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Theorem cvmlift2lem13 27134
Description: Lemma for cvmlift2 27135. (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 5688 . . . . . 6  |-  ( a  =  z  ->  (
( ( II  tX  II )  CnP  C ) `
 a )  =  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
98eleq2d 2508 . . . . 5  |-  ( a  =  z  ->  ( K  e.  ( (
( II  tX  II )  CnP  C ) `  a )  <->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) ) )
109cbvrabv 2969 . . . 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 3884 . . . . . . 7  |-  ( z  =  b  ->  { z }  =  { b } )
1211xpeq2d 4860 . . . . . 6  |-  ( z  =  b  ->  (
( 0 [,] 1
)  X.  { z } )  =  ( ( 0 [,] 1
)  X.  { b } ) )
1312sseq1d 3380 . . . . 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 2969 . . . 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 458 . . . . . . 7  |-  ( ( c  =  r  /\  d  =  t )  ->  d  =  t )
1615eleq1d 2507 . . . . . 6  |-  ( ( c  =  r  /\  d  =  t )  ->  ( d  e.  ( 0 [,] 1 )  <-> 
t  e.  ( 0 [,] 1 ) ) )
17 xpeq1 4850 . . . . . . . . . 10  |-  ( v  =  u  ->  (
v  X.  { b } )  =  ( u  X.  { b } ) )
1817sseq1d 3380 . . . . . . . . 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 4850 . . . . . . . . . 10  |-  ( v  =  u  ->  (
v  X.  { d } )  =  ( u  X.  { d } ) )
2019sseq1d 3380 . . . . . . . . 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 321 . . . . . . . 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 2946 . . . . . . 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 454 . . . . . . . . . 10  |-  ( ( c  =  r  /\  d  =  t )  ->  c  =  r )
2423sneqd 3886 . . . . . . . . 9  |-  ( ( c  =  r  /\  d  =  t )  ->  { c }  =  { r } )
2524fveq2d 5692 . . . . . . . 8  |-  ( ( c  =  r  /\  d  =  t )  ->  ( ( nei `  II ) `  { c } )  =  ( ( nei `  II ) `  { r } ) )
2615sneqd 3886 . . . . . . . . . . 11  |-  ( ( c  =  r  /\  d  =  t )  ->  { d }  =  { t } )
2726xpeq2d 4860 . . . . . . . . . 10  |-  ( ( c  =  r  /\  d  =  t )  ->  ( u  X.  {
d } )  =  ( u  X.  {
t } ) )
2827sseq1d 3380 . . . . . . . . 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 318 . . . . . . . 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 2930 . . . . . . 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 257 . . . . . 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 705 . . . . 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 4358 . . . 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 27133 . . 3  |-  ( ph  ->  K  e.  ( ( II  tX  II )  Cn  C ) )
351, 2, 3, 4, 5, 6, 7cvmlift2lem7 27128 . . 3  |-  ( ph  ->  ( F  o.  K
)  =  G )
36 0elunit 11399 . . . . 5  |-  0  e.  ( 0 [,] 1
)
371, 2, 3, 4, 5, 6, 7cvmlift2lem8 27129 . . . . 5  |-  ( (
ph  /\  0  e.  ( 0 [,] 1
) )  ->  (
0 K 0 )  =  ( H ` 
0 ) )
3836, 37mpan2 666 . . . 4  |-  ( ph  ->  ( 0 K 0 )  =  ( H `
 0 ) )
391, 2, 3, 4, 5, 6cvmlift2lem2 27123 . . . . 5  |-  ( ph  ->  ( H  e.  ( II  Cn  C )  /\  ( F  o.  H )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( H `
 0 )  =  P ) )
4039simp3d 997 . . . 4  |-  ( ph  ->  ( H `  0
)  =  P )
4138, 40eqtrd 2473 . . 3  |-  ( ph  ->  ( 0 K 0 )  =  P )
42 coeq2 4994 . . . . . 6  |-  ( g  =  K  ->  ( F  o.  g )  =  ( F  o.  K ) )
4342eqeq1d 2449 . . . . 5  |-  ( g  =  K  ->  (
( F  o.  g
)  =  G  <->  ( F  o.  K )  =  G ) )
44 oveq 6096 . . . . . 6  |-  ( g  =  K  ->  (
0 g 0 )  =  ( 0 K 0 ) )
4544eqeq1d 2449 . . . . 5  |-  ( g  =  K  ->  (
( 0 g 0 )  =  P  <->  ( 0 K 0 )  =  P ) )
4643, 45anbi12d 705 . . . 4  |-  ( g  =  K  ->  (
( ( F  o.  g )  =  G  /\  ( 0 g 0 )  =  P )  <->  ( ( F  o.  K )  =  G  /\  ( 0 K 0 )  =  P ) ) )
4746rspcev 3070 . . 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 1211 . 2  |-  ( ph  ->  E. g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) )
49 iitop 20415 . . . . 5  |-  II  e.  Top
50 iiuni 20416 . . . . 5  |-  ( 0 [,] 1 )  = 
U. II
5149, 49, 50, 50txunii 19125 . . . 4  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
52 iicon 20422 . . . . . 6  |-  II  e.  Con
53 txcon 19221 . . . . . 6  |-  ( ( II  e.  Con  /\  II  e.  Con )  -> 
( II  tX  II )  e.  Con )
5452, 52, 53mp2an 667 . . . . 5  |-  ( II 
tX  II )  e. 
