · 𝐵) d𝑥)
1083a1i 11 . . . . 5 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐺 = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿)))))
109 simpr 488 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 = 𝐴)
11057cnfldtopon 23388 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
11135, 36iccssred 12812 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
112111, 46sstrdi 3927 . . . . . . . . . . . . . 14 (𝜑 → (𝑋[,]𝑌) ⊆ ℂ)
113 resttopon 21766 . . . . . . . . . . . . . 14 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝑋[,]𝑌) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)))
114110, 112, 113sylancr 590 . . . . . . . . . . . . 13 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) ∈ (TopOn‘(𝑋[,]𝑌)))
115 resttopon 21766 . . . . . . . . . . . . . 14 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐾[,]𝐿) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)))
116110, 60, 115sylancr 590 . . . . . . . . . . . . 13 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿)) ∈ (TopOn‘(𝐾[,]𝐿)))
117 eqid 2798 . . . . . . . . . . . . . . . 16 ((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) = ((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌))
11857, 117, 62cncfcn 23515 . . . . . . . . . . . . . . 15 (((𝑋[,]𝑌) ⊆ ℂ ∧ (𝐾[,]𝐿) ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) = (((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿))))
119112, 60, 118syl2anc 587 . . . . . . . . . . . . . 14 (𝜑 → ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)) = (((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿))))
12050, 119eleqtrd 2892 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ (((TopOpen‘ℂfld) ↾t (𝑋[,]𝑌)) Cn ((TopOpen‘ℂfld) ↾t (𝐾[,]𝐿))))
121 cnf2 21854 . . . . . . . . . . . . 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))
122114, 116, 120, 121syl3anc 1368 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
123122adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
124 eqid 2798 . . . . . . . . . . . 12 (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)
125124fmpt 6851 . . . . . . . . . . 11 (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿) ↔ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝐾[,]𝐿))
126123, 125sylibr 237 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿))
127 ioossicc 12811 . . . . . . . . . . . 12 (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
128127sseli 3911 . . . . . . . . . . 11 (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
129128adantl 485 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝑥 ∈ (𝑋[,]𝑌))
130 rsp 3170 . . . . . . . . . 10 (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝐾[,]𝐿) → (𝑥 ∈ (𝑋[,]𝑌) → 𝐴 ∈ (𝐾[,]𝐿)))
131126, 129, 130sylc 65 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ (𝐾[,]𝐿))
132131adantr 484 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐴 ∈ (𝐾[,]𝐿))
133109, 132eqeltrd 2890 . . . . . . 7 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝑢 ∈ (𝐾[,]𝐿))
134133iftrued 4433 . . . . . 6 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))) = (𝐹𝑢))
135 simpll 766 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝜑)
136135, 133, 25syl2anc 587 . . . . . . 7 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 ∈ ℂ)
137133, 136, 27syl2anc 587 . . . . . 6 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → (𝐹𝑢) = 𝐶)
138 itgsubsticclem.13 . . . . . . 7 (𝑢 = 𝐴𝐶 = 𝐸)
139138adantl 485 . . . . . 6 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐶 = 𝐸)
140134, 137, 1393eqtrd 2837 . . . . 5 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))) = 𝐸)
14112adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → (𝐾[,]𝐿) ⊆ ℝ)
142141, 131sseldd 3916 . . . . 5 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ ℝ)
143 elex 3459 . . . . . . . 8 (𝐴 ∈ (𝐾[,]𝐿) → 𝐴 ∈ V)
144131, 143syl 17 . . . . . . 7 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ V)
145 isset 3453 . . . . . . 7 (𝐴 ∈ V ↔ ∃𝑢 𝑢 = 𝐴)
146144, 145sylib 221 . . . . . 6 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → ∃𝑢 𝑢 = 𝐴)
147139, 136eqeltrrd 2891 . . . . . 6 (((𝜑𝑥 ∈ (𝑋(,)𝑌)) ∧ 𝑢 = 𝐴) → 𝐸 ∈ ℂ)
148146, 147exlimddv 1936 . . . . 5 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐸 ∈ ℂ)
149108, 140, 142, 148fvmptd 6752 . . . 4 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → (𝐺𝐴) = 𝐸)
150149oveq1d 7150 . . 3 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → ((𝐺𝐴) · 𝐵) = (𝐸 · 𝐵))
15137, 150ditgeq3d 42604 . 2 (𝜑 → ⨜[𝑋𝑌]((𝐺𝐴) · 𝐵) d𝑥 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
152107, 151eqtrd 2833 1 (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  ifcif 4425  ∪ cuni 4800   class class class wbr 5030   ↦ cmpt 5110  ran crn 5520   ↾ cres 5521  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ℂcc 10524  ℝcr 10525   · cmul 10531  +∞cpnf 10661  -∞cmnf 10662  ℝ*cxr 10663   < clt 10664   ≤ cle 10665  (,)cioo 12726  [,]cicc 12729   ↾t crest 16686  TopOpenctopn 16687  topGenctg 16703  ℂfldccnfld 20091  Topctop 21498  TopOnctopon 21515   Cn ccn 21829  –cn→ccncf 23481  𝐿1cibl 24221  ⨜cdit 24449   D cdv 24466 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cc 9846  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605  ax-mulf 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-symdif 4169  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-disj 4996  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-ofr 7390  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-omul 8090  df-er 8272  df-map 8391  df-pm 8392  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-fi 8859  df-sup 8890  df-inf 8891  df-oi 8958  df-dju 9314  df-card 9352  df-acn 9355  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12886  df-fzo 13029  df-fl 13157  df-mod 13233  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-limsup 14820  df-clim 14837  df-rlim 14838  df-sum 15035  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-pt 16710  df-prds 16713  df-xrs 16767  df-qtop 16772  df-imas 16773  df-xps 16775  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-mulg 18217  df-cntz 18439  df-cmn 18900  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-fbas 20088  df-fg 20089  df-cnfld 20092  df-top 21499  df-topon 21516  df-topsp 21538  df-bases 21551  df-cld 21624  df-ntr 21625  df-cls 21626  df-nei 21703  df-lp 21741  df-perf 21742  df-cn 21832  df-cnp 21833  df-haus 21920  df-cmp 21992  df-tx 22167  df-hmeo 22360  df-fil 22451  df-fm 22543  df-flim 22544  df-flf 22545  df-xms 22927  df-ms 22928  df-tms 22929  df-cncf 23483  df-ovol 24068  df-vol 24069  df-mbf 24223  df-itg1 24224  df-itg2 24225  df-ibl 24226  df-itg 24227  df-0p 24274  df-ditg 24450  df-limc 24469  df-dv 24470 This theorem is referenced by:  itgsubsticc  42616
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