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Theorem cvmlift2lem13 29826
Description: Lemma for cvmlift2 29827. (Contributed by Mario Carneiro, 7-May-2015.)
Hypotheses
Ref Expression
cvmlift2.b  |-  B  = 
U. C
cvmlift2.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmlift2.g  |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )
cvmlift2.p  |-  ( ph  ->  P  e.  B )
cvmlift2.i  |-  ( ph  ->  ( F `  P
)  =  ( 0 G 0 ) )
cvmlift2.h  |-  H  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( f `
 0 )  =  P ) )
cvmlift2.k  |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( iota_ f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
f `  0 )  =  ( H `  x ) ) ) `
 y ) )
Assertion
Ref Expression
cvmlift2lem13  |-  ( ph  ->  E! g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) )
Distinct variable groups:    f, g, x, y, z, F    ph, f,
g, x, y, z   
f, J, g, x, y, z    f, G, g, x, y, z   
f, H, x, y, z    C, f, g, x, y, z    P, f, g, x, y, z   
x, B, y, z   
f, K, g, x, y, z
Allowed substitution hints:    B( f, g)    H( g)

Proof of Theorem cvmlift2lem13
Dummy variables  b 
c  d  u  v  a  r  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmlift2.b . . . 4  |-  B  = 
U. C
2 cvmlift2.f . . . 4  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
3 cvmlift2.g . . . 4  |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )
4 cvmlift2.p . . . 4  |-  ( ph  ->  P  e.  B )
5 cvmlift2.i . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( 0 G 0 ) )
6 cvmlift2.h . . . 4  |-  H  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( f `
 0 )  =  P ) )
7 cvmlift2.k . . . 4  |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( iota_ f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
f `  0 )  =  ( H `  x ) ) ) `
 y ) )
8 fveq2 5881 . . . . . 6  |-  ( a  =  z  ->  (
( ( II  tX  II )  CnP  C ) `
 a )  =  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
98eleq2d 2499 . . . . 5  |-  ( a  =  z  ->  ( K  e.  ( (
( II  tX  II )  CnP  C ) `  a )  <->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) ) )
109cbvrabv 3086 . . . 4  |-  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  =  {
z  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  |  K  e.  ( ( ( II  tX  II )  CnP  C ) `  z ) }
11 sneq 4012 . . . . . . 7  |-  ( z  =  b  ->  { z }  =  { b } )
1211xpeq2d 4878 . . . . . 6  |-  ( z  =  b  ->  (
( 0 [,] 1
)  X.  { z } )  =  ( ( 0 [,] 1
)  X.  { b } ) )
1312sseq1d 3497 . . . . 5  |-  ( z  =  b  ->  (
( ( 0 [,] 1 )  X.  {
z } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  |  K  e.  ( ( ( II  tX  II )  CnP  C ) `  a ) }  <->  ( (
0 [,] 1 )  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) )
1413cbvrabv 3086 . . . 4  |-  { z  e.  ( 0 [,] 1 )  |  ( ( 0 [,] 1
)  X.  { z } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } }  =  { b  e.  ( 0 [,] 1 )  |  ( ( 0 [,] 1 )  X. 
{ b } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } }
15 simpr 462 . . . . . . 7  |-  ( ( c  =  r  /\  d  =  t )  ->  d  =  t )
1615eleq1d 2498 . . . . . 6  |-  ( ( c  =  r  /\  d  =  t )  ->  ( d  e.  ( 0 [,] 1 )  <-> 
t  e.  ( 0 [,] 1 ) ) )
17 xpeq1 4868 . . . . . . . . . 10  |-  ( v  =  u  ->  (
v  X.  { b } )  =  ( u  X.  { b } ) )
1817sseq1d 3497 . . . . . . . . 9  |-  ( v  =  u  ->  (
( v  X.  {
b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  |  K  e.  ( ( ( II  tX  II )  CnP  C ) `  a ) }  <->  ( u  X.  { b } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) )
19 xpeq1 4868 . . . . . . . . . 10  |-  ( v  =  u  ->  (
v  X.  { d } )  =  ( u  X.  { d } ) )
2019sseq1d 3497 . . . . . . . . 9  |-  ( v  =  u  ->  (
( v  X.  {
d } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  |  K  e.  ( ( ( II  tX  II )  CnP  C ) `  a ) }  <->  ( u  X.  { d } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) )
2118, 20bibi12d 322 . . . . . . . 8  |-  ( v  =  u  ->  (
( ( v  X. 
