MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt11 Structured version   Visualization version   GIF version

Theorem cnmpt11 23387
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 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmpt11.a (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
cnmpt11.k (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
cnmpt11.b (πœ‘ β†’ (𝑦 ∈ π‘Œ ↦ 𝐡) ∈ (𝐾 Cn 𝐿))
cnmpt11.c (𝑦 = 𝐴 β†’ 𝐡 = 𝐢)
Assertion
Ref Expression
cnmpt11 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) ∈ (𝐽 Cn 𝐿))
Distinct variable groups:   𝑦,𝐴   π‘₯,𝑦   πœ‘,π‘₯   π‘₯,𝐽,𝑦   π‘₯,𝑋,𝑦   π‘₯,π‘Œ,𝑦   π‘₯,𝐾,𝑦   π‘₯,𝐿,𝑦   π‘₯,𝐡   𝑦,𝐢
Allowed substitution hints:   πœ‘(𝑦)   𝐴(π‘₯)   𝐡(𝑦)   𝐢(π‘₯)

Proof of Theorem cnmpt11
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simpr 483 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
2 cnmptid.j . . . . . . . . . . 11 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
3 cnmpt11.k . . . . . . . . . . 11 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
4 cnmpt11.a . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
5 cnf2 22973 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆπ‘Œ)
62, 3, 4, 5syl3anc 1369 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆπ‘Œ)
76fvmptelcdm 7113 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ π‘Œ)
8 eqid 2730 . . . . . . . . . 10 (π‘₯ ∈ 𝑋 ↦ 𝐴) = (π‘₯ ∈ 𝑋 ↦ 𝐴)
98fvmpt2 7008 . . . . . . . . 9 ((π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = 𝐴)
101, 7, 9syl2anc 582 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = 𝐴)
1110fveq2d 6894 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯)) = ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π΄))
12 eqid 2730 . . . . . . . 8 (𝑦 ∈ π‘Œ ↦ 𝐡) = (𝑦 ∈ π‘Œ ↦ 𝐡)
13 cnmpt11.c . . . . . . . 8 (𝑦 = 𝐴 β†’ 𝐡 = 𝐢)
1413eleq1d 2816 . . . . . . . . 9 (𝑦 = 𝐴 β†’ (𝐡 ∈ βˆͺ 𝐿 ↔ 𝐢 ∈ βˆͺ 𝐿))
15 cnmpt11.b . . . . . . . . . . . . . 14 (πœ‘ β†’ (𝑦 ∈ π‘Œ ↦ 𝐡) ∈ (𝐾 Cn 𝐿))
16 cntop2 22965 . . . . . . . . . . . . . 14 ((𝑦 ∈ π‘Œ ↦ 𝐡) ∈ (𝐾 Cn 𝐿) β†’ 𝐿 ∈ Top)
1715, 16syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐿 ∈ Top)
18 toptopon2 22640 . . . . . . . . . . . . 13 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
1917, 18sylib 217 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
20 cnf2 22973 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿) ∧ (𝑦 ∈ π‘Œ ↦ 𝐡) ∈ (𝐾 Cn 𝐿)) β†’ (𝑦 ∈ π‘Œ ↦ 𝐡):π‘ŒβŸΆβˆͺ 𝐿)
213, 19, 15, 20syl3anc 1369 . . . . . . . . . . 11 (πœ‘ β†’ (𝑦 ∈ π‘Œ ↦ 𝐡):π‘ŒβŸΆβˆͺ 𝐿)
2212fmpt 7110 . . . . . . . . . . 11 (βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ βˆͺ 𝐿 ↔ (𝑦 ∈ π‘Œ ↦ 𝐡):π‘ŒβŸΆβˆͺ 𝐿)
2321, 22sylibr 233 . . . . . . . . . 10 (πœ‘ β†’ βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ βˆͺ 𝐿)
2423adantr 479 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ βˆͺ 𝐿)
2514, 24, 7rspcdva 3612 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐢 ∈ βˆͺ 𝐿)
2612, 13, 7, 25fvmptd3 7020 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π΄) = 𝐢)
2711, 26eqtrd 2770 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯)) = 𝐢)
28 fvco3 6989 . . . . . . 7 (((π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆπ‘Œ ∧ π‘₯ ∈ 𝑋) β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯)))
296, 28sylan 578 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯)))
30 eqid 2730 . . . . . . . 8 (π‘₯ ∈ 𝑋 ↦ 𝐢) = (π‘₯ ∈ 𝑋 ↦ 𝐢)
