Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmlift2lem13 Structured version   Unicode version

Theorem cvmlift2lem13 28511
Description: Lemma for cvmlift2 28512. (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 5866 . . . . . 6  |-  ( a  =  z  ->  (
( ( II  tX  II )  CnP  C ) `
 a )  =  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
98eleq2d 2537 . . . . 5  |-  ( a  =  z  ->  ( K  e.  ( (
( II  tX  II )  CnP  C ) `  a )  <->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) ) )
109cbvrabv 3112 . . . 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 4037 . . . . . . 7  |-  ( z  =  b  ->  { z }  =  { b } )
1211xpeq2d 5023 . . . . . 6  |-  ( z  =  b  ->  (
( 0 [,] 1
)  X.  { z } )  =  ( ( 0 [,] 1
)  X.  { b } ) )
1312sseq1d 3531 . . . . 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 3112 . . . 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 461 . . . . . . 7  |-  ( ( c  =  r  /\  d  =  t )  ->  d  =  t )
1615eleq1d 2536 . . . . . 6  |-  ( ( c  =  r  /\  d  =  t )  ->  ( d  e.  ( 0 [,] 1 )  <-> 
t  e.  ( 0 [,] 1 ) ) )
17 xpeq1 5013 . . . . . . . . . 10  |-  ( v  =  u  ->  (
v  X.  { b } )  =  ( u  X.  { b } ) )
1817sseq1d 3531 . . . . . . . . 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 5013 . . . . . . . . . 10  |-  ( v  =  u  ->  (
v  X.  { d } )  =  ( u  X.  { d } ) )
2019sseq1d 3531 . . . . . . . . 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 3089 . . . . . . 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 457 . . . . . . . . . 10  |-  ( ( c  =  r  /\  d  =  t )  ->  c  =  r )
2423sneqd 4039 . . . . . . . . 9  |-  ( ( c  =  r  /\  d  =  t )  ->  { c }  =  { r } )
2524fveq2d 5870 . . . . . . . 8  |-  ( ( c  =  r  /\  d  =  t )  ->  ( ( nei `  II ) `  { c } )  =  ( ( nei `  II ) `  { r } ) )
2615sneqd 4039 . . . . . . . . . . 11  |-  ( ( c  =  r  /\  d  =  t )  ->  { d }  =  { t } )
2726xpeq2d 5023 . . . . . . . . . 10  |-  ( ( c  =  r  /\  d  =  t )  ->  ( u  X.  {
d } )  =  ( u  X.  {
t } ) )
2827sseq1d 3531 . . . . . . . . 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 3073 . . . . . . 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 710 . . . . 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 4516 . . . 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 28510 . . 3  |-  ( ph  ->  K  e.  ( ( II  tX  II )  Cn  C ) )
351, 2, 3, 4, 5, 6, 7cvmlift2lem7 28505 . . 3  |-  ( ph  ->  ( F  o.  K
)  =  G )
36 0elunit 11639 . . . . 5  |-  0  e.  ( 0 [,] 1
)
371, 2, 3, 4, 5, 6, 7cvmlift2lem8 28506 . . . . 5  |-  ( (
ph  /\  0  e.  ( 0 [,] 1
) )  ->  (
0 K 0 )  =  ( H ` 
0 ) )
3836, 37mpan2 671 . . . 4  |-  ( ph  ->  ( 0 K 0 )  =  ( H `
 0 ) )
391, 2, 3, 4, 5, 6cvmlift2lem2 28500 . . . . 5  |-  ( ph  ->  ( H  e.  ( II  Cn  C )  /\  ( F  o.  H )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( H `
 0 )  =  P ) )
4039simp3d 1010 . . . 4  |-  ( ph  ->  ( H `  0
)  =  P )
4138, 40eqtrd 2508 . . 3  |-  ( ph  ->  ( 0 K 0 )  =  P )
42 coeq2 5161 . . . . . 6  |-  ( g  =  K  ->  ( F  o.  g )  =  ( F  o.  K ) )
4342eqeq1d 2469 . . . . 5  |-  ( g  =  K  ->  (
( F  o.  g
)  =  G  <->  ( F  o.  K )  =  G ) )
44 oveq 6291 . . . . . 6  |-  ( g  =  K  ->  (
0 g 0 )  =  ( 0 K 0 ) )
4544eqeq1d 2469 . . . . 5  |-  ( g  =  K  ->  (
( 0 g 0 )  =  P  <->  ( 0 K 0 )  =  P ) )
4643, 45anbi12d 710 . . . 4  |-  ( g  =  K  ->  (
( ( F  o.  g )  =  G  /\  ( 0 g 0 )  =  P )  <->  ( ( F  o.  K )  =  G  /\  ( 0 K 0 )  =  P ) ) )
4746rspcev 3214 . . 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 1226 . 2  |-  ( ph  ->  E. g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P ) )
49 iitop 21211 . . . . 5  |-  II  e.  Top
50 iiuni 21212 . . . . 5  |-  ( 0 [,] 1 )  = 
U. II
5149, 49, 50, 50txunii 19921 . . . 4  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
52 iicon 21218 . . . . . 6  |-  II  e.  Con
53 txcon 20017 . . . . . 6  |-  ( ( II  e.  Con  /\  II  e.  Con )  -> 
( II  tX  II )  e.  Con )
5452, 52, 53mp2an 672 . . . . 5  |-  ( II 
tX  II )  e. 
