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Theorem cnmpt22 23669
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))
cnmpt22.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmpt22.m (𝜑𝑀 ∈ (TopOn‘𝑊))
cnmpt22.c (𝜑 → (𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
cnmpt22.d ((𝑧 = 𝐴𝑤 = 𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
cnmpt22 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
Distinct variable groups:   𝑧,𝑤,𝐴   𝑤,𝐵   𝑤,𝐷,𝑧   𝑧,𝐽   𝑥,𝑤,𝑦,𝑧,𝐿   𝜑,𝑥,𝑦,𝑧   𝑤,𝑋,𝑥,𝑦,𝑧   𝑤,𝑀,𝑥,𝑦,𝑧   𝑤,𝑁,𝑥,𝑦,𝑧   𝑤,𝑌,𝑥,𝑦,𝑧   𝑧,𝐾   𝑤,𝑊,𝑥,𝑦,𝑧   𝑤,𝑍,𝑥,𝑦,𝑧   𝑧,𝐵   𝑥,𝐶,𝑦
Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑧,𝑤)   𝐷(𝑥,𝑦)   𝐽(𝑥,𝑦,𝑤)   𝐾(𝑥,𝑦,𝑤)

Proof of Theorem cnmpt22
StepHypRef Expression
1 df-ov 7427 . . . 4 (𝐴(𝑧𝑍, 𝑤𝑊𝐶)𝐵) = ((𝑧𝑍, 𝑤𝑊𝐶)‘⟨𝐴, 𝐵⟩)
2 cnmpt21.j . . . . . . . . . 10 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 cnmpt21.k . . . . . . . . . 10 (𝜑𝐾 ∈ (TopOn‘𝑌))
4 txtopon 23586 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
52, 3, 4syl2anc 582 . . . . . . . . 9 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6 cnmpt22.l . . . . . . . . 9 (𝜑𝐿 ∈ (TopOn‘𝑍))
7 cnmpt21.a . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
8 cnf2 23244 . . . . . . . . 9 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
95, 6, 7, 8syl3anc 1368 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
10 eqid 2726 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1110fmpo 8082 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴𝑍 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
129, 11sylibr 233 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴𝑍)
13 rsp2 3265 . . . . . . 7 (∀𝑥𝑋𝑦𝑌 𝐴𝑍 → ((𝑥𝑋𝑦𝑌) → 𝐴𝑍))
1412, 13syl 17 . . . . . 6 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐴𝑍))
15143impib 1113 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → 𝐴𝑍)
16 cnmpt22.m . . . . . . . . 9 (𝜑𝑀 ∈ (TopOn‘𝑊))
17 cnmpt2t.b . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
18 cnf2 23244 . . . . . . . . 9 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑀 ∈ (TopOn‘𝑊) ∧ (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶𝑊)
195, 16, 17, 18syl3anc 1368 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶𝑊)
20 eqid 2726 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
2120fmpo 8082 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵𝑊 ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶𝑊)
2219, 21sylibr 233 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵𝑊)
23 rsp2 3265 . . . . . . 7 (∀𝑥𝑋𝑦𝑌 𝐵𝑊 → ((𝑥𝑋𝑦𝑌) → 𝐵𝑊))
2422, 23syl 17 . . . . . 6 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐵𝑊))
25243impib 1113 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → 𝐵𝑊)
2615, 25jca 510 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴𝑍𝐵𝑊))
27 txtopon 23586 . . . . . . . . . . 11 ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘𝑊)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)))