3130fvmpt2 7008 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ 𝐢 ∈ βˆͺ 𝐿) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) = 𝐢)
321, 25, 31syl2anc 582 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) = 𝐢)
3327, 29, 323eqtr4d 2780 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯))
3433ralrimiva 3144 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯))
35 nfv 1915 . . . . 5 Ⅎ𝑧(((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯)
36 nfcv 2901 . . . . . . . 8 β„²π‘₯(𝑦 ∈ π‘Œ ↦ 𝐡)
37 nfmpt1 5255 . . . . . . . 8 β„²π‘₯(π‘₯ ∈ 𝑋 ↦ 𝐴)
3836, 37nfco 5864 . . . . . . 7 β„²π‘₯((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))
39 nfcv 2901 . . . . . . 7 β„²π‘₯𝑧
4038, 39nffv 6900 . . . . . 6 β„²π‘₯(((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§)
41 nfmpt1 5255 . . . . . . 7 β„²π‘₯(π‘₯ ∈ 𝑋 ↦ 𝐢)
4241, 39nffv 6900 . . . . . 6 β„²π‘₯((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§)
4340, 42nfeq 2914 . . . . 5 β„²π‘₯(((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§)
44 fveq2 6890 . . . . . 6 (π‘₯ = 𝑧 β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§))
45 fveq2 6890 . . . . . 6 (π‘₯ = 𝑧 β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§))
4644, 45eqeq12d 2746 . . . . 5 (π‘₯ = 𝑧 β†’ ((((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) ↔ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§)))
4735, 43, 46cbvralw 3301 . . . 4 (βˆ€π‘₯ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) ↔ βˆ€π‘§ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§))
4834, 47sylib 217 . . 3 (πœ‘ β†’ βˆ€π‘§ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§))
49 fco 6740 . . . . . 6 (((𝑦 ∈ π‘Œ ↦ 𝐡):π‘ŒβŸΆβˆͺ 𝐿 ∧ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆπ‘Œ) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)):π‘‹βŸΆβˆͺ 𝐿)
5021, 6, 49syl2anc 582 . . . . 5 (πœ‘ β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)):π‘‹βŸΆβˆͺ 𝐿)
5150ffnd 6717 . . . 4 (πœ‘ β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) Fn 𝑋)
5225fmpttd 7115 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢):π‘‹βŸΆβˆͺ 𝐿)
5352ffnd 6717 . . . 4 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) Fn 𝑋)
54 eqfnfv 7031 . . . 4 ((((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) Fn 𝑋 ∧ (π‘₯ ∈ 𝑋 ↦ 𝐢) Fn 𝑋) β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐢) ↔ βˆ€π‘§ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§)))
5551, 53, 54syl2anc 582 . . 3 (πœ‘ β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐢) ↔ βˆ€π‘§ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§)))
5648, 55mpbird 256 . 2 (πœ‘ β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐢))
57 cnco 22990 . . 3 (((π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑦 ∈ π‘Œ ↦ 𝐡) ∈ (𝐾 Cn 𝐿)) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) ∈ (𝐽 Cn 𝐿))
584, 15, 57syl2anc 582 . 2 (πœ‘ β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) ∈ (𝐽 Cn 𝐿))
5956, 58eqeltrrd 2832 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆͺ cuni 4907   ↦ cmpt 5230   ∘ ccom 5679   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  Topctop 22615  TopOnctopon 22632   Cn ccn 22948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-top 22616  df-topon 22633  df-cn 22951
This theorem is referenced by:  cnmpt11f  23388  cnmptkp  23404  cnmptk1  23405  cnmpt1k  23406  ptunhmeo  23532  tmdgsum  23819  icchmeo  24685  icchmeoOLD  24686  evth2  24706  sinccvglem  34955  poimir  36824  broucube  36825
  Copyright terms: Public domain W3C validator