Con
5554a1i 11 . . . 4  |-  ( ph  ->  ( II  tX  II )  e.  Con )
56 iinllycon 27073 . . . . . 6  |-  II  e. 𝑛Locally  Con
57 txcon 19221 . . . . . . 7  |-  ( ( x  e.  Con  /\  y  e.  Con )  ->  ( x  tX  y
)  e.  Con )
5857txnlly 19169 . . . . . 6  |-  ( ( II  e. 𝑛Locally  Con  /\  II  e. 𝑛Locally  Con )  ->  ( II  tX  II )  e. 𝑛Locally  Con )
5956, 56, 58mp2an 667 . . . . 5  |-  ( II 
tX  II )  e. 𝑛Locally  Con
6059a1i 11 . . . 4  |-  ( ph  ->  ( II  tX  II )  e. 𝑛Locally  Con )
61 opelxpi 4867 . . . . . 6  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  <. 0 ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
6236, 36, 61mp2an 667 . . . . 5  |-  <. 0 ,  0 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) )
6362a1i 11 . . . 4  |-  ( ph  -> 
<. 0 ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
64 df-ov 6093 . . . . 5  |-  ( 0 G 0 )  =  ( G `  <. 0 ,  0 >. )
655, 64syl6eq 2489 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( G `
 <. 0 ,  0
>. ) )
661, 51, 2, 55, 60, 63, 3, 4, 65cvmliftmo 27103 . . 3  |-  ( ph  ->  E* g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( g `  <. 0 ,  0 >. )  =  P ) )
67 df-ov 6093 . . . . . 6  |-  ( 0 g 0 )  =  ( g `  <. 0 ,  0 >. )
6867eqeq1i 2448 . . . . 5  |-  ( ( 0 g 0 )  =  P  <->  ( g `  <. 0 ,  0
>. )  =  P
)
6968anbi2i 689 . . . 4  |-  ( ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P )  <-> 
( ( F  o.  g )  =  G  /\  ( g `  <. 0 ,  0 >.
)  =  P ) )
7069rmobii 2910 . . 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 212 . 2  |-  ( ph  ->  E* g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) )
72 reu5 2934 . 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 659 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 184    /\ wa 369    = wceq 1364    e. wcel 1761   E.wrex 2714   E!wreu 2715   E*wrmo 2716   {crab 2717    C_ wss 3325   {csn 3874   <.cop 3880   U.cuni 4088   {copab 4346    e. cmpt 4347    X. cxp 4834    o. ccom 4840   ` cfv 5415   iota_crio 6048  (class class class)co 6090    e. cmpt2 6092   0cc0 9278   1c1 9279   [,]cicc 11299   neicnei 18660    Cn ccn 18787    CnP ccnp 18788   Conccon 18974  𝑛Locally cnlly 19028    tX ctx 19092   IIcii 20410   CovMap ccvm 27074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-ec 7099  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-cn 18790  df-cnp 18791  df-cmp 18949  df-con 18975  df-lly 19029  df-nlly 19030  df-tx 19094  df-hmeo 19287  df-xms 19854  df-ms 19855  df-tms 19856  df-ii 20412  df-htpy 20501  df-phtpy 20502  df-phtpc 20523  df-pcon 27040  df-scon 27041  df-cvm 27075
This theorem is referenced by:  cvmlift2  27135
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