MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt22 Structured version   Visualization version   GIF version

Theorem cnmpt22 21382
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmpt21.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmpt21.a (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
cnmpt2t.b (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
cnmpt22.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmpt22.m (𝜑𝑀 ∈ (TopOn‘𝑊))
cnmpt22.c (𝜑 → (𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
cnmpt22.d ((𝑧 = 𝐴𝑤 = 𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
cnmpt22 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
Distinct variable groups:   𝑧,𝑤,𝐴   𝑤,𝐵   𝑤,𝐷,𝑧   𝑧,𝐽   𝑥,𝑤,𝑦,𝑧,𝐿   𝜑,𝑥,𝑦,𝑧   𝑤,𝑋,𝑥,𝑦,𝑧   𝑤,𝑀,𝑥,𝑦,𝑧   𝑤,𝑁,𝑥,𝑦,𝑧   𝑤,𝑌,𝑥,𝑦,𝑧   𝑧,𝐾   𝑤,𝑊,𝑥,𝑦,𝑧   𝑤,𝑍,𝑥,𝑦,𝑧   𝑧,𝐵   𝑥,𝐶,𝑦
Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑧,𝑤)   𝐷(𝑥,𝑦)   𝐽(𝑥,𝑦,𝑤)   𝐾(𝑥,𝑦,𝑤)

Proof of Theorem cnmpt22
StepHypRef Expression
1 df-ov 6608 . . . 4 (𝐴(𝑧𝑍, 𝑤𝑊𝐶)𝐵) = ((𝑧𝑍, 𝑤𝑊𝐶)‘⟨𝐴, 𝐵⟩)
2 cnmpt21.j . . . . . . . . . 10 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 cnmpt21.k . . . . . . . . . 10 (𝜑𝐾 ∈ (TopOn‘𝑌))
4 txtopon 21299 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
52, 3, 4syl2anc 692 . . . . . . . . 9 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6 cnmpt22.l . . . . . . . . 9 (𝜑𝐿 ∈ (TopOn‘𝑍))
7 cnmpt21.a . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
8 cnf2 20958 . . . . . . . . 9 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
95, 6, 7, 8syl3anc 1323 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
10 eqid 2626 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1110fmpt2 7183 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴𝑍 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
129, 11sylibr 224 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴𝑍)
13 rsp2 2936 . . . . . . 7 (∀𝑥𝑋𝑦𝑌 𝐴𝑍 → ((𝑥𝑋𝑦𝑌) → 𝐴𝑍))
1412, 13syl 17 . . . . . 6 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐴𝑍))
15143impib 1259 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → 𝐴𝑍)
16 cnmpt22.m . . . . . . . . 9 (𝜑𝑀 ∈ (TopOn‘𝑊))
17 cnmpt2t.b . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
18 cnf2 20958 . . . . . . . . 9 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑀 ∈ (TopOn‘𝑊) ∧ (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶𝑊)
195, 16, 17, 18syl3anc 1323 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶𝑊)
20 eqid 2626 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
2120fmpt2 7183 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵𝑊 ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶𝑊)
2219, 21sylibr 224 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵𝑊)
23 rsp2 2936 . . . . . . 7 (∀𝑥𝑋𝑦𝑌 𝐵𝑊 → ((𝑥𝑋𝑦𝑌) → 𝐵𝑊))
2422, 23syl 17 . . . . . 6 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐵𝑊))
25243impib 1259 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → 𝐵𝑊)
2615, 25jca 554 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴𝑍𝐵𝑊))