286, 16, 27syl2anc 582 . . . . . . . . . 10 (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)))
29 cnmpt22.c . . . . . . . . . . . 12 (𝜑 → (𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
30 cntop2 23236 . . . . . . . . . . . 12 ((𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
3129, 30syl 17 . . . . . . . . . . 11 (𝜑𝑁 ∈ Top)
32 toptopon2 22911 . . . . . . . . . . 11 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘ 𝑁))
3331, 32sylib 217 . . . . . . . . . 10 (𝜑𝑁 ∈ (TopOn‘ 𝑁))
34 cnf2 23244 . . . . . . . . . 10 (((𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)) ∧ 𝑁 ∈ (TopOn‘ 𝑁) ∧ (𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → (𝑧𝑍, 𝑤𝑊𝐶):(𝑍 × 𝑊)⟶ 𝑁)
3528, 33, 29, 34syl3anc 1368 . . . . . . . . 9 (𝜑 → (𝑧𝑍, 𝑤𝑊𝐶):(𝑍 × 𝑊)⟶ 𝑁)
36 eqid 2726 . . . . . . . . . 10 (𝑧𝑍, 𝑤𝑊𝐶) = (𝑧𝑍, 𝑤𝑊𝐶)
3736fmpo 8082 . . . . . . . . 9 (∀𝑧𝑍𝑤𝑊 𝐶 𝑁 ↔ (𝑧𝑍, 𝑤𝑊𝐶):(𝑍 × 𝑊)⟶ 𝑁)
3835, 37sylibr 233 . . . . . . . 8 (𝜑 → ∀𝑧𝑍𝑤𝑊 𝐶 𝑁)
39 r2al 3185 . . . . . . . 8 (∀𝑧𝑍𝑤𝑊 𝐶 𝑁 ↔ ∀𝑧𝑤((𝑧𝑍𝑤𝑊) → 𝐶 𝑁))
4038, 39sylib 217 . . . . . . 7 (𝜑 → ∀𝑧𝑤((𝑧𝑍𝑤𝑊) → 𝐶 𝑁))
41403ad2ant1 1130 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → ∀𝑧𝑤((𝑧𝑍𝑤𝑊) → 𝐶 𝑁))
42 eleq1 2814 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑧𝑍𝐴𝑍))
43 eleq1 2814 . . . . . . . . 9 (𝑤 = 𝐵 → (𝑤𝑊𝐵𝑊))
4442, 43bi2anan9 636 . . . . . . . 8 ((𝑧 = 𝐴𝑤 = 𝐵) → ((𝑧𝑍𝑤𝑊) ↔ (𝐴𝑍𝐵𝑊)))
45 cnmpt22.d . . . . . . . . 9 ((𝑧 = 𝐴𝑤 = 𝐵) → 𝐶 = 𝐷)
4645eleq1d 2811 . . . . . . . 8 ((𝑧 = 𝐴𝑤 = 𝐵) → (𝐶 𝑁𝐷 𝑁))
4744, 46imbi12d 343 . . . . . . 7 ((𝑧 = 𝐴𝑤 = 𝐵) → (((𝑧𝑍𝑤𝑊) → 𝐶 𝑁) ↔ ((𝐴𝑍𝐵𝑊) → 𝐷 𝑁)))
4847spc2gv 3586 . . . . . 6 ((𝐴𝑍𝐵𝑊) → (∀𝑧𝑤((𝑧𝑍𝑤𝑊) → 𝐶 𝑁) → ((𝐴𝑍𝐵𝑊) → 𝐷 𝑁)))
4926, 41, 26, 48syl3c 66 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → 𝐷 𝑁)
5045, 36ovmpoga 7580 . . . . 5 ((𝐴𝑍𝐵𝑊𝐷 𝑁) → (𝐴(𝑧𝑍, 𝑤𝑊𝐶)𝐵) = 𝐷)
5115, 25, 49, 50syl3anc 1368 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴(𝑧𝑍, 𝑤𝑊𝐶)𝐵) = 𝐷)
521, 51eqtr3id 2780 . . 3 ((𝜑𝑥𝑋𝑦𝑌) → ((𝑧𝑍, 𝑤𝑊𝐶)‘⟨𝐴, 𝐵⟩) = 𝐷)
5352mpoeq3dva 7502 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ((𝑧𝑍, 𝑤𝑊𝐶)‘⟨𝐴, 𝐵⟩)) = (𝑥𝑋, 𝑦𝑌𝐷))
542, 3, 7, 17cnmpt2t 23668 . . 3 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
552, 3, 54, 29cnmpt21f 23667 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ((𝑧𝑍, 𝑤𝑊𝐶)‘⟨𝐴, 𝐵⟩)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
5653, 55eqeltrrd 2827 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084  wal 1532   = wceq 1534  wcel 2099  wral 3051  cop 4639   cuni 4913   × cxp 5680  wf 6550  cfv 6554  (class class class)co 7424  cmpo 7426  Topctop 22886  TopOnctopon 22903   Cn ccn 23219   ×t ctx 23555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-map 8857  df-topgen 17458  df-top 22887  df-topon 22904  df-bases 22940  df-cn 23222  df-tx 23557
This theorem is referenced by:  cnmpt22f  23670  xkofvcn  23679  cnmptk2  23681  divcn  24877  cnrehmeo  24969  reparphti  25014  pcorevlem  25044  cxpcn  26772  cvxsconn  35071
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