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Theorem cvmlift2lem13 23846
Description: Lemma for cvmlift2 23847. (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 5525 . . . . . 6  |-  ( a  =  z  ->  (
( ( II  tX  II )  CnP  C ) `
 a )  =  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
98eleq2d 2350 . . . . 5  |-  ( a  =  z  ->  ( K  e.  ( (
( II  tX  II )  CnP  C ) `  a )  <->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) ) )
109cbvrabv 2787 . . . 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 3651 . . . . . . 7  |-  ( z  =  b  ->  { z }  =  { b } )
1211xpeq2d 4713 . . . . . 6  |-  ( z  =  b  ->  (
( 0 [,] 1
)  X.  { z } )  =  ( ( 0 [,] 1
)  X.  { b } ) )
1312sseq1d 3205 . . . . 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 2787 . . . 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 447 . . . . . . 7  |-  ( ( c  =  r  /\  d  =  t )  ->  d  =  t )
1615eleq1d 2349 . . . . . 6  |-  ( ( c  =  r  /\  d  =  t )  ->  ( d  e.  ( 0 [,] 1 )  <-> 
t  e.  ( 0 [,] 1 ) ) )
17 xpeq1 4703 . . . . . . . . . 10  |-  ( v  =  u  ->  (
v  X.  { b } )  =  ( u  X.  { b } ) )
1817sseq1d 3205 . . . . . . . . 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 4703 . . . . . . . . . 10  |-  ( v  =  u  ->  (
v  X.  { d } )  =  ( u  X.  { d } ) )
2019sseq1d 3205 . . . . . . . . 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 312 . . . . . . . 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 2765 . . . . . . 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 443 . . . . . . . . . 10  |-  ( ( c  =  r  /\  d  =  t )  ->  c  =  r )
2423sneqd 3653 . . . . . . . . 9  |-  ( ( c  =  r  /\  d  =  t )  ->  { c }  =  { r } )
2524fveq2d 5529 . . . . . . . 8  |-  ( ( c  =  r  /\  d  =  t )  ->  ( ( nei `  II ) `  { c } )  =  ( ( nei `  II ) `  { r } ) )
2615sneqd 3653 . . . . . . . . . . 11  |-  ( ( c  =  r  /\  d  =  t )  ->  { d }  =  { t } )
2726xpeq2d 4713 . . . . . . . . . 10  |-  ( ( c  =  r  /\  d  =  t )  ->  ( u  X.  {
d } )  =  ( u  X.  {
t } ) )
2827sseq1d 3205 . . . . . . . . 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 309 . . . . . . . 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 2749 . . . . . . 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 248 . . . . . 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 691 . . . . 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 4088 . . . 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 23845 . . 3  |-  ( ph  ->  K  e.  ( ( II  tX  II )  Cn  C ) )
351, 2, 3, 4, 5, 6, 7cvmlift2lem7 23840 . . 3  |-  ( ph  ->  ( F  o.  K
)  =  G )
36 0elunit 10754 . . . . 5  |-  0  e.  ( 0 [,] 1
)
371, 2, 3, 4, 5, 6, 7cvmlift2lem8 23841 . . . . 5  |-  ( (
ph  /\  0  e.  ( 0 [,] 1
) )  ->  (
0 K 0 )  =  ( H ` 
0 ) )
3836, 37mpan2 652 . . . 4  |-  ( ph  ->  ( 0 K 0 )  =  ( H `
 0 ) )
391, 2, 3, 4, 5, 6cvmlift2lem2 23835 . . . . 5  |-  ( ph  ->  ( H  e.  ( II  Cn  C )  /\  ( F  o.  H )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( H `
 0 )  =  P ) )
4039simp3d 969 . . . 4  |-  ( ph  ->  ( H `  0
)  =  P )
4138, 40eqtrd 2315 . . 3  |-  ( ph  ->  ( 0 K 0 )  =  P )
42 coeq2 4842 . . . . . 6  |-  ( g  =  K  ->  ( F  o.  g )  =  ( F  o.  K ) )
4342eqeq1d 2291 . . . . 5  |-  ( g  =  K  ->  (
( F  o.  g
)  =  G  <->  ( F  o.  K )  =  G ) )
44 oveq 5864 . . . . . 6  |-  ( g  =  K  ->  (
0 g 0 )  =  ( 0 K 0 ) )
4544eqeq1d 2291 . . . . 5  |-  ( g  =  K  ->  (
( 0 g 0 )  =  P  <->  ( 0 K 0 )  =  P ) )
4643, 45anbi12d 691 . . . 4  |-  ( g  =  K  ->  (
( ( F  o.  g )  =  G  /\  ( 0 g 0 )  =  P )  <->  ( ( F  o.  K )  =  G  /\  ( 0 K 0 )  =  P ) ) )
4746rspcev 2884 . . 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 1180 . 2  |-  ( ph  ->  E. g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) )
49 iitop 18384 . . . . 5  |-  II  e.  Top
50 iiuni 18385 . . . . 5  |-  ( 0 [,] 1 )  = 
U. II
5149, 49, 50, 50txunii 17288 . . . 4  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
52 iicon 18391 . . . . . 6  |-  II  e.  Con
53 txcon 17383 . . . . . 6  |-  ( ( II  e.  Con  /\  II  e.  Con )  -> 
( II  tX  II )  e.  Con )
5452, 52, 53mp2an 653 . . . . 5  |-  ( II 
tX  II )  e. 
