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Theorem cnmpt2t 13008
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 5494 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧) = ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑢, 𝑣⟩))
2 df-ov 5853 . . . . . . 7 (𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣) = ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑢, 𝑣⟩)
31, 2eqtr4di 2221 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧) = (𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣))
4 fveq2 5494 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧) = ((𝑥𝑋, 𝑦𝑌𝐵)‘⟨𝑢, 𝑣⟩))
5 df-ov 5853 . . . . . . 7 (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣) = ((𝑥𝑋, 𝑦𝑌𝐵)‘⟨𝑢, 𝑣⟩)
64, 5eqtr4di 2221 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧) = (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣))
73, 6opeq12d 3771 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩ = ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩)
87mpompt 5942 . . . 4 (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑢𝑋, 𝑣𝑌 ↦ ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩)
9 nfcv 2312 . . . . . . 7 𝑥𝑢
10 nfmpo1 5917 . . . . . . 7 𝑥(𝑥𝑋, 𝑦𝑌𝐴)
11 nfcv 2312 . . . . . . 7 𝑥𝑣
129, 10, 11nfov 5880 . . . . . 6 𝑥(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣)
13 nfmpo1 5917 . . . . . . 7 𝑥(𝑥𝑋, 𝑦𝑌𝐵)
149, 13, 11nfov 5880 . . . . . 6 𝑥(𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)
1512, 14nfop 3779 . . . . 5 𝑥⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩
16 nfcv 2312 . . . . . . 7 𝑦𝑢
17 nfmpo2 5918 . . . . . . 7 𝑦(𝑥𝑋, 𝑦𝑌𝐴)
18 nfcv 2312 . . . . . . 7 𝑦𝑣
1916, 17, 18nfov 5880 . . . . . 6 𝑦(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣)
20 nfmpo2 5918 . . . . . . 7 𝑦(𝑥𝑋, 𝑦𝑌𝐵)
2116, 20, 18nfov 5880 . . . . . 6 𝑦(𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)
2219, 21nfop 3779 . . . . 5 𝑦⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩
23 nfcv 2312 . . . . 5 𝑢⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩
24 nfcv 2312 . . . . 5 𝑣⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩
25 oveq12 5859 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣) = (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦))
26 oveq12 5859 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣) = (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦))
2725, 26opeq12d 3771 . . . . 5 ((𝑢 = 𝑥𝑣 = 𝑦) → ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩ = ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩)
2815, 22, 23, 24, 27cbvmpo 5929 . . . 4 (𝑢𝑋, 𝑣𝑌 ↦ ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩)
298, 28eqtri 2191 . . 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 12977 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
3330, 31, 32syl2anc 409 . . . 4 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
34 toponuni 12728 . . . 4 ((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
35 mpteq1 4071 . . . 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 17 . . 3 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩))
37 simp2 993 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝑥𝑋)
38 simp3 994 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝑦𝑌)
39 cnmpt21.a . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
40 cntop2 12917 . . . . . . . . . . . 12 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
4139, 40syl 14 . . . . . . . . . . 11 (𝜑𝐿 ∈ Top)
42 toptopon2 12732 . . . . . . . . . . 11 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
4341, 42sylib 121 . . . . . . . . . 10 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
44 cnf2 12920 . . . . . . . . . 10 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
4533, 43, 39, 44syl3anc 1233 . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
46 eqid 2170 . . . . . . . . . 10 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
4746fmpo 6177 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
4845, 47sylibr 133 . . . . . . . 8 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 𝐿)
49 rsp2 2520 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 → ((𝑥𝑋𝑦𝑌) → 𝐴 𝐿))
5048, 49syl 14 . . . . . . 7 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐴 𝐿))
51503impib 1196 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝐴 𝐿)
5246ovmpt4g 5972 . . . . . 6 ((𝑥𝑋𝑦𝑌𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
5337, 38, 51, 52syl3anc 1233 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
54 cnmpt2t.b . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
55 cntop2 12917 . . . . . . . . . . . 12 ((𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
5654, 55syl 14 . . . . . . . . . . 11 (𝜑𝑀 ∈ Top)
57 toptopon2 12732 . . . . . . . . . . 11 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
5856, 57sylib 121 . . . . . . . . . 10 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
59 cnf2 12920 . . . . . . . . . 10 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑀 ∈ (TopOn‘ 𝑀) ∧ (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶ 𝑀)
6033, 58, 54, 59syl3anc 1233 . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶ 𝑀)
61 eqid 2170 . . . . . . . . . 10 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
6261fmpo 6177 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝐵 𝑀 ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶ 𝑀)
6360, 62sylibr 133 . . . . . . . 8 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 𝑀)
64 rsp2 2520 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 𝑀 → ((𝑥𝑋𝑦𝑌) → 𝐵 𝑀))
6563, 64syl 14 . . . . . . 7 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐵 𝑀))
66653impib 1196 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝐵 𝑀)
6761ovmpt4g 5972 . . . . . 6 ((𝑥𝑋𝑦𝑌𝐵 𝑀) → (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦) = 𝐵)
6837, 38, 66, 67syl3anc 1233 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦) = 𝐵)
6953, 68opeq12d 3771 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩ = ⟨𝐴, 𝐵⟩)
7069mpoeq3dva 5914 . . 3 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩))
7129, 36, 703eqtr3a 2227 . 2 (𝜑 → (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩))
72 eqid 2170 . . . 4 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
73 eqid 2170 . . . 4 (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩)
7472, 73txcnmpt 12988 . . 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)))
7539, 54, 74syl2anc 409 . 2 (𝜑 → (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
7671, 75eqeltrrd 2248 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 973   = wceq 1348  wcel 2141  wral 2448  cop 3584   cuni 3794  cmpt 4048   × cxp 4607  wf 5192  cfv 5196  (class class class)co 5850  cmpo 5852  Topctop 12710  TopOnctopon 12723   Cn ccn 12900   ×t ctx 12967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-map 6624  df-topgen 12586  df-top 12711  df-topon 12724  df-bases 12756  df-cn 12903  df-tx 12968
This theorem is referenced by:  cnmpt22  13009  txhmeo  13034  txswaphmeo  13036
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