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Theorem cnmpt22f 23683
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 23249 . . . 4 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
63, 5syl 17 . . 3 (𝜑𝐿 ∈ Top)
7 toptopon2 22924 . . 3 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
86, 7sylib 218 . 2 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
9 cntop2 23249 . . . 4 ((𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
104, 9syl 17 . . 3 (𝜑𝑀 ∈ Top)
11 toptopon2 22924 . . 3 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
1210, 11sylib 218 . 2 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
13 txtopon 23599 . . . . . . 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 23249 . . . . . . . 8 (𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
1715, 16syl 17 . . . . . . 7 (𝜑𝑁 ∈ Top)
18 toptopon2 22924 . . . . . . 7 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘ 𝑁))
1917, 18sylib 218 . . . . . 6 (𝜑𝑁 ∈ (TopOn‘ 𝑁))
20 cnf2 23257 . . . . . 6 (((𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)) ∧ 𝑁 ∈ (TopOn‘ 𝑁) ∧ 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → 𝐹:( 𝐿 × 𝑀)⟶ 𝑁)
2114, 19, 15, 20syl3anc 1373 . . . . 5 (𝜑𝐹:( 𝐿 × 𝑀)⟶ 𝑁)
2221ffnd 6737 . . . 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 2842 . 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 23682 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108   cuni 4907   × cxp 5683   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  Topctop 22899  TopOnctopon 22916   Cn ccn 23232   ×t ctx 23568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cn 23235  df-tx 23570
This theorem is referenced by:  cnmptcom  23686  cnmpt2plusg  24096  istgp2  24099  cnmpt2vsca  24203  cnmpt2ds  24865  divcnOLD  24890  cnrehmeo  24984  cnrehmeoOLD  24985  htpycom  25008  htpyco1  25010  htpycc  25012  reparphti  25029  reparphtiOLD  25030  pcohtpylem  25052  cnmpt2ip  25282  cxpcnOLD  26788  vmcn  30718  dipcn  30739  mndpluscn  33925  cvxsconn  35248
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