Con
5554a1i 10 . . . 4  |-  ( ph  ->  ( II  tX  II )  e.  Con )
56 iinllycon 23785 . . . . . 6  |-  II  e. 𝑛Locally  Con
57 txcon 17383 . . . . . . 7  |-  ( ( x  e.  Con  /\  y  e.  Con )  ->  ( x  tX  y
)  e.  Con )
5857txnlly 17331 . . . . . 6  |-  ( ( II  e. 𝑛Locally  Con  /\  II  e. 𝑛Locally  Con )  ->  ( II  tX  II )  e. 𝑛Locally  Con )
5956, 56, 58mp2an 653 . . . . 5  |-  ( II 
tX  II )  e. 𝑛Locally  Con
6059a1i 10 . . . 4  |-  ( ph  ->  ( II  tX  II )  e. 𝑛Locally  Con )
61 opelxpi 4721 . . . . . 6  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  <. 0 ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
6236, 36, 61mp2an 653 . . . . 5  |-  <. 0 ,  0 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) )
6362a1i 10 . . . 4  |-  ( ph  -> 
<. 0 ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
64 df-ov 5861 . . . . 5  |-  ( 0 G 0 )  =  ( G `  <. 0 ,  0 >. )
655, 64syl6eq 2331 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( G `
 <. 0 ,  0
>. ) )
661, 51, 2, 55, 60, 63, 3, 4, 65cvmliftmo 23815 . . 3  |-  ( ph  ->  E* g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( g `  <. 0 ,  0 >. )  =  P ) )
67 df-ov 5861 . . . . . 6  |-  ( 0 g 0 )  =  ( g `  <. 0 ,  0 >. )
6867eqeq1i 2290 . . . . 5  |-  ( ( 0 g 0 )  =  P  <->  ( g `  <. 0 ,  0
>. )  =  P
)
6968anbi2i 675 . . . 4  |-  ( ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P )  <-> 
( ( F  o.  g )  =  G  /\  ( g `  <. 0 ,  0 >.
)  =  P ) )
7069rmobii 2731 . . 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 203 . 2  |-  ( ph  ->  E* g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) )
72 reu5 2753 . 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 645 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   E!wreu 2545   E*wrmo 2546   {crab 2547    C_ wss 3152   {csn 3640   <.cop 3643   U.cuni 3827   {copab 4076    e. cmpt 4077    X. cxp 4687    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   iota_crio 6297   0cc0 8737   1c1 8738   [,]cicc 10659   neicnei 16834    Cn ccn 16954    CnP ccnp 16955   Conccon 17137  𝑛Locally cnlly 17191    tX ctx 17255   IIcii 18379   CovMap ccvm 23786
This theorem is referenced by:  cvmlift2  23847
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-cn 16957  df-cnp 16958  df-cmp 17114  df-con 17138  df-lly 17192  df-nlly 17193  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-ii 18381  df-htpy 18468  df-phtpy 18469  df-phtpc 18490  df-pcon 23752  df-scon 23753  df-cvm 23787
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