MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt12 Structured version   Visualization version   GIF version

Theorem cnmpt12 21280
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 20863 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋𝑌)
51, 2, 3, 4syl3anc 1318 . . . . . 6 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
6 eqid 2610 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
76fmpt 6289 . . . . . 6 (∀𝑥𝑋 𝐴𝑌 ↔ (𝑥𝑋𝐴):𝑋𝑌)
85, 7sylibr 223 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
98r19.21bi 2916 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
10 cnmpt12.l . . . . . . 7 (𝜑𝐿 ∈ (TopOn‘𝑍))
11 cnmpt1t.b . . . . . . 7 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
12 cnf2 20863 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐵):𝑋𝑍)
131, 10, 11, 12syl3anc 1318 . . . . . 6 (𝜑 → (𝑥𝑋𝐵):𝑋𝑍)
14 eqid 2610 . . . . . . 7 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
1514fmpt 6289 . . . . . 6 (∀𝑥𝑋 𝐵𝑍 ↔ (𝑥𝑋𝐵):𝑋𝑍)
1613, 15sylibr 223 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐵𝑍)
1716r19.21bi 2916 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑍)
189, 17jca 553 . . . . 5 ((𝜑𝑥𝑋) → (𝐴𝑌𝐵𝑍))
19 txtopon 21204 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
202, 10, 19syl2anc 691 . . . . . . . . 9 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
21 cnmpt12.c . . . . . . . . . . 11 (𝜑 → (𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
22 cntop2 20855 . . . . . . . . . . 11 ((𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀) → 𝑀 ∈ Top)
2321, 22syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ Top)
24 eqid 2610 . . . . . . . . . . 11 𝑀 = 𝑀
2524toptopon 20548 . . . . . . . . . 10 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
2623, 25sylib 207 . . . . . . . . 9 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
27 cnf2 20863 . . . . . . . . 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 1318 . . . . . . . 8 (𝜑 → (𝑦𝑌, 𝑧𝑍𝐶):(𝑌 × 𝑍)⟶ 𝑀)
29 eqid 2610 . . . . . . . . 9 (𝑦𝑌, 𝑧𝑍𝐶) = (𝑦𝑌, 𝑧𝑍𝐶)
3029fmpt2 7126 . . . . . . . 8 (∀𝑦𝑌𝑧𝑍 𝐶 𝑀 ↔ (𝑦𝑌, 𝑧𝑍𝐶):(𝑌 × 𝑍)⟶ 𝑀)
3128, 30sylibr 223 . . . . . . 7 (𝜑 → ∀𝑦𝑌𝑧𝑍 𝐶 𝑀)
32 r2al 2923 . . . . . . 7 (∀𝑦𝑌𝑧𝑍 𝐶 𝑀 ↔ ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
3331, 32sylib 207 . . . . . 6 (𝜑 → ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
3433adantr 480 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀))
35 eleq1 2676 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑌𝐴𝑌))
36 eleq1 2676 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑍𝐵𝑍))
3735, 36bi2anan9 913 . . . . . . 7 ((𝑦 = 𝐴𝑧 = 𝐵) → ((𝑦𝑌𝑧𝑍) ↔ (𝐴𝑌𝐵𝑍)))
38 cnmpt12.d . . . . . . . 8 ((𝑦 = 𝐴𝑧 = 𝐵) → 𝐶 = 𝐷)
3938eleq1d 2672 . . . . . . 7 ((𝑦 = 𝐴𝑧 = 𝐵) → (𝐶 𝑀𝐷 𝑀))
4037, 39imbi12d 333 . . . . . 6 ((𝑦 = 𝐴𝑧 = 𝐵) → (((𝑦𝑌𝑧𝑍) → 𝐶 𝑀) ↔ ((𝐴𝑌𝐵𝑍) → 𝐷 𝑀)))
4140spc2gv 3269 . . . . 5 ((𝐴𝑌𝐵𝑍) → (∀𝑦𝑧((𝑦𝑌𝑧𝑍) → 𝐶 𝑀) → ((𝐴𝑌𝐵𝑍) → 𝐷 𝑀)))
4218, 34, 18, 41syl3c 64 . . . 4 ((𝜑𝑥𝑋) → 𝐷 𝑀)
4338, 29ovmpt2ga 6688 . . . 4 ((𝐴𝑌𝐵𝑍𝐷 𝑀) → (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵) = 𝐷)
449, 17, 42, 43syl3anc 1318 . . 3 ((𝜑𝑥𝑋) → (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵) = 𝐷)
4544mpteq2dva 4672 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵)) = (𝑥𝑋𝐷))
461, 3, 11, 21cnmpt12f 21279 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(𝑦𝑌, 𝑧𝑍𝐶)𝐵)) ∈ (𝐽 Cn 𝑀))
4745, 46eqeltrrd 2689 1 (𝜑 → (𝑥𝑋𝐷) ∈ (𝐽 Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1473   = wceq 1475  wcel 1977  wral 2896   cuni 4372  cmpt 4643   × cxp 5036  wf 5800  cfv 5804  (class class class)co 6549  cmpt2 6551  Topctop 20517  TopOnctopon 20518   Cn ccn 20838   ×t ctx 21173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841  df-tx 21175
This theorem is referenced by:  cnmptkk  21296  cnmptk1p  21298  pcocn  22625  pcopt  22630  pcopt2  22631  pcoass  22632  resqrtcn  24290  sqrtcn  24291  rmulccn  29302  pl1cn  29329  cxpcncf2  38786
  Copyright terms: Public domain W3C validator