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Theorem cuspcvg 20929
 Description: In a complete uniform space, any Cauchy filter has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.)
Hypotheses
Ref Expression
cuspcvg.1
cuspcvg.2
Assertion
Ref Expression
cuspcvg CUnifSp CauFiluUnifSt

Proof of Theorem cuspcvg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . 5 CUnifSp
2 cuspcvg.1 . . . . . 6
32fveq2i 5875 . . . . 5
41, 3syl6eleq 2555 . . . 4 CUnifSp
5 iscusp 20927 . . . . . 6 CUnifSp UnifSp CauFiluUnifSt
65simprbi 464 . . . . 5 CUnifSp CauFiluUnifSt
76adantr 465 . . . 4 CUnifSp CauFiluUnifSt
8 eleq1 2529 . . . . . 6 CauFiluUnifSt CauFiluUnifSt
9 cuspcvg.2 . . . . . . . . . 10
109eqcomi 2470 . . . . . . . . 9
1110a1i 11 . . . . . . . 8
12 id 22 . . . . . . . 8
1311, 12oveq12d 6314 . . . . . . 7
1413neeq1d 2734 . . . . . 6
158, 14imbi12d 320 . . . . 5 CauFiluUnifSt CauFiluUnifSt
1615rspcva 3208 . . . 4 CauFiluUnifSt CauFiluUnifSt
174, 7, 16syl2anc 661 . . 3 CUnifSp CauFiluUnifSt
18173impia 1193 . 2 CUnifSp CauFiluUnifSt
19183com23 1202 1 CUnifSp CauFiluUnifSt
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1395   wcel 1819   wne 2652  wral 2807  c0 3793  cfv 5594  (class class class)co 6296  cbs 14643  ctopn 14838  cfil 20471   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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