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Theorem cnmpt2t 23410
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 6891 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧) = ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑢, 𝑣⟩))
2 df-ov 7415 . . . . . . 7 (𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣) = ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑢, 𝑣⟩)
31, 2eqtr4di 2789 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧) = (𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣))
4 fveq2 6891 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧) = ((𝑥𝑋, 𝑦𝑌𝐵)‘⟨𝑢, 𝑣⟩))
5 df-ov 7415 . . . . . . 7 (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣) = ((𝑥𝑋, 𝑦𝑌𝐵)‘⟨𝑢, 𝑣⟩)
64, 5eqtr4di 2789 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧) = (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣))
73, 6opeq12d 4881 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩ = ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩)
87mpompt 7525 . . . 4 (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑢𝑋, 𝑣𝑌 ↦ ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩)
9 nfcv 2902 . . . . . . 7 𝑥𝑢
10 nfmpo1 7492 . . . . . . 7 𝑥(𝑥𝑋, 𝑦𝑌𝐴)
11 nfcv 2902 . . . . . . 7 𝑥𝑣
129, 10, 11nfov 7442 . . . . . 6 𝑥(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣)
13 nfmpo1 7492 . . . . . . 7 𝑥(𝑥𝑋, 𝑦𝑌𝐵)
149, 13, 11nfov 7442 . . . . . 6 𝑥(𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)
1512, 14nfop 4889 . . . . 5 𝑥⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩
16 nfcv 2902 . . . . . . 7 𝑦𝑢
17 nfmpo2 7493 . . . . . . 7 𝑦(𝑥𝑋, 𝑦𝑌𝐴)
18 nfcv 2902 . . . . . . 7 𝑦𝑣
1916, 17, 18nfov 7442 . . . . . 6 𝑦(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣)
20 nfmpo2 7493 . . . . . . 7 𝑦(𝑥𝑋, 𝑦𝑌𝐵)
2116, 20, 18nfov 7442 . . . . . 6 𝑦(𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)
2219, 21nfop 4889 . . . . 5 𝑦⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩
23 nfcv 2902 . . . . 5 𝑢⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩
24 nfcv 2902 . . . . 5 𝑣⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩
25 oveq12 7421 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣) = (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦))
26 oveq12 7421 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣) = (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦))
2725, 26opeq12d 4881 . . . . 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 7506 . . . 4 (𝑢𝑋, 𝑣𝑌 ↦ ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩)
298, 28eqtri 2759 . . 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 23328 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
3330, 31, 32syl2anc 583 . . . 4 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
34 toponuni 22649 . . . 4 ((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
35 mpteq1 5241 . . . 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 1136 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝑥𝑋)
38 simp3 1137 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝑦𝑌)
39 cnmpt21.a . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
40 cntop2 22978 . . . . . . . . . . . 12 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
4139, 40syl 17 . . . . . . . . . . 11 (𝜑𝐿 ∈ Top)
42 toptopon2 22653 . . . . . . . . . . 11 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
4341, 42sylib 217 . . . . . . . . . 10 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
44 cnf2 22986 . . . . . . . . . 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 1370 . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
46 eqid 2731 . . . . . . . . . 10 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
4746fmpo 8058 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
4845, 47sylibr 233 . . . . . . . 8 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 𝐿)
49 rsp2 3273 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 → ((𝑥𝑋𝑦𝑌) → 𝐴 𝐿))
5048, 49syl 17 . . . . . . 7 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐴 𝐿))
51503impib 1115 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝐴 𝐿)
5246ovmpt4g 7558 . . . . . 6 ((𝑥𝑋𝑦𝑌𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
5337, 38, 51, 52syl3anc 1370 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
54 cnmpt2t.b . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
55 cntop2 22978 . . . . . . . . . . . 12 ((𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
5654, 55syl 17 . . . . . . . . . . 11 (𝜑𝑀 ∈ Top)
57 toptopon2 22653 . . . . . . . . . . 11 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
5856, 57sylib 217 . . . . . . . . . 10 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
59 cnf2 22986 . . . . . . . . . 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 1370 . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶ 𝑀)
61 eqid 2731 . . . . . . . . . 10 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
6261fmpo 8058 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝐵 𝑀 ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶ 𝑀)
6360, 62sylibr 233 . . . . . . . 8 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 𝑀)
64 rsp2 3273 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 𝑀 → ((𝑥𝑋𝑦𝑌) → 𝐵 𝑀))
6563, 64syl 17 . . . . . . 7 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐵 𝑀))
66653impib 1115 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝐵 𝑀)
6761ovmpt4g 7558 . . . . . 6 ((𝑥𝑋𝑦𝑌𝐵 𝑀) → (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦) = 𝐵)
6837, 38, 66, 67syl3anc 1370 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦) = 𝐵)
6953, 68opeq12d 4881 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩ = ⟨𝐴, 𝐵⟩)
7069mpoeq3dva 7489 . . 3 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩))
7129, 36, 703eqtr3a 2795 . 2 (𝜑 → (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩))
72 eqid 2731 . . . 4 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
73 eqid 2731 . . . 4 (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩)
7472, 73txcnmpt 23361 . . 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 583 . 2 (𝜑 → (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
7671, 75eqeltrrd 2833 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  cop 4634   cuni 4908  cmpt 5231   × cxp 5674  wf 6539  cfv 6543  (class class class)co 7412  cmpo 7414  Topctop 22628  TopOnctopon 22645   Cn ccn 22961   ×t ctx 23297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-map 8828  df-topgen 17396  df-top 22629  df-topon 22646  df-bases 22682  df-cn 22964  df-tx 23299
This theorem is referenced by:  cnmpt22  23411  txhmeo  23540  txswaphmeo  23542  txsconnlem  34544
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