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Theorem cnmpt12 21959
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 21541 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋𝑌)
51, 2, 3, 4syl3anc 1364 . . . . 5 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
65fvmptelrn 6740 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
7 cnmpt12.l . . . . . 6 (𝜑𝐿 ∈ (TopOn‘𝑍))
8 cnmpt1t.b . . . . . 6 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
9 cnf2 21541 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐵):𝑋𝑍)
101, 7, 8, 9syl3anc 1364 . . . . 5 (𝜑 → (𝑥𝑋𝐵):𝑋𝑍)
1110fvmptelrn 6740 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑍)
126, 11jca 512 . . . . 5 ((𝜑𝑥𝑋) → (𝐴𝑌𝐵𝑍))
13 txtopon 21883 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
142, 7, 13syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
15 cnmpt12.c . . . . . . . . . . 11 (𝜑 → (𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
16 cntop2 21533 . . . . . . . . . . 11 ((𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀) → 𝑀 ∈ Top)
1715, 16syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ Top)
18 toptopon2 21210 . . . . . . . . . 10 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
1917, 18sylib 219 . . . . . . . . 9 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
20 cnf2 21541 . . . . . . . . 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 1364 . . . . . . . 8 (𝜑 → (𝑦𝑌, 𝑧𝑍𝐶):(𝑌 × 𝑍)⟶ 𝑀)
22 eqid 2795 . . . . . . . . 9 (𝑦𝑌, 𝑧𝑍𝐶) = (𝑦𝑌, 𝑧𝑍𝐶)
2322fmpo 7622 . . . . . . . 8 (∀𝑦𝑌𝑧𝑍 𝐶 𝑀 ↔ (𝑦𝑌, 𝑧𝑍𝐶):(𝑌 × 𝑍)⟶ 𝑀)
2421, 23sylibr 235 . . . . . . 7 (𝜑 → ∀𝑦𝑌𝑧𝑍 𝐶 𝑀)
25 r2al 3168 . . . . . . 7 (∀𝑦𝑌𝑧𝑍 𝐶 𝑀 ↔ ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
2624, 25sylib 219 . . . . . 6 (𝜑 → ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
2726adantr 481 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
28 eleq1 2870 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑌𝐴𝑌))
29 eleq1 2870 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑍𝐵𝑍))
3028, 29bi2anan9 635 . . . . . . 7 ((𝑦 = 𝐴𝑧 = 𝐵) → ((𝑦𝑌𝑧𝑍) ↔ (𝐴𝑌𝐵𝑍)))
31 cnmpt12.d . . . . . . . 8 ((𝑦 = 𝐴𝑧 = 𝐵) → 𝐶 = 𝐷)
3231eleq1d 2867 . . . . . . 7 ((𝑦 = 𝐴𝑧 = 𝐵) → (𝐶 𝑀𝐷 𝑀))
3330, 32imbi12d 346 . . . . . 6 ((𝑦 = 𝐴𝑧 = 𝐵) → (((𝑦𝑌𝑧𝑍) → 𝐶 𝑀) ↔ ((𝐴𝑌𝐵𝑍) → 𝐷 𝑀)))
3433spc2gv 3543 . . . . 5 ((𝐴𝑌𝐵𝑍) → (∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀) → ((𝐴𝑌𝐵𝑍) → 𝐷 𝑀)))
3512, 27, 12, 34syl3c 66 . . . 4 ((𝜑𝑥𝑋) → 𝐷 𝑀)
3631, 22ovmpoga 7160 . . . 4 ((𝐴𝑌𝐵𝑍𝐷 𝑀) → (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵) = 𝐷)
376, 11, 35, 36syl3anc 1364 . . 3 ((𝜑𝑥𝑋) → (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵) = 𝐷)
3837mpteq2dva 5055 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵)) = (𝑥𝑋𝐷))
391, 3, 8, 15cnmpt12f 21958 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵)) ∈ (𝐽 Cn 𝑀))
4038, 39eqeltrrd 2884 1 (𝜑 → (𝑥𝑋𝐷) ∈ (𝐽 Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1520   = wceq 1522  wcel 2081  wral 3105   cuni 4745  cmpt 5041   × cxp 5441  wf 6221  cfv 6225  (class class class)co 7016  cmpo 7018  Topctop 21185  TopOnctopon 21202   Cn ccn 21516   ×t ctx 21852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-map 8258  df-topgen 16546  df-top 21186  df-topon 21203  df-bases 21238  df-cn 21519  df-tx 21854
This theorem is referenced by:  cnmptkk  21975  cnmptk1p  21977  pcocn  23304  pcopt  23309  pcopt2  23310  pcoass  23311  resqrtcn  25011  sqrtcn  25012  rmulccn  30788  pl1cn  30815  cxpcncf1  31483  cxpcncf2  41744
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