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Theorem cvmlift3 27222
Description: A general version of cvmlift 27193. If  K is simply connected and weakly locally path-connected, then there is a unique lift of functions on  K which commutes with the covering map. (Contributed by Mario Carneiro, 9-Jul-2015.)
Hypotheses
Ref Expression
cvmlift3.b  |-  B  = 
U. C
cvmlift3.y  |-  Y  = 
U. K
cvmlift3.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmlift3.k  |-  ( ph  ->  K  e. SCon )
cvmlift3.l  |-  ( ph  ->  K  e. 𝑛Locally PCon )
cvmlift3.o  |-  ( ph  ->  O  e.  Y )
cvmlift3.g  |-  ( ph  ->  G  e.  ( K  Cn  J ) )
cvmlift3.p  |-  ( ph  ->  P  e.  B )
cvmlift3.e  |-  ( ph  ->  ( F `  P
)  =  ( G `
 O ) )
Assertion
Ref Expression
cvmlift3  |-  ( ph  ->  E! f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
Distinct variable groups:    f, J    f, F    B, f    f, G    C, f    ph, f    f, K    P, f    f, O   
f, Y

Proof of Theorem cvmlift3
Dummy variables  b 
c  d  k  s  z  g  a  u  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmlift3.b . . 3  |-  B  = 
U. C
2 cvmlift3.y . . 3  |-  Y  = 
U. K
3 cvmlift3.f . . 3  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
4 cvmlift3.k . . 3  |-  ( ph  ->  K  e. SCon )
5 cvmlift3.l . . 3  |-  ( ph  ->  K  e. 𝑛Locally PCon )
6 cvmlift3.o . . 3  |-  ( ph  ->  O  e.  Y )
7 cvmlift3.g . . 3  |-  ( ph  ->  G  e.  ( K  Cn  J ) )
8 cvmlift3.p . . 3  |-  ( ph  ->  P  e.  B )
9 cvmlift3.e . . 3  |-  ( ph  ->  ( F `  P
)  =  ( G `
 O ) )
10 eqeq2 2452 . . . . . . . 8  |-  ( b  =  z  ->  (
( ( iota_ d  e.  ( II  Cn  C
) ( ( F  o.  d )  =  ( G  o.  c
)  /\  ( d `  0 )  =  P ) ) ` 
1 )  =  b  <-> 
( ( iota_ d  e.  ( II  Cn  C
) ( ( F  o.  d )  =  ( G  o.  c
)  /\  ( d `  0 )  =  P ) ) ` 
1 )  =  z ) )
11103anbi3d 1295 . . . . . . 7  |-  ( b  =  z  ->  (
( ( c ` 
0 )  =  O  /\  ( c ` 
1 )  =  a  /\  ( ( iota_ d  e.  ( II  Cn  C ) ( ( F  o.  d )  =  ( G  o.  c )  /\  (
d `  0 )  =  P ) ) ` 
1 )  =  b )  <->  ( ( c `
 0 )  =  O  /\  ( c `
 1 )  =  a  /\  ( (
iota_ d  e.  (
II  Cn  C )
( ( F  o.  d )  =  ( G  o.  c )  /\  ( d ` 
0 )  =  P ) ) `  1
)  =  z ) ) )
1211rexbidv 2741 . . . . . 6  |-  ( b  =  z  ->  ( E. c  e.  (
II  Cn  K )
( ( c ` 
0 )  =  O  /\  ( c ` 
1 )  =  a  /\  ( ( iota_ d  e.  ( II  Cn  C ) ( ( F  o.  d )  =  ( G  o.  c )  /\  (
d `  0 )  =  P ) ) ` 
1 )  =  b )  <->  E. c  e.  ( II  Cn  K ) ( ( c ` 
0 )  =  O  /\  ( c ` 
1 )  =  a  /\  ( ( iota_ d  e.  ( II  Cn  C ) ( ( F  o.  d )  =  ( G  o.  c )  /\  (
d `  0 )  =  P ) ) ` 
1 )  =  z ) ) )
1312cbvriotav 6068 . . . . 5  |-  ( iota_ b  e.  B  E. c  e.  ( II  Cn  K
) ( ( c `
 0 )  =  O  /\  ( c `
 1 )  =  a  /\  ( (
iota_ d  e.  (
II  Cn  C )
( ( F  o.  d )  =  ( G  o.  c )  /\  ( d ` 
0 )  =  P ) ) `  1
)  =  b ) )  =  ( iota_ z  e.  B  E. c  e.  ( II  Cn  K
