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Theorem cnmpt12 23691
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 23273 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋𝑌)
51, 2, 3, 4syl3anc 1370 . . . . 5 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
65fvmptelcdm 7133 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
7 cnmpt12.l . . . . . 6 (𝜑𝐿 ∈ (TopOn‘𝑍))
8 cnmpt1t.b . . . . . 6 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
9 cnf2 23273 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐵):𝑋𝑍)
101, 7, 8, 9syl3anc 1370 . . . . 5 (𝜑 → (𝑥𝑋𝐵):𝑋𝑍)
1110fvmptelcdm 7133 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑍)
126, 11jca 511 . . . . 5 ((𝜑𝑥𝑋) → (𝐴𝑌𝐵𝑍))
13 txtopon 23615 . . . . . . . . . 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 23265 . . . . . . . . . . 11 ((𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀) → 𝑀 ∈ Top)
1715, 16syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ Top)
18 toptopon2 22940 . . . . . . . . . 10 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
1917, 18sylib 218 . . . . . . . . 9 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
20 cnf2 23273 . . . . . . . . 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 1370 . . . . . . . 8 (𝜑 → (𝑦𝑌, 𝑧𝑍𝐶):(𝑌 × 𝑍)⟶ 𝑀)
22 eqid 2735 . . . . . . . . 9 (𝑦𝑌, 𝑧𝑍𝐶) = (𝑦𝑌, 𝑧𝑍𝐶)
2322fmpo 8092 . . . . . . . 8 (∀𝑦𝑌𝑧𝑍 𝐶 𝑀 ↔ (𝑦𝑌, 𝑧𝑍𝐶):(𝑌 × 𝑍)⟶ 𝑀)
2421, 23sylibr 234 . . . . . . 7 (𝜑 → ∀𝑦𝑌𝑧𝑍 𝐶 𝑀)
25 r2al 3193 . . . . . . 7 (∀𝑦𝑌𝑧𝑍 𝐶 𝑀 ↔ ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
2624, 25sylib 218 . . . . . 6 (𝜑 → ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
2726adantr 480 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
28 eleq1 2827 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑌𝐴𝑌))
29 eleq1 2827 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑍𝐵𝑍))
3028, 29bi2anan9 638 . . . . . . 7 ((𝑦 = 𝐴𝑧 = 𝐵) → ((𝑦𝑌𝑧𝑍) ↔ (𝐴𝑌𝐵𝑍)))
31 cnmpt12.d . . . . . . . 8 ((𝑦 = 𝐴𝑧 = 𝐵) → 𝐶 = 𝐷)
3231eleq1d 2824 . . . . . . 7 ((𝑦 = 𝐴𝑧 = 𝐵) → (𝐶 𝑀𝐷 𝑀))
3330, 32imbi12d 344 . . . . . 6 ((𝑦 = 𝐴𝑧 = 𝐵) → (((𝑦𝑌𝑧𝑍) → 𝐶 𝑀) ↔ ((𝐴𝑌𝐵𝑍) → 𝐷 𝑀)))
3433spc2gv 3600 . . . . 5 ((𝐴𝑌𝐵𝑍) → (∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀) → ((𝐴𝑌𝐵𝑍) → 𝐷 𝑀)))
3512, 27, 12, 34syl3c 66 . . . 4 ((𝜑𝑥𝑋) → 𝐷 𝑀)
3631, 22ovmpoga 7587 . . . 4 ((𝐴𝑌𝐵𝑍𝐷 𝑀) → (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵) = 𝐷)
376, 11, 35, 36syl3anc 1370 . . 3 ((𝜑𝑥𝑋) → (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵) = 𝐷)
3837mpteq2dva 5248 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵)) = (𝑥𝑋𝐷))
391, 3, 8, 15cnmpt12f 23690 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵)) ∈ (𝐽 Cn 𝑀))
4038, 39eqeltrrd 2840 1 (𝜑 → (𝑥𝑋𝐷) ∈ (𝐽 Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  wral 3059   cuni 4912  cmpt 5231   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  Topctop 22915  TopOnctopon 22932   Cn ccn 23248   ×t ctx 23584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cn 23251  df-tx 23586
This theorem is referenced by:  cnmptkk  23707  cnmptk1p  23709  divccn  24911  iihalf1cn  24973  iihalf2cn  24976  icchmeo  24985  pcocn  25064  pcopt  25069  pcopt2  25070  pcoass  25071  mulcncf  25494  plycn  26315  psercn2  26481  resqrtcn  26807  sqrtcn  26808  efrlim  27027  rmulccn  33889  pl1cn  33916  cxpcncf1  34589  cvxpconn  35227  knoppcnlem10  36485  fprodcnlem  45555  cxpcncf2  45855
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