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Theorem cnmpt22 13047
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 5853 . . . 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 13015 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
52, 3, 4syl2anc 409 . . . . . . . . 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 12958 . . . . . . . . 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 1233 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
10 eqid 2170 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1110fmpo 6177 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴𝑍 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
129, 11sylibr 133 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴𝑍)
13 rsp2 2520 . . . . . . 7 (∀𝑥𝑋𝑦𝑌 𝐴𝑍 → ((𝑥𝑋𝑦𝑌) → 𝐴𝑍))
1412, 13syl 14 . . . . . 6 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐴𝑍))
15143impib 1196 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → 𝐴𝑍)
16 cnmpt22.m . . . . . . . . 9 (𝜑𝑀 ∈ (TopOn‘𝑊))
17 cnmpt2t.b . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
18 cnf2 12958 . . . . . . . . 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 1233 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶𝑊)
20 eqid 2170 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
2120fmpo 6177 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵𝑊 ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶𝑊)
2219, 21sylibr 133 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵𝑊)
23 rsp2 2520 . . . . . . 7 (∀𝑥𝑋𝑦𝑌 𝐵𝑊 → ((𝑥𝑋𝑦𝑌) → 𝐵𝑊))
2422, 23syl 14 . . . . . 6 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐵𝑊))
25243impib 1196 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → 𝐵𝑊)
2615, 25jca 304 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴𝑍𝐵𝑊))
27 txtopon 13015 . . . . . . . . . . 11 ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘𝑊)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)))
286, 16, 27syl2anc 409 . . . . . . . . . 10 (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)))
29 cnmpt22.c . . . . . . . . . . . 12 (𝜑 → (𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
30 cntop2 12955 . . . . . . . . . . . 12 ((𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
3129, 30syl 14 . . . . . . . . . . 11 (𝜑𝑁 ∈ Top)
32 toptopon2 12770 . . . . . . . . . . 11 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘ 𝑁))
3331, 32sylib 121 . . . . . . . . . 10 (𝜑𝑁 ∈ (TopOn‘ 𝑁))
34 cnf2 12958 . . . . . . . . . 10 (((𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)) ∧ 𝑁 ∈ (TopOn‘ 𝑁) ∧ (𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → (𝑧𝑍, 𝑤𝑊𝐶):(𝑍 × 𝑊)⟶ 𝑁)
3528, 33, 29, 34syl3anc 1233 . . . . . . . . 9 (𝜑 → (𝑧𝑍, 𝑤𝑊𝐶):(𝑍 × 𝑊)⟶ 𝑁)
36 eqid 2170 . . . . . . . . . 10 (𝑧𝑍, 𝑤𝑊𝐶) = (𝑧𝑍, 𝑤𝑊𝐶)
3736fmpo 6177 . . . . . . . . 9 (∀𝑧𝑍𝑤𝑊 𝐶 𝑁 ↔ (𝑧𝑍, 𝑤𝑊𝐶):(𝑍 × 𝑊)⟶ 𝑁)
3835, 37sylibr 133 . . . . . . . 8 (𝜑 → ∀𝑧𝑍𝑤𝑊 𝐶 𝑁)
39 r2al 2489 . . . . . . . 8 (∀𝑧𝑍𝑤𝑊 𝐶 𝑁 ↔ ∀𝑧𝑤((𝑧𝑍𝑤𝑊) → 𝐶 𝑁))
4038, 39sylib 121 . . . . . . 7 (𝜑 → ∀𝑧𝑤((𝑧𝑍𝑤𝑊) → 𝐶 𝑁))
41403ad2ant1 1013 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → ∀𝑧𝑤((𝑧𝑍𝑤𝑊) → 𝐶 𝑁))
42 eleq1 2233 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑧𝑍𝐴𝑍))
43 eleq1 2233 . . . . . . . . 9 (𝑤 = 𝐵 → (𝑤𝑊𝐵𝑊))
4442, 43bi2anan9 601 . . . . . . . 8 ((𝑧 = 𝐴𝑤 = 𝐵) → ((𝑧𝑍𝑤𝑊) ↔ (𝐴𝑍𝐵𝑊)))
45 cnmpt22.d . . . . . . . . 9 ((𝑧 = 𝐴𝑤 = 𝐵) → 𝐶 = 𝐷)
4645eleq1d 2239 . . . . . . . 8 ((𝑧 = 𝐴𝑤 = 𝐵) → (𝐶 𝑁𝐷 𝑁))
4744, 46imbi12d 233 . . . . . . 7 ((𝑧 = 𝐴𝑤 = 𝐵) → (((𝑧𝑍𝑤𝑊) → 𝐶 𝑁) ↔ ((𝐴𝑍𝐵𝑊) → 𝐷 𝑁)))
4847spc2gv 2821 . . . . . 6 ((𝐴𝑍𝐵𝑊) → (∀𝑧𝑤((𝑧𝑍𝑤𝑊) → 𝐶 𝑁) → ((𝐴𝑍𝐵𝑊) → 𝐷 𝑁)))
4926, 41, 26, 48syl3c 63 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → 𝐷 𝑁)
5045, 36ovmpoga 5979 . . . . 5 ((𝐴𝑍𝐵𝑊𝐷 𝑁) → (𝐴(𝑧𝑍, 𝑤𝑊𝐶)𝐵) = 𝐷)
5115, 25, 49, 50syl3anc 1233 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴(𝑧𝑍, 𝑤𝑊𝐶)𝐵) = 𝐷)
521, 51eqtr3id 2217 . . 3 ((𝜑𝑥𝑋𝑦𝑌) → ((𝑧𝑍, 𝑤𝑊𝐶)‘⟨𝐴, 𝐵⟩) = 𝐷)
5352mpoeq3dva 5914 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ((𝑧𝑍, 𝑤𝑊𝐶)‘⟨𝐴, 𝐵⟩)) = (𝑥𝑋, 𝑦𝑌𝐷))
542, 3, 7, 17cnmpt2t 13046 . . 3 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
552, 3, 54, 29cnmpt21f 13045 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ((𝑧𝑍, 𝑤𝑊𝐶)‘⟨𝐴, 𝐵⟩)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
5653, 55eqeltrrd 2248 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 973  wal 1346   = wceq 1348  wcel 2141  wral 2448  cop 3584   cuni 3794   × cxp 4607  wf 5192  cfv 5196  (class class class)co 5850  cmpo 5852  Topctop 12748  TopOnctopon 12761   Cn ccn 12938   ×t ctx 13005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-map 6624  df-topgen 12587  df-top 12749  df-topon 12762  df-bases 12794  df-cn 12941  df-tx 13006
This theorem is referenced by:  cnmpt22f  13048
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