) ( ( c `
 0 )  =  O  /\  ( c `
 1 )  =  a  /\  ( (
iota_ d  e.  (
II  Cn  C )
( ( F  o.  d )  =  ( G  o.  c )  /\  ( d ` 
0 )  =  P ) ) `  1
)  =  z ) )
14 fveq1 5695 . . . . . . . . . 10  |-  ( c  =  f  ->  (
c `  0 )  =  ( f ` 
0 ) )
1514eqeq1d 2451 . . . . . . . . 9  |-  ( c  =  f  ->  (
( c `  0
)  =  O  <->  ( f `  0 )  =  O ) )
16 fveq1 5695 . . . . . . . . . 10  |-  ( c  =  f  ->  (
c `  1 )  =  ( f ` 
1 ) )
1716eqeq1d 2451 . . . . . . . . 9  |-  ( c  =  f  ->  (
( c `  1
)  =  a  <->  ( f `  1 )  =  a ) )
18 coeq2 5003 . . . . . . . . . . . . . . 15  |-  ( d  =  g  ->  ( F  o.  d )  =  ( F  o.  g ) )
1918eqeq1d 2451 . . . . . . . . . . . . . 14  |-  ( d  =  g  ->  (
( F  o.  d
)  =  ( G  o.  c )  <->  ( F  o.  g )  =  ( G  o.  c ) ) )
20 fveq1 5695 . . . . . . . . . . . . . . 15  |-  ( d  =  g  ->  (
d `  0 )  =  ( g ` 
0 ) )
2120eqeq1d 2451 . . . . . . . . . . . . . 14  |-  ( d  =  g  ->  (
( d `  0
)  =  P  <->  ( g `  0 )  =  P ) )
2219, 21anbi12d 710 . . . . . . . . . . . . 13  |-  ( d  =  g  ->  (
( ( F  o.  d )  =  ( G  o.  c )  /\  ( d ` 
0 )  =  P )  <->  ( ( F  o.  g )  =  ( G  o.  c
)  /\  ( g `  0 )  =  P ) ) )
2322cbvriotav 6068 . . . . . . . . . . . 12  |-  ( iota_ d  e.  ( II  Cn  C ) ( ( F  o.  d )  =  ( G  o.  c )  /\  (
d `  0 )  =  P ) )  =  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  c )  /\  ( g ` 
0 )  =  P ) )
24 coeq2 5003 . . . . . . . . . . . . . . 15  |-  ( c  =  f  ->  ( G  o.  c )  =  ( G  o.  f ) )
2524eqeq2d 2454 . . . . . . . . . . . . . 14  |-  ( c  =  f  ->  (
( F  o.  g
)  =  ( G  o.  c )  <->  ( F  o.  g )  =  ( G  o.  f ) ) )
2625anbi1d 704 . . . . . . . . . . . . 13  |-  ( c  =  f  ->  (
( ( F  o.  g )  =  ( G  o.  c )  /\  ( g ` 
0 )  =  P )  <->  ( ( F  o.  g )  =  ( G  o.  f
)  /\  ( g `  0 )  =  P ) ) )
2726riotabidv 6059 . . . . . . . . . . . 12  |-  ( c  =  f  ->  ( iota_ g  e.  ( II 
Cn  C ) ( ( F  o.  g
)  =  ( G  o.  c )  /\  ( g `  0
)  =  P ) )  =  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) )
2823, 27syl5eq 2487 . . . . . . . . . . 11  |-  ( c  =  f  ->  ( iota_ d  e.  ( II 
Cn  C ) ( ( F  o.  d
)  =  ( G  o.  c )  /\  ( d `  0
)  =  P ) )  =  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) )
2928fveq1d 5698 . . . . . . . . . 10  |-  ( c  =  f  ->  (
( iota_ d  e.  ( II  Cn  C ) ( ( F  o.  d )  =  ( G  o.  c )  /\  ( d ` 
0 )  =  P ) ) `  1
)  =  ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
) )
3029eqeq1d 2451 . . . . . . . . 9  |-  ( c  =  f  ->  (
( ( iota_ d  e.  ( II  Cn  C
) ( ( F  o.  d )  =  ( G  o.  c
)  /\  ( d `  0 )  =  P ) ) ` 
1 )  =  z  <-> 
( ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  f
)  /\  ( g `  0 )  =  P ) ) ` 
1 )  =  z ) )
3115, 17, 303anbi123d 1289 . . . . . . . 8  |-  ( c  =  f  ->  (
( ( c ` 
0 )  =  O  /\  ( c ` 