Con
5554a1i 11 . . . 4  |-  ( ph  ->  ( II  tX  II )  e.  Con )
56 iinllycon 28450 . . . . . 6  |-  II  e. 𝑛Locally  Con
57 txcon 20017 . . . . . . 7  |-  ( ( x  e.  Con  /\  y  e.  Con )  ->  ( x  tX  y
)  e.  Con )
5857txnlly 19965 . . . . . 6  |-  ( ( II  e. 𝑛Locally  Con  /\  II  e. 𝑛Locally  Con )  ->  ( II  tX  II )  e. 𝑛Locally  Con )
5956, 56, 58mp2an 672 . . . . 5  |-  ( II 
tX  II )  e. 𝑛Locally  Con
6059a1i 11 . . . 4  |-  ( ph  ->  ( II  tX  II )  e. 𝑛Locally  Con )
61 opelxpi 5031 . . . . . 6  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  <. 0 ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
6236, 36, 61mp2an 672 . . . . 5  |-  <. 0 ,  0 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) )
6362a1i 11 . . . 4  |-  ( ph  -> 
<. 0 ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
64 df-ov 6288 . . . . 5  |-  ( 0 G 0 )  =  ( G `  <. 0 ,  0 >. )
655, 64syl6eq 2524 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( G `
 <. 0 ,  0
>. ) )
661, 51, 2, 55, 60, 63, 3, 4, 65cvmliftmo 28480 . . 3  |-  ( ph  ->  E* g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g
)  =  G  /\  ( g `  <. 0 ,  0 >. )  =  P ) )
67 df-ov 6288 . . . . . 6  |-  ( 0 g 0 )  =  ( g `  <. 0 ,  0 >. )
6867eqeq1i 2474 . . . . 5  |-  ( ( 0 g 0 )  =  P  <->  ( g `  <. 0 ,  0
>. )  =  P
)
6968anbi2i 694 . . . 4  |-  ( ( ( F  o.  g
)  =  G  /\  ( 0 g 0 )  =  P )  <-> 
( ( F  o.  g )  =  G  /\  ( g `  <. 0 ,  0 >.
)  =  P ) )
7069rmobii 3053 . . 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 3077 . 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 664 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 1379    e. wcel 1767   E.wrex 2815   E!wreu 2816   E*wrmo 2817   {crab 2818    C_ wss 3476   {csn 4027   <.cop 4033   U.cuni 4245   {copab 4504    |-> cmpt 4505    X. cxp 4997    o. ccom 5003   ` cfv 5588   iota_crio 6245  (class class class)co 6285    |-> cmpt2 6287   0cc0 9493   1c1 9494   [,]cicc 11533   neicnei 19404    Cn ccn 19531    CnP ccnp 19532   Conccon 19718  𝑛Locally cnlly 19772    tX ctx 19888   IIcii 21206   CovMap ccvm 28451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571  ax-addf 9572  ax-mulf 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-ec 7314  df-map 7423  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-fi 7872  df-sup 7902  df-oi 7936  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-q 11184  df-rp 11222  df-xneg 11319  df-xadd 11320  df-xmul 11321  df-ioo 11534  df-ico 11536  df-icc 11537  df-fz 11674  df-fzo 11794  df-fl 11898  df-seq 12077  df-exp 12136  df-hash 12375  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-clim 13277  df-sum 13475  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-starv 14573  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-unif 14581  df-hom 14582  df-cco 14583  df-rest 14681  df-topn 14682  df-0g 14700  df-gsum 14701  df-topgen 14702  df-pt 14703  df-prds 14706  df-xrs 14760  df-qtop 14765  df-imas 14766  df-xps 14768  df-mre 14844  df-mrc 14845  df-acs 14847  df-mnd 15735  df-submnd 15790  df-mulg 15874  df-cntz 16169  df-cmn 16615  df-psmet 18222  df-xmet 18223  df-met 18224  df-bl 18225  df-mopn 18226  df-cnfld 18232  df-top 19206  df-bases 19208  df-topon 19209  df-topsp 19210  df-cld 19326  df-ntr 19327  df-cls 19328  df-nei 19405  df-cn 19534  df-cnp 19535  df-cmp 19693  df-con 19719  df-lly 19773  df-nlly 19774  df-tx 19890  df-hmeo 20083  df-xms 20650  df-ms 20651  df-tms 20652  df-ii 21208  df-htpy 21297  df-phtpy 21298  df-phtpc 21319  df-pcon 28417  df-scon 28418  df-cvm 28452
This theorem is referenced by:  cvmlift2  28512
  Copyright terms: Public domain W3C validator