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Theorem cnmpt12 22272
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 12-Jun-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))
cnmpt12.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmpt12.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmpt12.c (𝜑 → (𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
cnmpt12.d ((𝑦 = 𝐴𝑧 = 𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
cnmpt12 (𝜑 → (𝑥𝑋𝐷) ∈ (𝐽 Cn 𝑀))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑧,𝐵   𝑦,𝐷,𝑧   𝑥,𝑦   𝜑,𝑥   𝑥,𝐽,𝑦   𝑥,𝑧,𝑀,𝑦   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦,𝑧)   𝐷(𝑥)   𝐽(𝑧)   𝐾(𝑧)   𝐿(𝑧)

Proof of Theorem cnmpt12
StepHypRef Expression
1 cnmptid.j . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 cnmpt12.k . . . . . 6 (𝜑𝐾 ∈ (TopOn‘𝑌))
3 cnmpt11.a . . . . . 6 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
4 cnf2 21854 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋𝑌)
51, 2, 3, 4syl3anc 1368 . . . . 5 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
65fvmptelrn 6854 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
7 cnmpt12.l . . . . . 6 (𝜑𝐿 ∈ (TopOn‘𝑍))
8 cnmpt1t.b . . . . . 6 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
9 cnf2 21854 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐵):𝑋𝑍)
101, 7, 8, 9syl3anc 1368 . . . . 5 (𝜑 → (𝑥𝑋𝐵):𝑋𝑍)
1110fvmptelrn 6854 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑍)
126, 11jca 515 . . . . 5 ((𝜑𝑥𝑋) → (𝐴𝑌𝐵𝑍))
13 txtopon 22196 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
142, 7, 13syl2anc 587 . . . . . . . . 9 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
15 cnmpt12.c . . . . . . . . . . 11 (𝜑 → (𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
16 cntop2 21846 . . . . . . . . . . 11 ((𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀) → 𝑀 ∈ Top)
1715, 16syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ Top)
18 toptopon2 21523 . . . . . . . . . 10 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
1917, 18sylib 221 . . . . . . . . 9 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
20 cnf2 21854 . . . . . . . . 9 (((𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)) ∧ 𝑀 ∈ (TopOn‘ 𝑀) ∧ (𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) → (𝑦𝑌, 𝑧𝑍𝐶):(𝑌 × 𝑍)⟶ 𝑀)
2114, 19, 15, 20syl3anc 1368 . . . . . . . 8 (𝜑 → (𝑦𝑌, 𝑧𝑍𝐶):(𝑌 × 𝑍)⟶ 𝑀)
22 eqid 2798 . . . . . . . . 9 (𝑦𝑌, 𝑧𝑍𝐶) = (𝑦𝑌, 𝑧𝑍𝐶)
2322fmpo 7748 . . . . . . . 8 (∀𝑦𝑌𝑧𝑍 𝐶 𝑀 ↔ (𝑦𝑌, 𝑧𝑍𝐶):(𝑌 × 𝑍)⟶ 𝑀)
2421, 23sylibr 237 . . . . . . 7 (𝜑 → ∀𝑦𝑌𝑧𝑍 𝐶 𝑀)
25 r2al 3166 . . . . . . 7 (∀𝑦𝑌𝑧𝑍 𝐶 𝑀 ↔ ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
2624, 25sylib 221 . . . . . 6 (𝜑 → ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
2726adantr 484 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
28 eleq1 2877 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑌𝐴𝑌))
29 eleq1 2877 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑍𝐵𝑍))
3028, 29bi2anan9 638 . . . . . . 7 ((𝑦 = 𝐴𝑧 = 𝐵) → ((𝑦𝑌𝑧𝑍) ↔ (𝐴𝑌𝐵𝑍)))
31 cnmpt12.d . . . . . . . 8 ((𝑦 = 𝐴𝑧 = 𝐵) → 𝐶 = 𝐷)
3231eleq1d 2874 . . . . . . 7 ((𝑦 = 𝐴𝑧 = 𝐵) → (𝐶 𝑀𝐷 𝑀))
3330, 32imbi12d 348 . . . . . 6 ((𝑦 = 𝐴𝑧 = 𝐵) → (((𝑦𝑌𝑧𝑍) → 𝐶 𝑀) ↔ ((𝐴𝑌𝐵𝑍) → 𝐷 𝑀)))
3433spc2gv 3549 . . . . 5 ((𝐴𝑌𝐵𝑍) → (∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀) → ((𝐴𝑌𝐵𝑍) → 𝐷 𝑀)))
3512, 27, 12, 34syl3c 66 . . . 4 ((𝜑𝑥𝑋) → 𝐷 𝑀)
3631, 22ovmpoga 7283 . . . 4 ((𝐴𝑌𝐵𝑍𝐷 𝑀) → (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵) = 𝐷)
376, 11, 35, 36syl3anc 1368 . . 3 ((𝜑𝑥𝑋) → (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵) = 𝐷)
3837mpteq2dva 5125 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵)) = (𝑥𝑋𝐷))
391, 3, 8, 15cnmpt12f 22271 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵)) ∈ (𝐽 Cn 𝑀))
4038, 39eqeltrrd 2891 1 (𝜑 → (𝑥𝑋𝐷) ∈ (𝐽 Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536   = wceq 1538  wcel 2111  wral 3106   cuni 4800  cmpt 5110   × cxp 5517  wf 6320  cfv 6324  (class class class)co 7135  cmpo 7137  Topctop 21498  TopOnctopon 21515   Cn ccn 21829   ×t ctx 22165
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391  df-topgen 16709  df-top 21499  df-topon 21516  df-bases 21551  df-cn 21832  df-tx 22167
This theorem is referenced by:  cnmptkk  22288  cnmptk1p  22290  pcocn  23622  pcopt  23627  pcopt2  23628  pcoass  23629  resqrtcn  25338  sqrtcn  25339  rmulccn  31281  pl1cn  31308  cxpcncf1  31976  cxpcncf2  42539
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