1 )  =  a  /\  ( ( iota_ d  e.  ( II  Cn  C ) ( ( F  o.  d )  =  ( G  o.  c )  /\  (
d `  0 )  =  P ) ) ` 
1 )  =  z )  <->  ( ( f `
 0 )  =  O  /\  ( f `
 1 )  =  a  /\  ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
)  =  z ) ) )
3231cbvrexv 2953 . . . . . . 7  |-  ( E. c  e.  ( II 
Cn  K ) ( ( c `  0
)  =  O  /\  ( c `  1
)  =  a  /\  ( ( iota_ d  e.  ( II  Cn  C
) ( ( F  o.  d )  =  ( G  o.  c
)  /\  ( d `  0 )  =  P ) ) ` 
1 )  =  z )  <->  E. f  e.  ( II  Cn  K ) ( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  a  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  z ) )
33 eqeq2 2452 . . . . . . . . 9  |-  ( a  =  x  ->  (
( f `  1
)  =  a  <->  ( f `  1 )  =  x ) )
34333anbi2d 1294 . . . . . . . 8  |-  ( a  =  x  ->  (
( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  a  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  z )  <->  ( ( f `
 0 )  =  O  /\  ( f `
 1 )  =  x  /\  ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
)  =  z ) ) )
3534rexbidv 2741 . . . . . . 7  |-  ( a  =  x  ->  ( E. f  e.  (
II  Cn  K )
( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  a  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  z )  <->  E. f  e.  ( II  Cn  K ) ( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  z ) ) )
3632, 35syl5bb 257 . . . . . 6  |-  ( a  =  x  ->  ( E. c  e.  (
II  Cn  K )
( ( c ` 
0 )  =  O  /\  ( c ` 
1 )  =  a  /\  ( ( iota_ d  e.  ( II  Cn  C ) ( ( F  o.  d )  =  ( G  o.  c )  /\  (
d `  0 )  =  P ) ) ` 
1 )  =  z )  <->  E. f  e.  ( II  Cn  K ) ( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  z ) ) )
3736riotabidv 6059 . . . . 5  |-  ( a  =  x  ->  ( iota_ z  e.  B  E. c  e.  ( II  Cn  K ) ( ( c `  0 )  =  O  /\  (
c `  1 )  =  a  /\  (
( iota_ d  e.  ( II  Cn  C ) ( ( F  o.  d )  =  ( G  o.  c )  /\  ( d ` 
0 )  =  P ) ) `  1
)  =  z ) )  =  ( iota_ z  e.  B  E. f  e.  ( II  Cn  K
) ( ( f `
 0 )  =  O  /\  ( f `
 1 )  =  x  /\  ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
)  =  z ) ) )
3813, 37syl5eq 2487 . . . 4  |-  ( a  =  x  ->  ( iota_ b  e.  B  E. c  e.  ( II  Cn  K ) ( ( c `  0 )  =  O  /\  (
c `  1 )  =  a  /\  (
( iota_ d  e.  ( II  Cn  C ) ( ( F  o.  d )  =  ( G  o.  c )  /\  ( d ` 
0 )  =  P ) ) `  1
)  =  b ) )  =  ( iota_ z  e.  B  E. f  e.  ( II  Cn  K
) ( ( f `
 0 )  =  O  /\  ( f `
 1 )  =  x  /\  ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
)  =  z ) ) )
3938cbvmptv 4388 . . 3  |-  ( a  e.  Y  |->  ( iota_ b  e.  B  E. c  e.  ( II  Cn  K
) ( ( c `
 0 )  =  O  /\  ( c `
 1 )  =  a  /\  ( (
iota_ d  e.  (
II  Cn  C )
( ( F  o.  d )  =  ( G  o.  c )  /\  ( d ` 
0 )  =  P ) ) `  1
)  =  b ) ) )  =  ( x  e.  Y  |->  (
iota_ z  e.  B  E. f  e.  (
II  Cn  K )
( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  z ) ) )