{ b } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( v  X.  { d } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } )  <->  ( (
u  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( u  X.  { d } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) ) )
2221cbvrexv 3063 . . . . . . 7  |-  ( E. v  e.  ( ( nei `  II ) `
 { c } ) ( ( v  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( v  X.  { d } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } )  <->  E. u  e.  ( ( nei `  II ) `  { c } ) ( ( u  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( u  X.  { d } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) )
23 simpl 458 . . . . . . . . . 10  |-  ( ( c  =  r  /\  d  =  t )  ->  c  =  r )
2423sneqd 4014 . . . . . . . . 9  |-  ( ( c  =  r  /\  d  =  t )  ->  { c }  =  { r } )
2524fveq2d 5885 . . . . . . . 8  |-  ( ( c  =  r  /\  d  =  t )  ->  ( ( nei `  II ) `  { c } )  =  ( ( nei `  II ) `  { r } ) )
2615sneqd 4014 . . . . . . . . . . 11  |-  ( ( c  =  r  /\  d  =  t )  ->  { d }  =  { t } )
2726xpeq2d 4878 . . . . . . . . . 10  |-  ( ( c  =  r  /\  d  =  t )  ->  ( u  X.  {
d } )  =  ( u  X.  {
t } ) )
2827sseq1d 3497 . . . . . . . . 9  |-  ( ( c  =  r  /\  d  =  t )  ->  ( ( u  X.  { d } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( u  X.  { t } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) )
2928bibi2d 319 . . . . . . . 8  |-  ( ( c  =  r  /\  d  =  t )  ->  ( ( ( u  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( u  X.  { d } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } )  <->  ( (
u  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( u  X.  { t } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) ) )
3025, 29rexeqbidv 3047 . . . . . . 7  |-  ( ( c  =  r  /\  d  =  t )  ->  ( E. u  e.  ( ( nei `  II ) `  { c } ) ( ( u  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( u  X.  { d } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } )  <->  E. u  e.  ( ( nei `  II ) `  { r } ) ( ( u  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( u  X.  { t } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) ) )
3122, 30syl5bb 260 . . . . . 6  |-  ( ( c  =  r  /\  d  =  t )  ->  ( E. v  e.  ( ( nei `  II ) `  { c } ) ( ( v  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( v  X.  { d } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } )  <->  E. u  e.  ( ( nei `  II ) `  { r } ) ( ( u  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( u  X.  { t } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) ) )
3216, 31anbi12d 715 . . . . 5  |-  ( ( c  =  r  /\  d  =  t )  ->  ( ( d  e.  ( 0 [,] 1
)  /\  E. v  e.  ( ( nei `  II ) `  { c } ) ( ( v  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( v  X.  { d } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) )  <-> 
( t  e.  ( 0 [,] 1 )  /\  E. u  e.  ( ( nei `  II ) `  { r } ) ( ( u  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( u  X.  { t } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) ) ) )
3332cbvopabv 4495 . . . 4  |-  { <. c ,  d >.  |  ( d  e.  ( 0 [,] 1 )  /\  E. v  e.  ( ( nei `  II ) `
 { c } ) ( ( v  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( v  X.  { d } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) ) }  =  { <. r ,  t >.  |  ( t  e.  ( 0 [,] 1 )  /\  E. u  e.  ( ( nei `  II ) `
 { r } ) ( ( u  X.  { b } )  C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) }  <->  ( u  X.  { t } ) 
C_  { a  e.  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  a
) } ) ) }
341, 2, 3, 4, 5, 6, 7, 10, 14, 33cvmlift2lem12 29825 . . 3  |-  ( ph  ->  K  e.  ( ( II  tX  II )  Cn  C ) )
351, 2, 3, 4, 5, 6, 7cvmlift2lem7 29820 . . 3  |-  ( ph  ->  ( F  o.  K
)  =  G )
36 0elunit 11748 . . . . 5  |-  0  e.  ( 0 [,] 1
)
371, 2, 3, 4, 5, 6, 7cvmlift2lem8 29821 . . . . 5  |-  ( (
ph  /\  0  e.  ( 0 [,] 1
) )  ->  (
0 K 0 )  =  ( H ` 
0 ) )
3836, 37mpan2 675 . . . 4  |-  ( ph  ->  ( 0 K 0 )  =  ( H `
 0 ) )
391, 2, 3, 4, 5, 6cvmlift2lem2 29815 . . . . 5  |-  ( ph  ->  ( H  e.  ( II  Cn  C )  /\  ( F  o.  H )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( H `
 0 )  =  P ) )
4039simp3d 1019 . . . 4  |-  ( ph  ->  ( H `  0
)  =  P )
4138, 40eqtrd 2470 . . 3  |-  ( ph  ->  ( 0 K 0 )  =  P )
42 coeq2 5013 . . . . . 6  |-  ( g  =  K  ->  ( F  o.  g )  =  ( F  o.  K ) )
4342eqeq1d 2431 . . . . 5  |-  ( g  =  K  ->  (
( F  o.  g
)  =  G  <->  ( F  o.  K )  =  G ) )
44 oveq 6311 . . . . . 6  |-  ( g  =  K  ->  (
0 g 0 )  =  ( 0 K 0 ) )
4544eqeq1d 2431 . . . . 5  |-  ( g  =  K  ->  (
( 0 g 0 )  =  P  <->  ( 0 K 0 )  =  P ) )
4643, 45anbi12d 715 . . . 4  |-  ( g  =  K  ->  (
( ( F  o.  g )  =  G  /\  ( 0 g 0 )  =  P )  <->  ( ( F  o.  K )  =  G  /\  ( 0 K 0 )  =  P ) ) )
4746rspcev 3188 . . 3  |-  ( ( K  e.  ( ( II  tX  II )  Cn  C )  /\  (
( F  o.  K
)  =  G  /\  ( 0 K 0 )  =  P ) )  ->  E. g  e.  ( ( II  tX  II )  Cn  C
) ( ( F  o.  g )  =  G  /\  ( 0 g 0 )  =  P ) )
4834, 35, 41, 47syl12anc 1262 . 2  |-  ( ph  ->  E. g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) )
49 iitop 21808 . . . . 5  |-  II  e.  Top
50 iiuni 21809 . . . . 5  |-  ( 0 [,] 1 )  = 
U. II
5149, 49, 50, 50txunii 20539 . . . 4  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
52 iicon 21815 . . . . . 6  |-  II  e.  Con
53 txcon 20635 . . . . . 6  |-  ( ( II  e.  Con  /\  II  e.  Con )  -> 
( II  tX  II )  e.  Con )
5452, 52, 53mp2an 676 . . . . 5  |-  ( II 
tX  II )  e. 
