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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  cc 10524  cr 10525   · cmul 10531  +∞cpnf 10661  -∞cmnf 10662  *cxr 10663   < clt 10664  cle 10665  (,)cioo 12726  [,]cicc 12729  t crest 16686  TopOpenctopn 16687  topGenctg 16703  fldccnfld 20091  Topctop 21498  TopOnctopon 21515   Cn ccn 21829  cnccncf 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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