Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmlift3 Structured version   Unicode version

Theorem cvmlift3 29501
Description: A general version of cvmlift 29472. 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 2415 . . . . . . . 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 1305 . . . . . . 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 2915 . . . . . 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 6205 . . . . 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 5802 . . . . . . . . . 10  |-  ( c  =  f  ->  (
c `  0 )  =  ( f ` 
0 ) )
1514eqeq1d 2402 . . . . . . . . 9  |-  ( c  =  f  ->  (
( c `  0
)  =  O  <->  ( f `  0 )  =  O ) )
16 fveq1 5802 . . . . . . . . . 10  |-  ( c  =  f  ->  (
c `  1 )  =  ( f ` 
1 ) )
1716eqeq1d 2402 . . . . . . . . 9  |-  ( c  =  f  ->  (
( c `  1
)  =  a  <->  ( f `  1 )  =  a ) )
18 coeq2 5101 . . . . . . . . . . . . . . 15  |-  ( d  =  g  ->  ( F  o.  d )  =  ( F  o.  g ) )
1918eqeq1d 2402 . . . . . . . . . . . . . 14  |-  ( d  =  g  ->  (
( F  o.  d
)  =  ( G  o.  c )  <->  ( F  o.  g )  =  ( G  o.  c ) ) )
20 fveq1 5802 . . . . . . . . . . . . . . 15  |-  ( d  =  g  ->  (
d `  0 )  =  ( g ` 
0 ) )
2120eqeq1d 2402 . . . . . . . . . . . . . 14  |-  ( d  =  g  ->  (
( d `  0
)  =  P  <->  ( g `  0 )  =  P ) )
2219, 21anbi12d 709 . . . . . . . . . . . . 13  |-  ( d  =  g  ->  (
( ( F  o.  d )  =  ( G  o.  c )  /\  ( d ` 
0 )  =  P )  <->  ( ( F  o.  g )  =  ( G  o.  c
)  /\  ( g `  0 )  =  P ) ) )
2322cbvriotav 6205 . . . . . . . . . . . 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 5101 . . . . . . . . . . . . . . 15  |-  ( c  =  f  ->  ( G  o.  c )  =  ( G  o.  f ) )
2524eqeq2d 2414 . . . . . . . . . . . . . 14  |-  ( c  =  f  ->  (
( F  o.  g
)  =  ( G  o.  c )  <->  ( F  o.  g )  =  ( G  o.  f ) ) )
2625anbi1d 703 . . . . . . . . . . . . 13  |-  ( c  =  f  ->  (
( ( F  o.  g )  =  ( G  o.  c )  /\  ( g ` 
0 )  =  P )  <->  ( ( F  o.  g )  =  ( G  o.  f
)  /\  ( g `  0 )  =  P ) ) )
2726riotabidv 6196 . . . . . . . . . . . 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 2453 . . . . . . . . . . 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 5805 . . . . . . . . . 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 2402 . . . . . . . . 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 1299 . . . . . . . 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 3032 . . . . . . 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 2415 . . . . . . . . 9  |-  ( a  =  x  ->  (
( f `  1
)  =  a  <->  ( f `  1 )  =  x ) )
34333anbi2d 1304 . . . . . . . 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 2915 . . . . . . 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 6196 . . . . 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 2453 . . . 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 4484 . . 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 2400 . . . 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 29431 . . 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 29500 . 2  |-  ( ph  ->  E. f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
43 sconpcon 29400 . . . 4  |-  ( K  e. SCon  ->  K  e. PCon )
44 pconcon 29404 . . . 4  |-  ( K  e. PCon  ->  K  e.  Con )
454, 43, 443syl 20 . . 3  |-  ( ph  ->  K  e.  Con )
46 pconcon 29404 . . . . . 6  |-  ( x  e. PCon  ->  x  e.  Con )
4746ssriv 3443 . . . . 5  |- PCon  C_  Con
48 nllyss 20163 . . . . 5  |-  (PCon  C_  Con  -> 𝑛Locally PCon  C_ 𝑛Locally  Con )
4947, 48ax-mp 5 . . . 4  |- 𝑛Locally PCon  C_ 𝑛Locally  Con
5049, 5sseldi 3437 . . 3  |-  ( ph  ->  K  e. 𝑛Locally  Con )
511, 2, 3, 45, 50, 6, 7, 8, 9cvmliftmo 29457 . 2  |-  ( ph  ->  E* f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
52 reu5 3020 . 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 662 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 367    /\ w3a 972    = wceq 1403    e. wcel 1840   A.wral 2751   E.wrex 2752   E!wreu 2753   E*wrmo 2754   {crab 2755    \ cdif 3408    i^i cin 3410    C_ wss 3411   (/)c0 3735   ~Pcpw 3952   {csn 3969   U.cuni 4188    |-> cmpt 4450   `'ccnv 4939    |` cres 4942   "cima 4943    o. ccom 4944   ` cfv 5523   iota_crio 6193  (class class class)co 6232   0cc0 9440   1c1 9441   ↾t crest 14925    Cn ccn 19908   Conccon 20094  𝑛Locally cnlly 20148   Homeochmeo 20436   IIcii 21561  PConcpcon 29392  SConcscon 29393   CovMap ccvm 29428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518  ax-addf 9519  ax-mulf 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-ec 7268  df-map 7377  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-fi 7823  df-sup 7853  df-oi 7887  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-q 11144  df-rp 11182  df-xneg 11287  df-xadd 11288  df-xmul 11289  df-ioo 11502  df-ico 11504  df-icc 11505  df-fz 11642  df-fzo 11766  df-fl 11877  df-seq 12060  df-exp 12119  df-hash 12358  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-clim 13365  df-sum 13563  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-starv 14814  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-unif 14822  df-hom 14823  df-cco 14824  df-rest 14927  df-topn 14928  df-0g 14946  df-gsum 14947  df-topgen 14948  df-pt 14949  df-prds 14952  df-xrs 15006  df-qtop 15011  df-imas 15012  df-xps 15014  df-mre 15090  df-mrc 15091  df-acs 15093  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-submnd 16181  df-mulg 16274  df-cntz 16569  df-cmn 17014  df-psmet 18621  df-xmet 18622  df-met 18623  df-bl 18624  df-mopn 18625  df-cnfld 18631  df-top 19581  df-bases 19583  df-topon 19584  df-topsp 19585  df-cld 19702  df-ntr 19703  df-cls 19704  df-nei 19782  df-cn 19911  df-cnp 19912  df-cmp 20070  df-con 20095  df-lly 20149  df-nlly 20150  df-tx 20245  df-hmeo 20438  df-xms 21005  df-ms 21006  df-tms 21007  df-ii 21563  df-htpy 21652  df-phtpy 21653  df-phtpc 21674  df-pco 21687  df-pcon 29394  df-scon 29395  df-cvm 29429
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator