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Theorem flfcnp 23508
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 770 . . . 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 23494 . . . . 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 482 . . . 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 2836 . . 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 772 . . 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 23503 . . 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 585 . 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 22747 . . . . . . . 8 (𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ 𝐾 ∈ Top)
98ad2antll 728 . . . . . . 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 22420 . . . . . . 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 22428 . . . . . 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 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†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
15 toponmax 22428 . . . . . 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 1193 . . . . . 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 23352 . . . . . 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 22754 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐺:π‘‹βŸΆβˆͺ 𝐾)
2114, 11, 5, 20syl3anc 1372 . . . . 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 1194 . . . . 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 23465 . . . . 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 1379 . . . 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 7425 . . 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 6742 . . . . 5 ((𝐺:π‘‹βŸΆβˆͺ 𝐾 ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐺 ∘ 𝐹):π‘ŒβŸΆβˆͺ 𝐾)
2721, 22, 26syl2anc 585 . . . 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 23494 . . . 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 1372 . . 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 23448 . . . . 5 ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBasβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝑋 FilMap 𝐹)β€˜πΏ) ∈ (Filβ€˜π‘‹))
3116, 19, 22, 30syl3anc 1372 . . . 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 23494 . . . 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 1372 . . 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 2783 . 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 2837 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆͺ cuni 4909   ∘ ccom 5681  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  fBascfbas 20932  Topctop 22395  TopOnctopon 22412   CnP ccnp 22729  Filcfil 23349   FilMap cfm 23437   fLim cflim 23438   fLimf cflf 23439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-ntr 22524  df-nei 22602  df-cnp 22732  df-fil 23350  df-fm 23442  df-flim 23443  df-flf 23444
This theorem is referenced by:  flfcnp2  23511  tsmsmhm  23650
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