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Theorem cnmpt12f 21690
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
cnmptid.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt11.a (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
cnmpt1t.b (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
cnmpt12f.f (𝜑𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
Assertion
Ref Expression
cnmpt12f (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀))
Distinct variable groups:   𝑥,𝐹   𝜑,𝑥   𝑥,𝐽   𝑥,𝑀   𝑥,𝑋   𝑥,𝐾   𝑥,𝐿
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem cnmpt12f
StepHypRef Expression
1 df-ov 6796 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
21mpteq2i 4875 . 2 (𝑥𝑋 ↦ (𝐴𝐹𝐵)) = (𝑥𝑋 ↦ (𝐹‘⟨𝐴, 𝐵⟩))
3 cnmptid.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
4 cnmpt11.a . . . 4 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
5 cnmpt1t.b . . . 4 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
63, 4, 5cnmpt1t 21689 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
7 cnmpt12f.f . . 3 (𝜑𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
83, 6, 7cnmpt11f 21688 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐹‘⟨𝐴, 𝐵⟩)) ∈ (𝐽 Cn 𝑀))
92, 8syl5eqel 2854 1 (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  cop 4322  cmpt 4863  cfv 6031  (class class class)co 6793  TopOnctopon 20935   Cn ccn 21249   ×t ctx 21584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-cn 21252  df-tx 21586
This theorem is referenced by:  cnmpt12  21691  cnmpt1plusg  22111  istgp2  22115  clsnsg  22133  tgpt0  22142  cnmpt1vsca  22217  cnmpt1ds  22865  fsumcn  22893  expcn  22895  divccn  22896  cncfmpt2f  22937  cdivcncf  22940  iirevcn  22949  iihalf1cn  22951  iihalf2cn  22953  icchmeo  22960  evth  22978  evth2  22979  pcoass  23043  cnmpt1ip  23265  dvcnvlem  23959  plycn  24237  psercn2  24397  atansopn  24880  efrlim  24917  ipasslem7  28031  occllem  28502  hmopidmchi  29350  cvxpconn  31562  cvmlift2lem2  31624  cvmlift2lem3  31625  cvmliftphtlem  31637  sinccvglem  31904  knoppcnlem10  32829  broucube  33776  areacirclem2  33833  fprodcnlem  40349
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