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Theorem cnmpt1t 23016
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
cnmptid.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt11.a (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
cnmpt1t.b (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
Assertion
Ref Expression
cnmpt1t (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐽   𝑥,𝑋   𝑥,𝐾   𝑥,𝐿
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem cnmpt1t
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnmptid.j . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 toponuni 22263 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3 mpteq1 5198 . . . 4 (𝑋 = 𝐽 → (𝑥𝑋 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩))
41, 2, 33syl 18 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩))
5 simpr 485 . . . . . 6 ((𝜑𝑥𝑋) → 𝑥𝑋)
6 cnmpt11.a . . . . . . . . . 10 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
7 cntop2 22592 . . . . . . . . . 10 ((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
86, 7syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Top)
9 toptopon2 22267 . . . . . . . . 9 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
108, 9sylib 217 . . . . . . . 8 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
11 cnf2 22600 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋 𝐾)
121, 10, 6, 11syl3anc 1371 . . . . . . 7 (𝜑 → (𝑥𝑋𝐴):𝑋 𝐾)
1312fvmptelcdm 7061 . . . . . 6 ((𝜑𝑥𝑋) → 𝐴 𝐾)
14 eqid 2736 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
1514fvmpt2 6959 . . . . . 6 ((𝑥𝑋𝐴 𝐾) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
165, 13, 15syl2anc 584 . . . . 5 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
17 cnmpt1t.b . . . . . . . . . 10 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
18 cntop2 22592 . . . . . . . . . 10 ((𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
1917, 18syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
20 toptopon2 22267 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
2119, 20sylib 217 . . . . . . . 8 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
22 cnf2 22600 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐵):𝑋 𝐿)
231, 21, 17, 22syl3anc 1371 . . . . . . 7 (𝜑 → (𝑥𝑋𝐵):𝑋 𝐿)
2423fvmptelcdm 7061 . . . . . 6 ((𝜑𝑥𝑋) → 𝐵 𝐿)
25 eqid 2736 . . . . . . 7 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
2625fvmpt2 6959 . . . . . 6 ((𝑥𝑋𝐵 𝐿) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
275, 24, 26syl2anc 584 . . . . 5 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
2816, 27opeq12d 4838 . . . 4 ((𝜑𝑥𝑋) → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
2928mpteq2dva 5205 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
304, 29eqtr3d 2778 . 2 (𝜑 → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
31 eqid 2736 . . . 4 𝐽 = 𝐽
32 nfcv 2907 . . . . 5 𝑦⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩
33 nffvmpt1 6853 . . . . . 6 𝑥((𝑥𝑋𝐴)‘𝑦)
34 nffvmpt1 6853 . . . . . 6 𝑥((𝑥𝑋𝐵)‘𝑦)
3533, 34nfop 4846 . . . . 5 𝑥⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩
36 fveq2 6842 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝑦))
37 fveq2 6842 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝑋𝐵)‘𝑥) = ((𝑥𝑋𝐵)‘𝑦))
3836, 37opeq12d 4838 . . . . 5 (𝑥 = 𝑦 → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩)
3932, 35, 38cbvmpt 5216 . . . 4 (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑦 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩)
4031, 39txcnmpt 22975 . . 3 (((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
416, 17, 40syl2anc 584 . 2 (𝜑 → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
4230, 41eqeltrrd 2839 1 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cop 4592   cuni 4865  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7357  Topctop 22242  TopOnctopon 22259   Cn ccn 22575   ×t ctx 22911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-tx 22913
This theorem is referenced by:  cnmpt12f  23017  xkoinjcn  23038  txconn  23040  imasnopn  23041  imasncld  23042  imasncls  23043  ptunhmeo  23159  xkohmeo  23166  cnrehmeo  24316
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