40 eqid 2443 . . . 4  |-  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
k )  /\  A. c  e.  s  ( A. d  e.  (
s  \  { c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c ) Homeo ( Jt  k ) ) ) ) } )  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/)
} )  |  ( U. s  =  ( `' F " k )  /\  A. c  e.  s  ( A. d  e.  ( s  \  {
c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c )
Homeo ( Jt  k ) ) ) ) } )
4140cvmscbv 27152 . . 3  |-  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
k )  /\  A. c  e.  s  ( A. d  e.  (
s  \  { c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c ) Homeo ( Jt  k ) ) ) ) } )  =  ( a  e.  J  |->  { b  e.  ( ~P C  \  { (/)
} )  |  ( U. b  =  ( `' F " a )  /\  A. v  e.  b  ( A. u  e.  ( b  \  {
v } ) ( v  i^i  u )  =  (/)  /\  ( F  |`  v )  e.  ( ( Ct  v )
Homeo ( Jt  a ) ) ) ) } )
421, 2, 3, 4, 5, 6, 7, 8, 9, 39, 41cvmlift3lem9 27221 . 2  |-  ( ph  ->  E. f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
43 sconpcon 27121 . . . 4  |-  ( K  e. SCon  ->  K  e. PCon )
44 pconcon 27125 . . . 4  |-  ( K  e. PCon  ->  K  e.  Con )
454, 43, 443syl 20 . . 3  |-  ( ph  ->  K  e.  Con )
46 pconcon 27125 . . . . . 6  |-  ( x  e. PCon  ->  x  e.  Con )
4746ssriv 3365 . . . . 5  |- PCon  C_  Con
48 nllyss 19089 . . . . 5  |-  (PCon  C_  Con  -> 𝑛Locally PCon  C_ 𝑛Locally  Con )
4947, 48ax-mp 5 . . . 4  |- 𝑛Locally PCon  C_ 𝑛Locally  Con
5049, 5sseldi 3359 . . 3  |-  ( ph  ->  K  e. 𝑛Locally  Con )
511, 2, 3, 45, 50, 6, 7, 8, 9cvmliftmo 27178 . 2  |-  ( ph  ->  E* f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
52 reu5 2941 . 2  |-  ( E! f  e.  ( K  Cn  C ) ( ( F  o.  f
)  =  G  /\  ( f `  O
)  =  P )  <-> 
( E. f  e.  ( K  Cn  C
) ( ( F  o.  f )  =  G  /\  ( f `
 O )  =  P )  /\  E* f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P ) ) )
5342, 51, 52sylanbrc 664 1  |-  ( ph  ->  E! f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   E!wreu 2722   E*wrmo 2723   {crab 2724    \ cdif 3330    i^i cin 3332    C_ wss 3333   (/)c0 3642   ~Pcpw 3865   {csn 3882   U.cuni 4096    e. cmpt 4355   `'ccnv 4844    |` cres 4847   "cima 4848    o. ccom 4849   ` cfv 5423   iota_crio 6056  (class class class)co 6096   0cc0 9287   1c1 9288   ↾t crest 14364    Cn ccn 18833   Conccon 19020  𝑛Locally cnlly 19074   Homeochmeo 19331   IIcii 20456  PConcpcon 27113  SConcscon 27114   CovMap ccvm 27149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-ec 7108  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-cn 18836  df-cnp 18837  df-cmp 18995  df-con 19021  df-lly 19075  df-nlly 19076  df-tx 19140  df-hmeo 19333  df-xms 19900  df-ms 19901  df-tms 19902  df-ii 20458  df-htpy 20547  df-phtpy 20548  df-phtpc 20569  df-pco 20582  df-pcon 27115  df-scon 27116  df-cvm 27150
This theorem is referenced by: (None)
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