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Theorem cnmpt12 21680
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 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 cnmpt12.k . . . . . . 7 (𝜑𝐾 ∈ (TopOn‘𝑌))
3 cnmpt11.a . . . . . . 7 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
4 cnf2 21263 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋𝑌)
51, 2, 3, 4syl3anc 1483 . . . . . 6 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
6 eqid 2805 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
76fmpt 6599 . . . . . 6 (∀𝑥𝑋 𝐴𝑌 ↔ (𝑥𝑋𝐴):𝑋𝑌)
85, 7sylibr 225 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
98r19.21bi 3119 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
10 cnmpt12.l . . . . . . 7 (𝜑𝐿 ∈ (TopOn‘𝑍))
11 cnmpt1t.b . . . . . . 7 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
12 cnf2 21263 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐵):𝑋𝑍)
131, 10, 11, 12syl3anc 1483 . . . . . 6 (𝜑 → (𝑥𝑋𝐵):𝑋𝑍)
14 eqid 2805 . . . . . . 7 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
1514fmpt 6599 . . . . . 6 (∀𝑥𝑋 𝐵𝑍 ↔ (𝑥𝑋𝐵):𝑋𝑍)
1613, 15sylibr 225 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐵𝑍)
1716r19.21bi 3119 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑍)
189, 17jca 503 . . . . 5 ((𝜑𝑥𝑋) → (𝐴𝑌𝐵𝑍))
19 txtopon 21604 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
202, 10, 19syl2anc 575 . . . . . . . . 9 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
21 cnmpt12.c . . . . . . . . . . 11 (𝜑 → (𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
22 cntop2 21255 . . . . . . . . . . 11 ((𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀) → 𝑀 ∈ Top)
2321, 22syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ Top)
24 eqid 2805 . . . . . . . . . . 11 𝑀 = 𝑀
2524toptopon 20931 . . . . . . . . . 10 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
2623, 25sylib 209 . . . . . . . . 9 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
27 cnf2 21263 . . . . . . . . 9 (((𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)) ∧ 𝑀 ∈ (TopOn‘ 𝑀) ∧ (𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) → (𝑦𝑌, 𝑧𝑍𝐶):(𝑌 × 𝑍)⟶ 𝑀)
2820, 26, 21, 27syl3anc 1483 . . . . . . . 8 (𝜑 → (𝑦𝑌, 𝑧𝑍𝐶):(𝑌 × 𝑍)⟶ 𝑀)
29 eqid 2805 . . . . . . . . 9 (𝑦𝑌, 𝑧𝑍𝐶) = (𝑦𝑌, 𝑧𝑍𝐶)
3029fmpt2 7467 . . . . . . . 8 (∀𝑦𝑌𝑧𝑍 𝐶 𝑀 ↔ (𝑦𝑌, 𝑧𝑍𝐶):(𝑌 × 𝑍)⟶ 𝑀)
3128, 30sylibr 225 . . . . . . 7 (𝜑 → ∀𝑦𝑌𝑧𝑍 𝐶 𝑀)
32 r2al 3126 . . . . . . 7 (∀𝑦𝑌𝑧𝑍 𝐶 𝑀 ↔ ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
3331, 32sylib 209 . . . . . 6 (𝜑 → ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
3433adantr 468 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
35 eleq1 2872 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑌𝐴𝑌))
36 eleq1 2872 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑍𝐵𝑍))
3735, 36bi2anan9 622 . . . . . . 7 ((𝑦 = 𝐴𝑧 = 𝐵) → ((𝑦𝑌𝑧𝑍) ↔ (𝐴𝑌𝐵𝑍)))
38 cnmpt12.d . . . . . . . 8 ((𝑦 = 𝐴𝑧 = 𝐵) → 𝐶 = 𝐷)
3938eleq1d 2869 . . . . . . 7 ((𝑦 = 𝐴𝑧 = 𝐵) → (𝐶 𝑀𝐷 𝑀))
4037, 39imbi12d 335 . . . . . 6 ((𝑦 = 𝐴𝑧 = 𝐵) → (((𝑦𝑌𝑧𝑍) → 𝐶 𝑀) ↔ ((𝐴𝑌𝐵𝑍) → 𝐷 𝑀)))
4140spc2gv 3488 . . . . 5 ((𝐴𝑌𝐵𝑍) → (∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀) → ((𝐴𝑌𝐵𝑍) → 𝐷 𝑀)))
4218, 34, 18, 41syl3c 66 . . . 4 ((𝜑𝑥𝑋) → 𝐷 𝑀)
4338, 29ovmpt2ga 7017 . . . 4 ((𝐴𝑌𝐵𝑍𝐷 𝑀) → (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵) = 𝐷)
449, 17, 42, 43syl3anc 1483 . . 3 ((𝜑𝑥𝑋) → (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵) = 𝐷)
4544mpteq2dva 4934 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵)) = (𝑥𝑋𝐷))
461, 3, 11, 21cnmpt12f 21679 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵)) ∈ (𝐽 Cn 𝑀))
4745, 46eqeltrrd 2885 1 (𝜑 → (𝑥𝑋𝐷) ∈ (𝐽 Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1635   = wceq 1637  wcel 2158  wral 3095   cuni 4626  cmpt 4919   × cxp 5306  wf 6094  cfv 6098  (class class class)co 6871  cmpt2 6873  Topctop 20907  TopOnctopon 20924   Cn ccn 21238   ×t ctx 21573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4627  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-id 5216  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-fv 6106  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-1st 7395  df-2nd 7396  df-map 8091  df-topgen 16305  df-top 20908  df-topon 20925  df-bases 20960  df-cn 21241  df-tx 21575
This theorem is referenced by:  cnmptkk  21696  cnmptk1p  21698  pcocn  23025  pcopt  23030  pcopt2  23031  pcoass  23032  resqrtcn  24700  sqrtcn  24701  rmulccn  30296  pl1cn  30323  cxpcncf1  30995  cxpcncf2  40590
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