Con
5554a1i 11 . . . 4  |-  ( ph  ->  ( II  tX  II )  e.  Con )
56 iinllycon 29765 . . . . . 6  |-  II  e. 𝑛Locally  Con
57 txcon 20635 . . . . . . 7  |-  ( ( x  e.  Con  /\  y  e.  Con )  ->  ( x  tX  y
)  e.  Con )
5857txnlly 20583 . . . . . 6  |-  ( ( II  e. 𝑛Locally  Con  /\  II  e. 𝑛Locally  Con )  ->  ( II  tX  II )  e. 𝑛Locally  Con )
5956, 56, 58mp2an 676 . . . . 5  |-  ( II 
tX  II )  e. 𝑛Locally  Con
6059a1i 11 . . . 4  |-  ( ph  ->  ( II  tX  II )  e. 𝑛Locally  Con )
61 opelxpi 4886 . . . . . 6  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  <. 0 ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
6236, 36, 61mp2an 676 . . . . 5  |-  <. 0 ,  0 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) )
6362a1i 11 . . . 4  |-  ( ph  -> 
<. 0 ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
64 df-ov 6308 . . . . 5  |-  ( 0 G 0 )  =  ( G `  <. 0 ,  0 >. )
655, 64syl6eq 2486 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( G `
 <. 0 ,  0
>. ) )
661, 51, 2, 55, 60, 63, 3, 4, 65cvmliftmo 29795 . . 3  |-  ( ph  ->  E* g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( g `  <. 0 ,  0 >. )  =  P ) )
67 df-ov 6308 . . . . . 6  |-  ( 0 g 0 )  =  ( g `  <. 0 ,  0 >. )
6867eqeq1i 2436 . . . . 5  |-  ( ( 0 g 0 )  =  P  <->  ( g `  <. 0 ,  0
>. )  =  P
)
6968anbi2i 698 . . . 4  |-  ( ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P )  <-> 
( ( F  o.  g )  =  G  /\  ( g `  <. 0 ,  0 >.
)  =  P ) )
7069rmobii 3027 . . 3  |-  ( E* g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g )  =  G  /\  (
0 g 0 )  =  P )  <->  E* g  e.  ( ( II  tX  II )  Cn  C
) ( ( F  o.  g )  =  G  /\  ( g `
 <. 0 ,  0
>. )  =  P
) )
7166, 70sylibr 215 . 2  |-  ( ph  ->  E* g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) )
72 reu5 3051 . 2  |-  ( E! g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g )  =  G  /\  (
0 g 0 )  =  P )  <->  ( E. g  e.  ( (
II  tX  II )  Cn  C ) ( ( F  o.  g )  =  G  /\  (
0 g 0 )  =  P )  /\  E* g  e.  (
( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) ) )
7348, 71, 72sylanbrc 668 1  |-  ( ph  ->  E! g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783   E!wreu 2784   E*wrmo 2785   {crab 2786    C_ wss 3442   {csn 4002   <.cop 4008   U.cuni 4222   {copab 4483    |-> cmpt 4484    X. cxp 4852    o. ccom 4858   ` cfv 5601   iota_crio 6266  (class class class)co 6305    |-> cmpt2 6307   0cc0 9538   1c1 9539   [,]cicc 11638   neicnei 20044    Cn ccn 20171    CnP ccnp 20172   Conccon 20357  𝑛Locally cnlly 20411    tX ctx 20506   IIcii 21803   CovMap ccvm 29766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-ec 7373  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-cn 20174  df-cnp 20175  df-cmp 20333  df-con 20358  df-lly 20412  df-nlly 20413  df-tx 20508  df-hmeo 20701  df-xms 21266  df-ms 21267  df-tms 21268  df-ii 21805  df-htpy 21894  df-phtpy 21895  df-phtpc 21916  df-pcon 29732  df-scon 29733  df-cvm 29767
This theorem is referenced by:  cvmlift2  29827
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