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Theorem cnmpt1t 13656
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 13384 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3 mpteq1 4086 . . . 4 (𝑋 = 𝐽 → (𝑥𝑋 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩))
41, 2, 33syl 17 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩))
5 simpr 110 . . . . . 6 ((𝜑𝑥𝑋) → 𝑥𝑋)
6 cnmpt11.a . . . . . . . . . 10 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
7 cntop2 13573 . . . . . . . . . 10 ((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
86, 7syl 14 . . . . . . . . 9 (𝜑𝐾 ∈ Top)
9 toptopon2 13388 . . . . . . . . 9 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
108, 9sylib 122 . . . . . . . 8 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
11 cnf2 13576 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋 𝐾)
121, 10, 6, 11syl3anc 1238 . . . . . . 7 (𝜑 → (𝑥𝑋𝐴):𝑋 𝐾)
1312fvmptelcdm 5668 . . . . . 6 ((𝜑𝑥𝑋) → 𝐴 𝐾)
14 eqid 2177 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
1514fvmpt2 5598 . . . . . 6 ((𝑥𝑋𝐴 𝐾) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
165, 13, 15syl2anc 411 . . . . 5 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
17 cnmpt1t.b . . . . . . . . . 10 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
18 cntop2 13573 . . . . . . . . . 10 ((𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
1917, 18syl 14 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
20 toptopon2 13388 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
2119, 20sylib 122 . . . . . . . 8 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
22 cnf2 13576 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐵):𝑋 𝐿)
231, 21, 17, 22syl3anc 1238 . . . . . . 7 (𝜑 → (𝑥𝑋𝐵):𝑋 𝐿)
2423fvmptelcdm 5668 . . . . . 6 ((𝜑𝑥𝑋) → 𝐵 𝐿)
25 eqid 2177 . . . . . . 7 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
2625fvmpt2 5598 . . . . . 6 ((𝑥𝑋𝐵 𝐿) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
275, 24, 26syl2anc 411 . . . . 5 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
2816, 27opeq12d 3786 . . . 4 ((𝜑𝑥𝑋) → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
2928mpteq2dva 4092 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
304, 29eqtr3d 2212 . 2 (𝜑 → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
31 eqid 2177 . . . 4 𝐽 = 𝐽
32 nfcv 2319 . . . . 5 𝑦⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩
33 nffvmpt1 5525 . . . . . 6 𝑥((𝑥𝑋𝐴)‘𝑦)
34 nffvmpt1 5525 . . . . . 6 𝑥((𝑥𝑋𝐵)‘𝑦)
3533, 34nfop 3794 . . . . 5 𝑥⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩
36 fveq2 5514 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝑦))
37 fveq2 5514 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝑋𝐵)‘𝑥) = ((𝑥𝑋𝐵)‘𝑦))
3836, 37opeq12d 3786 . . . . 5 (𝑥 = 𝑦 → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩)
3932, 35, 38cbvmpt 4097 . . . 4 (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑦 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩)
4031, 39txcnmpt 13644 . . 3 (((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
416, 17, 40syl2anc 411 . 2 (𝜑 → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
4230, 41eqeltrrd 2255 1 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  cop 3595   cuni 3809  cmpt 4063  wf 5211  cfv 5215  (class class class)co 5872  Topctop 13366  TopOnctopon 13379   Cn ccn 13556   ×t ctx 13623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-map 6647  df-topgen 12697  df-top 13367  df-topon 13380  df-bases 13412  df-cn 13559  df-tx 13624
This theorem is referenced by:  cnmpt12f  13657  imasnopn  13670  cnrehmeocntop  13964
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