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Theorem cnmpt1t 23689
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 22936 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3 mpteq1 5241 . . . 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 484 . . . . . 6 ((𝜑𝑥𝑋) → 𝑥𝑋)
6 cnmpt11.a . . . . . . . . . 10 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
7 cntop2 23265 . . . . . . . . . 10 ((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
86, 7syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Top)
9 toptopon2 22940 . . . . . . . . 9 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
108, 9sylib 218 . . . . . . . 8 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
11 cnf2 23273 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋 𝐾)
121, 10, 6, 11syl3anc 1370 . . . . . . 7 (𝜑 → (𝑥𝑋𝐴):𝑋 𝐾)
1312fvmptelcdm 7133 . . . . . 6 ((𝜑𝑥𝑋) → 𝐴 𝐾)
14 eqid 2735 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
1514fvmpt2 7027 . . . . . 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 23265 . . . . . . . . . 10 ((𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
1917, 18syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
20 toptopon2 22940 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
2119, 20sylib 218 . . . . . . . 8 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
22 cnf2 23273 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐵):𝑋 𝐿)
231, 21, 17, 22syl3anc 1370 . . . . . . 7 (𝜑 → (𝑥𝑋𝐵):𝑋 𝐿)
2423fvmptelcdm 7133 . . . . . 6 ((𝜑𝑥𝑋) → 𝐵 𝐿)
25 eqid 2735 . . . . . . 7 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
2625fvmpt2 7027 . . . . . 6 ((𝑥𝑋𝐵 𝐿) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
275, 24, 26syl2anc 584 . . . . 5 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
2816, 27opeq12d 4886 . . . 4 ((𝜑𝑥𝑋) → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
2928mpteq2dva 5248 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
304, 29eqtr3d 2777 . 2 (𝜑 → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
31 eqid 2735 . . . 4 𝐽 = 𝐽
32 nfcv 2903 . . . . 5 𝑦⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩
33 nffvmpt1 6918 . . . . . 6 𝑥((𝑥𝑋𝐴)‘𝑦)
34 nffvmpt1 6918 . . . . . 6 𝑥((𝑥𝑋𝐵)‘𝑦)
3533, 34nfop 4894 . . . . 5 𝑥⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩
36 fveq2 6907 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝑦))
37 fveq2 6907 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝑋𝐵)‘𝑥) = ((𝑥𝑋𝐵)‘𝑦))
3836, 37opeq12d 4886 . . . . 5 (𝑥 = 𝑦 → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩)
3932, 35, 38cbvmpt 5259 . . . 4 (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑦 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩)
4031, 39txcnmpt 23648 . . 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 2840 1 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cop 4637   cuni 4912  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  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:  cnmpt12f  23690  xkoinjcn  23711  txconn  23713  imasnopn  23714  imasncld  23715  imasncls  23716  ptunhmeo  23832  xkohmeo  23839  cnrehmeo  24998  cnrehmeoOLD  24999
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