MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt22f Structured version   Visualization version   GIF version

Theorem cnmpt22f 23699
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 23265 . . . 4 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
63, 5syl 17 . . 3 (𝜑𝐿 ∈ Top)
7 toptopon2 22940 . . 3 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
86, 7sylib 218 . 2 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
9 cntop2 23265 . . . 4 ((𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
104, 9syl 17 . . 3 (𝜑𝑀 ∈ Top)
11 toptopon2 22940 . . 3 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
1210, 11sylib 218 . 2 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
13 txtopon 23615 . . . . . . 7 ((𝐿 ∈ (TopOn‘ 𝐿) ∧ 𝑀 ∈ (TopOn‘ 𝑀)) → (𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)))
148, 12, 13syl2anc 584 . . . . . 6 (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)))
15 cnmpt22f.f . . . . . . . 8 (𝜑𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
16 cntop2 23265 . . . . . . . 8 (𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
1715, 16syl 17 . . . . . . 7 (𝜑𝑁 ∈ Top)
18 toptopon2 22940 . . . . . . 7 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘ 𝑁))
1917, 18sylib 218 . . . . . 6 (𝜑𝑁 ∈ (TopOn‘ 𝑁))
20 cnf2 23273 . . . . . 6 (((𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)) ∧ 𝑁 ∈ (TopOn‘ 𝑁) ∧ 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → 𝐹:( 𝐿 × 𝑀)⟶ 𝑁)
2114, 19, 15, 20syl3anc 1370 . . . . 5 (𝜑𝐹:( 𝐿 × 𝑀)⟶ 𝑁)
2221ffnd 6738 . . . 4 (𝜑𝐹 Fn ( 𝐿 × 𝑀))
23 fnov 7564 . . . 4 (𝐹 Fn ( 𝐿 × 𝑀) ↔ 𝐹 = (𝑧 𝐿, 𝑤 𝑀 ↦ (𝑧𝐹𝑤)))
2422, 23sylib 218 . . 3 (𝜑𝐹 = (𝑧 𝐿, 𝑤 𝑀 ↦ (𝑧𝐹𝑤)))
2524, 15eqeltrrd 2840 . 2 (𝜑 → (𝑧 𝐿, 𝑤 𝑀 ↦ (𝑧𝐹𝑤)) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
26 oveq12 7440 . 2 ((𝑧 = 𝐴𝑤 = 𝐵) → (𝑧𝐹𝑤) = (𝐴𝐹𝐵))
271, 2, 3, 4, 8, 12, 25, 26cnmpt22 23698 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106   cuni 4912   × cxp 5687   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  Topctop 22915  TopOnctopon 22932   Cn ccn 23248   ×t ctx 23584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cn 23251  df-tx 23586
This theorem is referenced by:  cnmptcom  23702  cnmpt2plusg  24112  istgp2  24115  cnmpt2vsca  24219  cnmpt2ds  24879  divcnOLD  24904  cnrehmeo  24998  cnrehmeoOLD  24999  htpycom  25022  htpyco1  25024  htpycc  25026  reparphti  25043  reparphtiOLD  25044  pcohtpylem  25066  cnmpt2ip  25296  cxpcnOLD  26803  vmcn  30728  dipcn  30749  mndpluscn  33887  cvxsconn  35228
  Copyright terms: Public domain W3C validator