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Theorem cvmlift3 25020
Description: A general version of cvmlift 24991. 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 2447 . . . . . . . 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 1261 . . . . . . 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 2728 . . . . . 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 6564 . . . . 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 5730 . . . . . . . . . 10  |-  ( c  =  f  ->  (
c `  0 )  =  ( f ` 
0 ) )
1514eqeq1d 2446 . . . . . . . . 9  |-  ( c  =  f  ->  (
( c `  0
)  =  O  <->  ( f `  0 )  =  O ) )
16 fveq1 5730 . . . . . . . . . 10  |-  ( c  =  f  ->  (
c `  1 )  =  ( f ` 
1 ) )
1716eqeq1d 2446 . . . . . . . . 9  |-  ( c  =  f  ->  (
( c `  1
)  =  a  <->  ( f `  1 )  =  a ) )
18 coeq2 5034 . . . . . . . . . . . . . . 15  |-  ( d  =  g  ->  ( F  o.  d )  =  ( F  o.  g ) )
1918eqeq1d 2446 . . . . . . . . . . . . . 14  |-  ( d  =  g  ->  (
( F  o.  d
)  =  ( G  o.  c )  <->  ( F  o.  g )  =  ( G  o.  c ) ) )
20 fveq1 5730 . . . . . . . . . . . . . . 15  |-  ( d  =  g  ->  (
d `  0 )  =  ( g ` 
0 ) )
2120eqeq1d 2446 . . . . . . . . . . . . . 14  |-  ( d  =  g  ->  (
( d `  0
)  =  P  <->  ( g `  0 )  =  P ) )
2219, 21anbi12d 693 . . . . . . . . . . . . 13  |-  ( d  =  g  ->  (
( ( F  o.  d )  =  ( G  o.  c )  /\  ( d ` 
0 )  =  P )  <->  ( ( F  o.  g )  =  ( G  o.  c
)  /\  ( g `  0 )  =  P ) ) )
2322cbvriotav 6564 . . . . . . . . . . . 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 5034 . . . . . . . . . . . . . . 15  |-  ( c  =  f  ->  ( G  o.  c )  =  ( G  o.  f ) )
2524eqeq2d 2449 . . . . . . . . . . . . . 14  |-  ( c  =  f  ->  (
( F  o.  g
)  =  ( G  o.  c )  <->  ( F  o.  g )  =  ( G  o.  f ) ) )
2625anbi1d 687 . . . . . . . . . . . . 13  |-  ( c  =  f  ->  (
( ( F  o.  g )  =  ( G  o.  c )  /\  ( g ` 
0 )  =  P )  <->  ( ( F  o.  g )  =  ( G  o.  f
)  /\  ( g `  0 )  =  P ) ) )
2726riotabidv 6554 . . . . . . . . . . . 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 2482 . . . . . . . . . . 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 5733 . . . . . . . . . 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 2446 . . . . . . . . 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 1255 . . . . . . . 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 2935 . . . . . . 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 2447 . . . . . . . . 9  |-  ( a  =  x  ->  (
( f `  1
)  =  a  <->  ( f `  1 )  =  x ) )
34333anbi2d 1260 . . . . . . . 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 2728 . . . . . . 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 250 . . . . . 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 6554 . . . . 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 2482 . . . 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 4303 . . 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 2438 . . . 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 24950 . . 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 25019 . 2  |-  ( ph  ->  E. f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
43 sconpcon 24919 . . . 4  |-  ( K  e. SCon  ->  K  e. PCon )
44 pconcon 24923 . . . 4  |-  ( K  e. PCon  ->  K  e.  Con )
454, 43, 443syl 19 . . 3  |-  ( ph  ->  K  e.  Con )
46 pconcon 24923 . . . . . 6  |-  ( x  e. PCon  ->  x  e.  Con )
4746ssriv 3354 . . . . 5  |- PCon  C_  Con
48 nllyss 17548 . . . . 5  |-  (PCon  C_  Con  -> 𝑛Locally PCon  C_ 𝑛Locally  Con )
4947, 48ax-mp 5 . . . 4  |- 𝑛Locally PCon  C_ 𝑛Locally  Con
5049, 5sseldi 3348 . . 3  |-  ( ph  ->  K  e. 𝑛Locally  Con )
511, 2, 3, 45, 50, 6, 7, 8, 9cvmliftmo 24976 . 2  |-  ( ph  ->  E* f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
52 reu5 2923 . 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 647 1  |-  ( ph  ->  E! f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   E!wreu 2709   E*wrmo 2710   {crab 2711    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   U.cuni 4017    e. cmpt 4269   `'ccnv 4880    |` cres 4883   "cima 4884    o. ccom 4885   ` cfv 5457  (class class class)co 6084   iota_crio 6545   0cc0 8995   1c1 8996   ↾t crest 13653    Cn ccn 17293   Conccon 17479  𝑛Locally cnlly 17533    Homeo chmeo 17790   IIcii 18910  PConcpcon 24911  SConcscon 24912   CovMap ccvm 24947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-ec 6910  df-map 7023  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-cn 17296  df-cnp 17297  df-cmp 17455  df-con 17480  df-lly 17534  df-nlly 17535  df-tx 17599  df-hmeo 17792  df-xms 18355  df-ms 18356  df-tms 18357  df-ii 18912  df-htpy 19000  df-phtpy 19001  df-phtpc 19022  df-pco 19035  df-pcon 24913  df-scon 24914  df-cvm 24948
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