27 txtopon 21299 . . . . . . . . . . 11 ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘𝑊)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)))
286, 16, 27syl2anc 692 . . . . . . . . . 10 (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)))
29 cnmpt22.c . . . . . . . . . . . 12 (𝜑 → (𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
30 cntop2 20950 . . . . . . . . . . . 12 ((𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
3129, 30syl 17 . . . . . . . . . . 11 (𝜑𝑁 ∈ Top)
32 eqid 2626 . . . . . . . . . . . 12 𝑁 = 𝑁
3332toptopon 20643 . . . . . . . . . . 11 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘ 𝑁))
3431, 33sylib 208 . . . . . . . . . 10 (𝜑𝑁 ∈ (TopOn‘ 𝑁))
35 cnf2 20958 . . . . . . . . . 10 (((𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)) ∧ 𝑁 ∈ (TopOn‘ 𝑁) ∧ (𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → (𝑧𝑍, 𝑤𝑊𝐶):(𝑍 × 𝑊)⟶ 𝑁)
3628, 34, 29, 35syl3anc 1323 . . . . . . . . 9 (𝜑 → (𝑧𝑍, 𝑤𝑊𝐶):(𝑍 × 𝑊)⟶ 𝑁)
37 eqid 2626 . . . . . . . . . 10 (𝑧𝑍, 𝑤𝑊𝐶) = (𝑧𝑍, 𝑤𝑊𝐶)
3837fmpt2 7183 . . . . . . . . 9 (∀𝑧𝑍𝑤𝑊 𝐶 𝑁 ↔ (𝑧𝑍, 𝑤𝑊𝐶):(𝑍 × 𝑊)⟶ 𝑁)
3936, 38sylibr 224 . . . . . . . 8 (𝜑 → ∀𝑧𝑍𝑤𝑊 𝐶 𝑁)
40 r2al 2939 . . . . . . . 8 (∀𝑧𝑍𝑤𝑊 𝐶 𝑁 ↔ ∀𝑧𝑤((𝑧𝑍𝑤𝑊) → 𝐶 𝑁))
4139, 40sylib 208 . . . . . . 7 (𝜑 → ∀𝑧𝑤((𝑧𝑍𝑤𝑊) → 𝐶 𝑁))
42413ad2ant1 1080 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → ∀𝑧𝑤((𝑧𝑍𝑤𝑊) → 𝐶 𝑁))
43 eleq1 2692 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑧𝑍𝐴𝑍))
44 eleq1 2692 . . . . . . . . 9 (𝑤 = 𝐵 → (𝑤𝑊𝐵𝑊))
4543, 44bi2anan9 916 . . . . . . . 8 ((𝑧 = 𝐴𝑤 = 𝐵) → ((𝑧𝑍𝑤𝑊) ↔ (𝐴𝑍𝐵𝑊)))
46 cnmpt22.d . . . . . . . . 9 ((𝑧 = 𝐴𝑤 = 𝐵) → 𝐶 = 𝐷)
4746eleq1d 2688 . . . . . . . 8 ((𝑧 = 𝐴𝑤 = 𝐵) → (𝐶 𝑁𝐷 𝑁))
4845, 47imbi12d 334 . . . . . . 7 ((𝑧 = 𝐴𝑤 = 𝐵) → (((𝑧𝑍𝑤𝑊) → 𝐶 𝑁) ↔ ((𝐴𝑍𝐵𝑊) → 𝐷 𝑁)))
4948spc2gv 3287 . . . . . 6 ((𝐴𝑍𝐵𝑊) → (∀𝑧𝑤((𝑧𝑍𝑤𝑊) → 𝐶 𝑁) → ((𝐴𝑍𝐵𝑊) → 𝐷 𝑁)))
5026, 42, 26, 49syl3c 66 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → 𝐷 𝑁)
5146, 37ovmpt2ga 6744 . . . . 5 ((𝐴𝑍𝐵𝑊𝐷 𝑁) → (𝐴(𝑧𝑍, 𝑤𝑊𝐶)𝐵) = 𝐷)
5215, 25, 50, 51syl3anc 1323 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴(𝑧𝑍, 𝑤𝑊𝐶)𝐵) = 𝐷)
531, 52syl5eqr 2674 . . 3 ((𝜑𝑥𝑋𝑦𝑌) → ((𝑧𝑍, 𝑤𝑊𝐶)‘⟨𝐴, 𝐵⟩) = 𝐷)
5453mpt2eq3dva 6673 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ((𝑧𝑍, 𝑤𝑊𝐶)‘⟨𝐴, 𝐵⟩)) = (𝑥𝑋, 𝑦𝑌𝐷))
552, 3, 7, 17cnmpt2t 21381 . . 3 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
562, 3, 55, 29cnmpt21f 21380 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ((𝑧𝑍, 𝑤𝑊𝐶)‘⟨𝐴, 𝐵⟩)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
5754, 56eqeltrrd 2705 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1992  wral 2912  cop 4159   cuni 4407   × cxp 5077  wf 5846  cfv 5850  (class class class)co 6605  cmpt2 6607  Topctop 20612  TopOnctopon 20613   Cn ccn 20933   ×t ctx 21268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-map 7805  df-topgen 16020  df-top 20616  df-bases 20617  df-topon 20618  df-cn 20936  df-tx 21270
This theorem is referenced by:  cnmpt22f  21383  xkofvcn  21392  cnmptk2  21394  pcorevlem  22729
  Copyright terms: Public domain W3C validator