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Theorem flfcnp 23371
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 23357 . . . . 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 2840 . . 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 23366 . . 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 22610 . . . . . . . 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 22283 . . . . . . 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 22291 . . . . . 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 22291 . . . . . 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 23215 . . . . . 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 22617 . . . . . 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 23328 . . . . 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 7378 . . 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 6697 . . . . 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 23357 . . . 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 23311 . . . . 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 23357 . . . 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 2787 . 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 2841 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 4870   ∘ ccom 5642  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  fBascfbas 20800  Topctop 22258  TopOnctopon 22275   CnP ccnp 22592  Filcfil 23212   FilMap cfm 23300   fLim cflim 23301   fLimf cflf 23302
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-map 8774  df-fbas 20809  df-fg 20810  df-top 22259  df-topon 22276  df-ntr 22387  df-nei 22465  df-cnp 22595  df-fil 23213  df-fm 23305  df-flim 23306  df-flf 23307
This theorem is referenced by:  flfcnp2  23374  tsmsmhm  23513
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