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Theorem flfcnp 23728
Description: A continuous function preserves filter limits. (Contributed by Mario Carneiro, 18-Sep-2015.)
Assertion
Ref Expression
flfcnp (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ (πΊβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜(𝐺 ∘ 𝐹)))
Proof of Theorem flfcnp
StepHypRef Expression
1 simprl 769 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ 𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ))
2 flfval 23714 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐽 fLimf 𝐿)β€˜πΉ) = (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
32adantr 481 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ ((𝐽 fLimf 𝐿)β€˜πΉ) = (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
41, 3eleqtrd 2835 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ 𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
5 simprr 771 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))
6 cnpflfi 23723 . . 3 ((𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΊβ€˜π΄) ∈ ((𝐾 fLimf ((𝑋 FilMap 𝐹)β€˜πΏ))β€˜πΊ))
74, 5, 6syl2anc 584 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ (πΊβ€˜π΄) ∈ ((𝐾 fLimf ((𝑋 FilMap 𝐹)β€˜πΏ))β€˜πΊ))
8 cnptop2 22967 . . . . . . . 8 (𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ 𝐾 ∈ Top)
98ad2antll 727 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ 𝐾 ∈ Top)
10 toptopon2 22640 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
119, 10sylib 217 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
12 toponmax 22648 . . . . . 6 (𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) β†’ βˆͺ 𝐾 ∈ 𝐾)
1311, 12syl 17 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ βˆͺ 𝐾 ∈ 𝐾)
14 simpl1 1191 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
15 toponmax 22648 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
1614, 15syl 17 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ 𝑋 ∈ 𝐽)
17 simpl2 1192 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ 𝐿 ∈ (Filβ€˜π‘Œ))
18 filfbas 23572 . . . . . 6 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ 𝐿 ∈ (fBasβ€˜π‘Œ))
1917, 18syl 17 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ 𝐿 ∈ (fBasβ€˜π‘Œ))
20 cnpf2 22974 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐺:π‘‹βŸΆβˆͺ 𝐾)
2114, 11, 5, 20syl3anc 1371 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ 𝐺:π‘‹βŸΆβˆͺ 𝐾)
22 simpl3 1193 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ 𝐹:π‘ŒβŸΆπ‘‹)
23 fmco 23685 . . . . 5 (((βˆͺ 𝐾 ∈ 𝐾 ∧ 𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBasβ€˜π‘Œ)) ∧ (𝐺:π‘‹βŸΆβˆͺ 𝐾 ∧ 𝐹:π‘ŒβŸΆπ‘‹)) β†’ ((βˆͺ 𝐾 FilMap (𝐺 ∘ 𝐹))β€˜πΏ) = ((βˆͺ 𝐾 FilMap 𝐺)β€˜((𝑋 FilMap 𝐹)β€˜πΏ)))
2413, 16, 19, 21, 22, 23syl32anc 1378 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ ((βˆͺ 𝐾 FilMap (𝐺 ∘ 𝐹))β€˜πΏ) = ((βˆͺ 𝐾 FilMap 𝐺)β€˜((𝑋 FilMap 𝐹)β€˜πΏ)))
2524oveq2d 7427 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ (𝐾 fLim ((βˆͺ 𝐾 FilMap (𝐺 ∘ 𝐹))β€˜πΏ)) = (𝐾 fLim ((βˆͺ 𝐾 FilMap 𝐺)β€˜((𝑋 FilMap 𝐹)β€˜πΏ))))
26 fco 6741 . . . . 5 ((𝐺:π‘‹βŸΆβˆͺ 𝐾 ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐺 ∘ 𝐹):π‘ŒβŸΆβˆͺ 𝐾)
2721, 22, 26syl2anc 584 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ (𝐺 ∘ 𝐹):π‘ŒβŸΆβˆͺ 𝐾)
28 flfval 23714 . . . 4 ((𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ (𝐺 ∘ 𝐹):π‘ŒβŸΆβˆͺ 𝐾) β†’ ((𝐾 fLimf 𝐿)β€˜(𝐺 ∘ 𝐹)) = (𝐾 fLim ((βˆͺ 𝐾 FilMap (𝐺 ∘ 𝐹))β€˜πΏ)))
2911, 17, 27, 28syl3anc 1371 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ ((𝐾 fLimf 𝐿)β€˜(𝐺 ∘ 𝐹)) = (𝐾 fLim ((βˆͺ 𝐾 FilMap (𝐺 ∘ 𝐹))β€˜πΏ)))
30 fmfil 23668 . . . . 5 ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝑋 FilMap 𝐹)β€˜πΏ) ∈ (Filβ€˜π‘‹))
3116, 19, 22, 30syl3anc 1371 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ ((𝑋 FilMap 𝐹)β€˜πΏ) ∈ (Filβ€˜π‘‹))
32 flfval 23714 . . . 4 ((𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ ((𝑋 FilMap 𝐹)β€˜πΏ) ∈ (Filβ€˜π‘‹) ∧ 𝐺:π‘‹βŸΆβˆͺ 𝐾) β†’ ((𝐾 fLimf ((𝑋 FilMap 𝐹)β€˜πΏ))β€˜πΊ) = (𝐾 fLim ((βˆͺ 𝐾 FilMap 𝐺)β€˜((𝑋 FilMap 𝐹)β€˜πΏ))))
3311, 31, 21, 32syl3anc 1371 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ ((𝐾 fLimf ((𝑋 FilMap 𝐹)β€˜πΏ))β€˜πΊ) = (𝐾 fLim ((βˆͺ 𝐾 FilMap 𝐺)β€˜((𝑋 FilMap 𝐹)β€˜πΏ))))
3425, 29, 333eqtr4d 2782 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ ((𝐾 fLimf 𝐿)β€˜(𝐺 ∘ 𝐹)) = ((𝐾 fLimf ((𝑋 FilMap 𝐹)β€˜πΏ))β€˜πΊ))
357, 34eleqtrrd 2836 1 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)β€˜πΉ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄))) β†’ (πΊβ€˜π΄) ∈ ((𝐾 fLimf 𝐿)β€˜(𝐺 ∘ 𝐹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆͺ cuni 4908   ∘ ccom 5680  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  fBascfbas 21132  Topctop 22615  TopOnctopon 22632   CnP ccnp 22949  Filcfil 23569   FilMap cfm 23657   fLim cflim 23658   fLimf cflf 23659
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-fbas 21141  df-fg 21142  df-top 22616  df-topon 22633  df-ntr 22744  df-nei 22822  df-cnp 22952  df-fil 23570  df-fm 23662  df-flim 23663  df-flf 23664
This theorem is referenced by:  flfcnp2  23731  tsmsmhm  23870
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