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Theorem cvmlift2lem13 24963
Description: Lemma for cvmlift2 24964. (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 5695 . . . . . 6  |-  ( a  =  z  ->  (
( ( II  tX  II )  CnP  C ) `
 a )  =  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
98eleq2d 2479 . . . . 5  |-  ( a  =  z  ->  ( K  e.  ( (
( II  tX  II )  CnP  C ) `  a )  <->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) ) )
109cbvrabv 2923 . . . 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 3793 . . . . . . 7  |-  ( z  =  b  ->  { z }  =  { b } )
1211xpeq2d 4869 . . . . . 6  |-  ( z  =  b  ->  (
( 0 [,] 1
)  X.  { z } )  =  ( ( 0 [,] 1
)  X.  { b } ) )
1312sseq1d 3343 . . . . 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 2923 . . . 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 448 . . . . . . 7  |-  ( ( c  =  r  /\  d  =  t )  ->  d  =  t )
1615eleq1d 2478 . . . . . 6  |-  ( ( c  =  r  /\  d  =  t )  ->  ( d  e.  ( 0 [,] 1 )  <-> 
t  e.  ( 0 [,] 1 ) ) )
17 xpeq1 4859 . . . . . . . . . 10  |-  ( v  =  u  ->  (
v  X.  { b } )  =  ( u  X.  { b } ) )
1817sseq1d 3343 . . . . . . . . 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 4859 . . . . . . . . . 10  |-  ( v  =  u  ->  (
v  X.  { d } )  =  ( u  X.  { d } ) )
2019sseq1d 3343 . . . . . . . . 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 313 . . . . . . . 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 2901 . . . . . . 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 444 . . . . . . . . . 10  |-  ( ( c  =  r  /\  d  =  t )  ->  c  =  r )
2423sneqd 3795 . . . . . . . . 9  |-  ( ( c  =  r  /\  d  =  t )  ->  { c }  =  { r } )
2524fveq2d 5699 . . . . . . . 8  |-  ( ( c  =  r  /\  d  =  t )  ->  ( ( nei `  II ) `  { c } )  =  ( ( nei `  II ) `  { r } ) )
2615sneqd 3795 . . . . . . . . . . 11  |-  ( ( c  =  r  /\  d  =  t )  ->  { d }  =  { t } )
2726xpeq2d 4869 . . . . . . . . . 10  |-  ( ( c  =  r  /\  d  =  t )  ->  ( u  X.  {
d } )  =  ( u  X.  {
t } ) )
2827sseq1d 3343 . . . . . . . . 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 310 . . . . . . . 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 2885 . . . . . . 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 249 . . . . . 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 692 . . . . 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 4245 . . . 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 24962 . . 3  |-  ( ph  ->  K  e.  ( ( II  tX  II )  Cn  C ) )
351, 2, 3, 4, 5, 6, 7cvmlift2lem7 24957 . . 3  |-  ( ph  ->  ( F  o.  K
)  =  G )
36 0elunit 10979 . . . . 5  |-  0  e.  ( 0 [,] 1
)
371, 2, 3, 4, 5, 6, 7cvmlift2lem8 24958 . . . . 5  |-  ( (
ph  /\  0  e.  ( 0 [,] 1
) )  ->  (
0 K 0 )  =  ( H ` 
0 ) )
3836, 37mpan2 653 . . . 4  |-  ( ph  ->  ( 0 K 0 )  =  ( H `
 0 ) )
391, 2, 3, 4, 5, 6cvmlift2lem2 24952 . . . . 5  |-  ( ph  ->  ( H  e.  ( II  Cn  C )  /\  ( F  o.  H )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( H `
 0 )  =  P ) )
4039simp3d 971 . . . 4  |-  ( ph  ->  ( H `  0
)  =  P )
4138, 40eqtrd 2444 . . 3  |-  ( ph  ->  ( 0 K 0 )  =  P )
42 coeq2 4998 . . . . . 6  |-  ( g  =  K  ->  ( F  o.  g )  =  ( F  o.  K ) )
4342eqeq1d 2420 . . . . 5  |-  ( g  =  K  ->  (
( F  o.  g
)  =  G  <->  ( F  o.  K )  =  G ) )
44 oveq 6054 . . . . . 6  |-  ( g  =  K  ->  (
0 g 0 )  =  ( 0 K 0 ) )
4544eqeq1d 2420 . . . . 5  |-  ( g  =  K  ->  (
( 0 g 0 )  =  P  <->  ( 0 K 0 )  =  P ) )
4643, 45anbi12d 692 . . . 4  |-  ( g  =  K  ->  (
( ( F  o.  g )  =  G  /\  ( 0 g 0 )  =  P )  <->  ( ( F  o.  K )  =  G  /\  ( 0 K 0 )  =  P ) ) )
4746rspcev 3020 . . 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 1182 . 2  |-  ( ph  ->  E. g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) )
49 iitop 18871 . . . . 5  |-  II  e.  Top
50 iiuni 18872 . . . . 5  |-  ( 0 [,] 1 )  = 
U. II
5149, 49, 50, 50txunii 17586 . . . 4  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
52 iicon 18878 . . . . . 6  |-  II  e.  Con
53 txcon 17682 . . . . . 6  |-  ( ( II  e.  Con  /\  II  e.  Con )  -> 
( II  tX  II )  e.  Con )
5452, 52, 53mp2an 654 . . . . 5  |-  ( II 
tX  II )  e. 
