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Theorem cuspcvg 20929
Description: In a complete uniform space, any Cauchy filter  C has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.)
Hypotheses
Ref Expression
cuspcvg.1  |-  B  =  ( Base `  W
)
cuspcvg.2  |-  J  =  ( TopOpen `  W )
Assertion
Ref Expression
cuspcvg  |-  ( ( W  e. CUnifSp  /\  C  e.  (CauFilu `  (UnifSt `  W
) )  /\  C  e.  ( Fil `  B
) )  ->  ( J  fLim  C )  =/=  (/) )

Proof of Theorem cuspcvg
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . 5  |-  ( ( W  e. CUnifSp  /\  C  e.  ( Fil `  B
) )  ->  C  e.  ( Fil `  B
) )
2 cuspcvg.1 . . . . . 6  |-  B  =  ( Base `  W
)
32fveq2i 5875 . . . . 5  |-  ( Fil `  B )  =  ( Fil `  ( Base `  W ) )
41, 3syl6eleq 2555 . . . 4  |-  ( ( W  e. CUnifSp  /\  C  e.  ( Fil `  B
) )  ->  C  e.  ( Fil `  ( Base `  W ) ) )
5 iscusp 20927 . . . . . 6  |-  ( W  e. CUnifSp 
<->  ( W  e. UnifSp  /\  A. c  e.  ( Fil `  ( Base `  W
) ) ( c  e.  (CauFilu `  (UnifSt `  W
) )  ->  (
( TopOpen `  W )  fLim  c )  =/=  (/) ) ) )
65simprbi 464 . . . . 5  |-  ( W  e. CUnifSp  ->  A. c  e.  ( Fil `  ( Base `  W ) ) ( c  e.  (CauFilu `  (UnifSt `  W ) )  -> 
( ( TopOpen `  W
)  fLim  c )  =/=  (/) ) )
76adantr 465 . . . 4  |-  ( ( W  e. CUnifSp  /\  C  e.  ( Fil `  B
) )  ->  A. c  e.  ( Fil `  ( Base `  W ) ) ( c  e.  (CauFilu `  (UnifSt `  W )
)  ->  ( ( TopOpen
`  W )  fLim  c )  =/=  (/) ) )
8 eleq1 2529 . . . . . 6  |-  ( c  =  C  ->  (
c  e.  (CauFilu `  (UnifSt `  W ) )  <->  C  e.  (CauFilu `  (UnifSt `  W )
) ) )
9 cuspcvg.2 . . . . . . . . . 10  |-  J  =  ( TopOpen `  W )
109eqcomi 2470 . . . . . . . . 9  |-  ( TopOpen `  W )  =  J
1110a1i 11 . . . . . . . 8  |-  ( c  =  C  ->  ( TopOpen
`  W )  =  J )
12 id 22 . . . . . . . 8  |-  ( c  =  C  ->  c  =  C )
1311, 12oveq12d 6314 . . . . . . 7  |-  ( c  =  C  ->  (
( TopOpen `  W )  fLim  c )  =  ( J  fLim  C )
)
1413neeq1d 2734 . . . . . 6  |-  ( c  =  C  ->  (
( ( TopOpen `  W
)  fLim  c )  =/=  (/)  <->  ( J  fLim  C )  =/=  (/) ) )
158, 14imbi12d 320 . . . . 5  |-  ( c  =  C  ->  (
( c  e.  (CauFilu `  (UnifSt `  W )
)  ->  ( ( TopOpen
`  W )  fLim  c )  =/=  (/) )  <->  ( C  e.  (CauFilu `  (UnifSt `  W
) )  ->  ( J  fLim  C )  =/=  (/) ) ) )
1615rspcva 3208 . . . 4  |-  ( ( C  e.  ( Fil `  ( Base `  W
) )  /\  A. c  e.  ( Fil `  ( Base `  W
) ) ( c  e.  (CauFilu `  (UnifSt `  W
) )  ->  (
( TopOpen `  W )  fLim  c )  =/=  (/) ) )  ->  ( C  e.  (CauFilu `  (UnifSt `  W
) )  ->  ( J  fLim  C )  =/=  (/) ) )
174, 7, 16syl2anc 661 . . 3  |-  ( ( W  e. CUnifSp  /\  C  e.  ( Fil `  B
) )  ->  ( C  e.  (CauFilu `  (UnifSt `  W ) )  -> 
( J  fLim  C
)  =/=  (/) ) )
18173impia 1193 . 2  |-  ( ( W  e. CUnifSp  /\  C  e.  ( Fil `  B
)  /\  C  e.  (CauFilu `  (UnifSt `  W )
) )  ->  ( J  fLim  C )  =/=  (/) )
19183com23 1202 1  |-  ( ( W  e. CUnifSp  /\  C  e.  (CauFilu `  (UnifSt `  W
) )  /\  C  e.  ( Fil `  B
) )  ->  ( J  fLim  C )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   (/)c0 3793   ` cfv 5594  (class class class)co 6296   Basecbs 14643   TopOpenctopn 14838   Filcfil 20471    fLim cflim 20560  UnifStcuss 20881  UnifSpcusp 20882  CauFiluccfilu 20914  CUnifSpccusp 20925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-cusp 20926
This theorem is referenced by:  cnextucn  20931
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