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Theorem cnmpt2t 21599
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmpt21.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmpt21.a (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
cnmpt2t.b (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
Assertion
Ref Expression
cnmpt2t (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
Distinct variable groups:   𝑥,𝑦,𝐿   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑀,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt2t
Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6304 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧) = ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑢, 𝑣⟩))
2 df-ov 6768 . . . . . . 7 (𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣) = ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑢, 𝑣⟩)
31, 2syl6eqr 2776 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧) = (𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣))
4 fveq2 6304 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧) = ((𝑥𝑋, 𝑦𝑌𝐵)‘⟨𝑢, 𝑣⟩))
5 df-ov 6768 . . . . . . 7 (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣) = ((𝑥𝑋, 𝑦𝑌𝐵)‘⟨𝑢, 𝑣⟩)
64, 5syl6eqr 2776 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧) = (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣))
73, 6opeq12d 4517 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩ = ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩)
87mpt2mpt 6869 . . . 4 (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑢𝑋, 𝑣𝑌 ↦ ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩)
9 nfcv 2866 . . . . . . 7 𝑥𝑢
10 nfmpt21 6839 . . . . . . 7 𝑥(𝑥𝑋, 𝑦𝑌𝐴)
11 nfcv 2866 . . . . . . 7 𝑥𝑣
129, 10, 11nfov 6791 . . . . . 6 𝑥(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣)
13 nfmpt21 6839 . . . . . . 7 𝑥(𝑥𝑋, 𝑦𝑌𝐵)
149, 13, 11nfov 6791 . . . . . 6 𝑥(𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)
1512, 14nfop 4525 . . . . 5 𝑥⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩
16 nfcv 2866 . . . . . . 7 𝑦𝑢
17 nfmpt22 6840 . . . . . . 7 𝑦(𝑥𝑋, 𝑦𝑌𝐴)
18 nfcv 2866 . . . . . . 7 𝑦𝑣
1916, 17, 18nfov 6791 . . . . . 6 𝑦(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣)
20 nfmpt22 6840 . . . . . . 7 𝑦(𝑥𝑋, 𝑦𝑌𝐵)
2116, 20, 18nfov 6791 . . . . . 6 𝑦(𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)
2219, 21nfop 4525 . . . . 5 𝑦⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩
23 nfcv 2866 . . . . 5 𝑢⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩
24 nfcv 2866 . . . . 5 𝑣⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩
25 oveq12 6774 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣) = (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦))
26 oveq12 6774 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣) = (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦))
2725, 26opeq12d 4517 . . . . 5 ((𝑢 = 𝑥𝑣 = 𝑦) → ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩ = ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩)
2815, 22, 23, 24, 27cbvmpt2 6851 . . . 4 (𝑢𝑋, 𝑣𝑌 ↦ ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩)
298, 28eqtri 2746 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩)
30 cnmpt21.j . . . . 5 (𝜑𝐽 ∈ (TopOn‘𝑋))
31 cnmpt21.k . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
32 txtopon 21517 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
3330, 31, 32syl2anc 696 . . . 4 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
34 toponuni 20842 . . . 4 ((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
35 mpteq1 4845 . . . 4 ((𝑋 × 𝑌) = (𝐽 ×t 𝐾) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩))
3633, 34, 353syl 18 . . 3 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩))
37 simp2 1129 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝑥𝑋)
38 simp3 1130 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝑦𝑌)
39 cnmpt21.a . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
40 cntop2 21168 . . . . . . . . . . . 12 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
4139, 40syl 17 . . . . . . . . . . 11 (𝜑𝐿 ∈ Top)
42 eqid 2724 . . . . . . . . . . . 12 𝐿 = 𝐿
4342toptopon 20845 . . . . . . . . . . 11 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
4441, 43sylib 208 . . . . . . . . . 10 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
45 cnf2 21176 . . . . . . . . . 10 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
4633, 44, 39, 45syl3anc 1439 . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
47 eqid 2724 . . . . . . . . . 10 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
4847fmpt2 7357 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
4946, 48sylibr 224 . . . . . . . 8 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 𝐿)
50 rsp2 3038 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 → ((𝑥𝑋𝑦𝑌) → 𝐴 𝐿))
5149, 50syl 17 . . . . . . 7 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐴 𝐿))
52513impib 1108 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝐴 𝐿)
5347ovmpt4g 6900 . . . . . 6 ((𝑥𝑋𝑦𝑌𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
5437, 38, 52, 53syl3anc 1439 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
55 cnmpt2t.b . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
56 cntop2 21168 . . . . . . . . . . . 12 ((𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
5755, 56syl 17 . . . . . . . . . . 11 (𝜑𝑀 ∈ Top)
58 eqid 2724 . . . . . . . . . . . 12 𝑀 = 𝑀
5958toptopon 20845 . . . . . . . . . . 11 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
6057, 59sylib 208 . . . . . . . . . 10 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
61 cnf2 21176 . . . . . . . . . 10 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑀 ∈ (TopOn‘ 𝑀) ∧ (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶ 𝑀)
6233, 60, 55, 61syl3anc 1439 . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶ 𝑀)
63 eqid 2724 . . . . . . . . . 10 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
6463fmpt2 7357 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝐵 𝑀 ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶ 𝑀)
6562, 64sylibr 224 . . . . . . . 8 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 𝑀)
66 rsp2 3038 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 𝑀 → ((𝑥𝑋𝑦𝑌) → 𝐵 𝑀))
6765, 66syl 17 . . . . . . 7 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐵 𝑀))
68673impib 1108 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝐵 𝑀)
6963ovmpt4g 6900 . . . . . 6 ((𝑥𝑋𝑦𝑌𝐵 𝑀) → (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦) = 𝐵)
7037, 38, 68, 69syl3anc 1439 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦) = 𝐵)
7154, 70opeq12d 4517 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩ = ⟨𝐴, 𝐵⟩)
7271mpt2eq3dva 6836 . . 3 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩))
7329, 36, 723eqtr3a 2782 . 2 (𝜑 → (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩))
74 eqid 2724 . . . 4 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
75 eqid 2724 . . . 4 (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩)
7674, 75txcnmpt 21550 . . 3 (((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) → (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
7739, 55, 76syl2anc 696 . 2 (𝜑 → (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
7873, 77eqeltrrd 2804 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  cop 4291   cuni 4544  cmpt 4837   × cxp 5216  wf 5997  cfv 6001  (class class class)co 6765  cmpt2 6767  Topctop 20821  TopOnctopon 20838   Cn ccn 21151   ×t ctx 21486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-map 7976  df-topgen 16227  df-top 20822  df-topon 20839  df-bases 20873  df-cn 21154  df-tx 21488
This theorem is referenced by:  cnmpt22  21600  txhmeo  21729  txswaphmeo  21731  txsconnlem  31450
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