Con
5554a1i 11 . . . 4  |-  ( ph  ->  ( II  tX  II )  e.  Con )
56 iinllycon 24902 . . . . . 6  |-  II  e. 𝑛Locally  Con
57 txcon 17682 . . . . . . 7  |-  ( ( x  e.  Con  /\  y  e.  Con )  ->  ( x  tX  y
)  e.  Con )
5857txnlly 17630 . . . . . 6  |-  ( ( II  e. 𝑛Locally  Con  /\  II  e. 𝑛Locally  Con )  ->  ( II  tX  II )  e. 𝑛Locally  Con )
5956, 56, 58mp2an 654 . . . . 5  |-  ( II 
tX  II )  e. 𝑛Locally  Con
6059a1i 11 . . . 4  |-  ( ph  ->  ( II  tX  II )  e. 𝑛Locally  Con )
61 opelxpi 4877 . . . . . 6  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  <. 0 ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
6236, 36, 61mp2an 654 . . . . 5  |-  <. 0 ,  0 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) )
6362a1i 11 . . . 4  |-  ( ph  -> 
<. 0 ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
64 df-ov 6051 . . . . 5  |-  ( 0 G 0 )  =  ( G `  <. 0 ,  0 >. )
655, 64syl6eq 2460 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( G `
 <. 0 ,  0
>. ) )
661, 51, 2, 55, 60, 63, 3, 4, 65cvmliftmo 24932 . . 3  |-  ( ph  ->  E* g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( g `  <. 0 ,  0 >. )  =  P ) )
67 df-ov 6051 . . . . . 6  |-  ( 0 g 0 )  =  ( g `  <. 0 ,  0 >. )
6867eqeq1i 2419 . . . . 5  |-  ( ( 0 g 0 )  =  P  <->  ( g `  <. 0 ,  0
>. )  =  P
)
6968anbi2i 676 . . . 4  |-  ( ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P )  <-> 
( ( F  o.  g )  =  G  /\  ( g `  <. 0 ,  0 >.
)  =  P ) )
7069rmobii 2867 . . 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 204 . 2  |-  ( ph  ->  E* g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) )
72 reu5 2889 . 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 646 1  |-  ( ph  ->  E! g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2675   E!wreu 2676   E*wrmo 2677   {crab 2678    C_ wss 3288   {csn 3782   <.cop 3785   U.cuni 3983   {copab 4233    e. cmpt 4234    X. cxp 4843    o. ccom 4849   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   iota_crio 6509   0cc0 8954   1c1 8955   [,]cicc 10883   neicnei 17124    Cn ccn 17250    CnP ccnp 17251   Conccon 17435  𝑛Locally cnlly 17489    tX ctx 17553   IIcii 18866   CovMap ccvm 24903
This theorem is referenced by:  cvmlift2  24964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-ec 6874  df-map 6987  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-sum 12443  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-cn 17253  df-cnp 17254  df-cmp 17412  df-con 17436  df-lly 17490  df-nlly 17491  df-tx 17555  df-hmeo 17748  df-xms 18311  df-ms 18312  df-tms 18313  df-ii 18868  df-htpy 18956  df-phtpy 18957  df-phtpc 18978  df-pcon 24869  df-scon 24870  df-cvm 24904
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