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Theorem cnmpt22f 21418
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))
cnmpt22f.f (𝜑𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
Assertion
Ref Expression
cnmpt22f (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt22f
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnmpt21.j . 2 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 cnmpt21.k . 2 (𝜑𝐾 ∈ (TopOn‘𝑌))
3 cnmpt21.a . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
4 cnmpt2t.b . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
5 cntop2 20985 . . . 4 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
63, 5syl 17 . . 3 (𝜑𝐿 ∈ Top)
7 eqid 2621 . . . 4 𝐿 = 𝐿
87toptopon 20662 . . 3 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
96, 8sylib 208 . 2 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
10 cntop2 20985 . . . 4 ((𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
114, 10syl 17 . . 3 (𝜑𝑀 ∈ Top)
12 eqid 2621 . . . 4 𝑀 = 𝑀
1312toptopon 20662 . . 3 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
1411, 13sylib 208 . 2 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
15 txtopon 21334 . . . . . . 7 ((𝐿 ∈ (TopOn‘ 𝐿) ∧ 𝑀 ∈ (TopOn‘ 𝑀)) → (𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)))
169, 14, 15syl2anc 692 . . . . . 6 (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)))
17 cnmpt22f.f . . . . . . . 8 (𝜑𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
18 cntop2 20985 . . . . . . . 8 (𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
1917, 18syl 17 . . . . . . 7 (𝜑𝑁 ∈ Top)
20 eqid 2621 . . . . . . . 8 𝑁 = 𝑁
2120toptopon 20662 . . . . . . 7 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘ 𝑁))
2219, 21sylib 208 . . . . . 6 (𝜑𝑁 ∈ (TopOn‘ 𝑁))
23 cnf2 20993 . . . . . 6 (((𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)) ∧ 𝑁 ∈ (TopOn‘ 𝑁) ∧ 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → 𝐹:( 𝐿 × 𝑀)⟶ 𝑁)
2416, 22, 17, 23syl3anc 1323 . . . . 5 (𝜑𝐹:( 𝐿 × 𝑀)⟶ 𝑁)
25 ffn 6012 . . . . 5 (𝐹:( 𝐿 × 𝑀)⟶ 𝑁𝐹 Fn ( 𝐿 × 𝑀))
2624, 25syl 17 . . . 4 (𝜑𝐹 Fn ( 𝐿 × 𝑀))
27 fnov 6733 . . . 4 (𝐹 Fn ( 𝐿 × 𝑀) ↔ 𝐹 = (𝑧 𝐿, 𝑤 𝑀 ↦ (𝑧𝐹𝑤)))
2826, 27sylib 208 . . 3 (𝜑𝐹 = (𝑧 𝐿, 𝑤 𝑀 ↦ (𝑧𝐹𝑤)))
2928, 17eqeltrrd 2699 . 2 (𝜑 → (𝑧 𝐿, 𝑤 𝑀 ↦ (𝑧𝐹𝑤)) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
30 oveq12 6624 . 2 ((𝑧 = 𝐴𝑤 = 𝐵) → (𝑧𝐹𝑤) = (𝐴𝐹𝐵))
311, 2, 3, 4, 9, 14, 29, 30cnmpt22 21417 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987   cuni 4409   × cxp 5082   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  cmpt2 6617  Topctop 20638  TopOnctopon 20655   Cn ccn 20968   ×t ctx 21303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-map 7819  df-topgen 16044  df-top 20639  df-topon 20656  df-bases 20690  df-cn 20971  df-tx 21305
This theorem is referenced by:  cnmptcom  21421  cnmpt2plusg  21832  istgp2  21835  cnmpt2vsca  21938  cnmpt2ds  22586  divcn  22611  cnrehmeo  22692  htpycom  22715  htpyco1  22717  htpycc  22719  reparphti  22737  pcohtpylem  22759  cnmpt2ip  22987  cxpcn  24420  vmcn  27442  dipcn  27463  mndpluscn  29796  cvxsconn  30986
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