Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnmpt22f GIF version

Theorem cnmpt22f 12523
 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 12430 . . . 4 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
63, 5syl 14 . . 3 (𝜑𝐿 ∈ Top)
7 toptopon2 12245 . . 3 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
86, 7sylib 121 . 2 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
9 cntop2 12430 . . . 4 ((𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
104, 9syl 14 . . 3 (𝜑𝑀 ∈ Top)
11 toptopon2 12245 . . 3 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
1210, 11sylib 121 . 2 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
13 txtopon 12490 . . . . . . 7 ((𝐿 ∈ (TopOn‘ 𝐿) ∧ 𝑀 ∈ (TopOn‘ 𝑀)) → (𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)))
148, 12, 13syl2anc 409 . . . . . 6 (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)))
15 cnmpt22f.f . . . . . . . 8 (𝜑𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
16 cntop2 12430 . . . . . . . 8 (𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
1715, 16syl 14 . . . . . . 7 (𝜑𝑁 ∈ Top)
18 toptopon2 12245 . . . . . . 7 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘ 𝑁))
1917, 18sylib 121 . . . . . 6 (𝜑𝑁 ∈ (TopOn‘ 𝑁))
20 cnf2 12433 . . . . . 6 (((𝐿 ×t 𝑀) ∈ (TopOn‘( 𝐿 × 𝑀)) ∧ 𝑁 ∈ (TopOn‘ 𝑁) ∧ 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → 𝐹:( 𝐿 × 𝑀)⟶ 𝑁)
2114, 19, 15, 20syl3anc 1217 . . . . 5 (𝜑𝐹:( 𝐿 × 𝑀)⟶ 𝑁)
2221ffnd 5282 . . . 4 (𝜑𝐹 Fn ( 𝐿 × 𝑀))
23 fnovim 5888 . . . 4 (𝐹 Fn ( 𝐿 × 𝑀) → 𝐹 = (𝑧 𝐿, 𝑤 𝑀 ↦ (𝑧𝐹𝑤)))
2422, 23syl 14 . . 3 (𝜑𝐹 = (𝑧 𝐿, 𝑤 𝑀 ↦ (𝑧𝐹𝑤)))
2524, 15eqeltrrd 2218 . 2 (𝜑 → (𝑧 𝐿, 𝑤 𝑀 ↦ (𝑧𝐹𝑤)) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
26 oveq12 5792 . 2 ((𝑧 = 𝐴𝑤 = 𝐵) → (𝑧𝐹𝑤) = (𝐴𝐹𝐵))
271, 2, 3, 4, 8, 12, 25, 26cnmpt22 12522 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  ∪ cuni 3745   × cxp 4546   Fn wfn 5127  ⟶wf 5128  ‘cfv 5132  (class class class)co 5783   ∈ cmpo 5785  Topctop 12223  TopOnctopon 12236   Cn ccn 12413   ×t ctx 12480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-ov 5786  df-oprab 5787  df-mpo 5788  df-1st 6047  df-2nd 6048  df-map 6553  df-topgen 12200  df-top 12224  df-topon 12237  df-bases 12269  df-cn 12416  df-tx 12481 This theorem is referenced by:  cnmptcom  12526  divcnap  12783  cnrehmeocntop  12821
 Copyright terms: Public domain W3C validator