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Theorem cnmpt22f 22284
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 21850 . . . 4 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
63, 5syl 17 . . 3 (𝜑𝐿 ∈ Top)
7 toptopon2 21527 . . 3 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
86, 7sylib 221 . 2 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
9 cntop2 21850 . . . 4 ((𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
104, 9syl 17 . . 3 (𝜑𝑀 ∈ Top)
11 toptopon2 21527 . . 3 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
1210, 11sylib 221 . 2 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
13 txtopon 22200 . . . . . . 7 ((𝐿 ∈ (TopOn‘ 𝐿) ∧ 𝑀 ∈ (TopOn‘ 𝑀)) → (𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)))
148, 12, 13syl2anc 587 . . . . . 6 (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)))
15 cnmpt22f.f . . . . . . . 8 (𝜑𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
16 cntop2 21850 . . . . . . . 8 (𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
1715, 16syl 17 . . . . . . 7 (𝜑𝑁 ∈ Top)
18 toptopon2 21527 . . . . . . 7 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘ 𝑁))
1917, 18sylib 221 . . . . . 6 (𝜑𝑁 ∈ (TopOn‘ 𝑁))
20 cnf2 21858 . . . . . 6 (((𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)) ∧ 𝑁 ∈ (TopOn‘ 𝑁) ∧ 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → 𝐹:( 𝐿 × 𝑀)⟶ 𝑁)
2114, 19, 15, 20syl3anc 1368 . . . . 5 (𝜑𝐹:( 𝐿 × 𝑀)⟶ 𝑁)
2221ffnd 6492 . . . 4 (𝜑𝐹 Fn ( 𝐿 × 𝑀))
23 fnov 7265 . . . 4 (𝐹 Fn ( 𝐿 × 𝑀) ↔ 𝐹 = (𝑧 𝐿, 𝑤 𝑀 ↦ (𝑧𝐹𝑤)))
2422, 23sylib 221 . . 3 (𝜑𝐹 = (𝑧 𝐿, 𝑤 𝑀 ↦ (𝑧𝐹𝑤)))
2524, 15eqeltrrd 2894 . 2 (𝜑 → (𝑧 𝐿, 𝑤 𝑀 ↦ (𝑧𝐹𝑤)) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
26 oveq12 7148 . 2 ((𝑧 = 𝐴𝑤 = 𝐵) → (𝑧𝐹𝑤) = (𝐴𝐹𝐵))
271, 2, 3, 4, 8, 12, 25, 26cnmpt22 22283 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112   cuni 4803   × cxp 5521   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  Topctop 21502  TopOnctopon 21519   Cn ccn 21833   ×t ctx 22169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-topgen 16713  df-top 21503  df-topon 21520  df-bases 21555  df-cn 21836  df-tx 22171
This theorem is referenced by:  cnmptcom  22287  cnmpt2plusg  22697  istgp2  22700  cnmpt2vsca  22804  cnmpt2ds  23452  divcn  23477  cnrehmeo  23562  htpycom  23585  htpyco1  23587  htpycc  23589  reparphti  23606  pcohtpylem  23628  cnmpt2ip  23856  cxpcn  25338  vmcn  28486  dipcn  28507  mndpluscn  31283  cvxsconn  32604
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