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Theorem flfcnp 23499
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 23485 . . . . 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 23494 . . 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 22738 . . . . . . . 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 22411 . . . . . . 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 22419 . . . . . 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 22419 . . . . . 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 23343 . . . . . 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 22745 . . . . . 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 23456 . . . . 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 7421 . . 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 6738 . . . . 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 23485 . . . 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 23439 . . . . 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 23485 . . . 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 4907   ∘ ccom 5679  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  fBascfbas 20924  Topctop 22386  TopOnctopon 22403   CnP ccnp 22720  Filcfil 23340   FilMap cfm 23428   fLim cflim 23429   fLimf cflf 23430
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 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-iun 4998  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 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-fbas 20933  df-fg 20934  df-top 22387  df-topon 22404  df-ntr 22515  df-nei 22593  df-cnp 22723  df-fil 23341  df-fm 23433  df-flim 23434  df-flf 23435
This theorem is referenced by:  flfcnp2  23502  tsmsmhm  23641
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