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Theorem txlly 20245
Description: If the property  A is preserved under topological products, then so is the property of being locally  A. (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypothesis
Ref Expression
txlly.1  |-  ( ( j  e.  A  /\  k  e.  A )  ->  ( j  tX  k
)  e.  A )
Assertion
Ref Expression
txlly  |-  ( ( R  e. Locally  A  /\  S  e. Locally  A )  -> 
( R  tX  S
)  e. Locally  A )
Distinct variable groups:    j, k, A    R, j, k    S, k
Allowed substitution hint:    S( j)

Proof of Theorem txlly
Dummy variables  r 
s  u  v  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 20081 . . 3  |-  ( R  e. Locally  A  ->  R  e. 
Top )
2 llytop 20081 . . 3  |-  ( S  e. Locally  A  ->  S  e. 
Top )
3 txtop 20178 . . 3  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( R  tX  S
)  e.  Top )
41, 2, 3syl2an 475 . 2  |-  ( ( R  e. Locally  A  /\  S  e. Locally  A )  -> 
( R  tX  S
)  e.  Top )
5 eltx 20177 . . . 4  |-  ( ( R  e. Locally  A  /\  S  e. Locally  A )  -> 
( x  e.  ( R  tX  S )  <->  A. y  e.  x  E. u  e.  R  E. v  e.  S  ( y  e.  ( u  X.  v )  /\  ( u  X.  v )  C_  x
) ) )
6 simpll 751 . . . . . . . . 9  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  R  e. Locally  A )
7 simprll 761 . . . . . . . . 9  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  u  e.  R )
8 simprrl 763 . . . . . . . . . 10  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  y  e.  ( u  X.  v
) )
9 xp1st 6751 . . . . . . . . . 10  |-  ( y  e.  ( u  X.  v )  ->  ( 1st `  y )  e.  u )
108, 9syl 16 . . . . . . . . 9  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  ( 1st `  y )  e.  u )
11 llyi 20083 . . . . . . . . 9  |-  ( ( R  e. Locally  A  /\  u  e.  R  /\  ( 1st `  y )  e.  u )  ->  E. r  e.  R  ( r  C_  u  /\  ( 1st `  y
)  e.  r  /\  ( Rt  r )  e.  A ) )
126, 7, 10, 11syl3anc 1226 . . . . . . . 8  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  E. r  e.  R  ( r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A ) )
13 simplr 753 . . . . . . . . 9  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  S  e. Locally  A )
14 simprlr 762 . . . . . . . . 9  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  v  e.  S )
15 xp2nd 6752 . . . . . . . . . 10  |-  ( y  e.  ( u  X.  v )  ->  ( 2nd `  y )  e.  v )
168, 15syl 16 . . . . . . . . 9  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  ( 2nd `  y )  e.  v )
17 llyi 20083 . . . . . . . . 9  |-  ( ( S  e. Locally  A  /\  v  e.  S  /\  ( 2nd `  y )  e.  v )  ->  E. s  e.  S  ( s  C_  v  /\  ( 2nd `  y
)  e.  s  /\  ( St  s )  e.  A ) )
1813, 14, 16, 17syl3anc 1226 . . . . . . . 8  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  E. s  e.  S  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) )
19 reeanv 2967 . . . . . . . . 9  |-  ( E. r  e.  R  E. s  e.  S  (
( r  C_  u  /\  ( 1st `  y
)  e.  r  /\  ( Rt  r )  e.  A )  /\  (
s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A
) )  <->  ( E. r  e.  R  (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  E. s  e.  S  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) )
201ad3antrrr 727 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  R  e.  Top )
212ad3antlr 728 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  S  e.  Top )
22 simprll 761 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  r  e.  R )
23 simprlr 762 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  s  e.  S )
24 txopn 20211 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( r  e.  R  /\  s  e.  S
) )  ->  (
r  X.  s )  e.  ( R  tX  S ) )
2520, 21, 22, 23, 24syl22anc 1227 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  (
r  X.  s )  e.  ( R  tX  S ) )
26 simprl1 1039 . . . . . . . . . . . . . . . 16  |-  ( ( ( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) )  ->  r  C_  u )
27 simprr1 1042 . . . . . . . . . . . . . . . 16  |-  ( ( ( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) )  ->  s  C_  v )
28 xpss12 5038 . . . . . . . . . . . . . . . 16  |-  ( ( r  C_  u  /\  s  C_  v )  -> 
( r  X.  s
)  C_  ( u  X.  v ) )
2926, 27, 28syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ( ( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) )  ->  ( r  X.  s )  C_  (
u  X.  v ) )
30 simprrr 764 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  (
u  X.  v ) 
C_  x )
3129, 30sylan9ssr 3448 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  (
r  X.  s ) 
C_  x )
32 vex 3054 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
3332elpw2 4546 . . . . . . . . . . . . . 14  |-  ( ( r  X.  s )  e.  ~P x  <->  ( r  X.  s )  C_  x
)
3431, 33sylibr 212 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  (
r  X.  s )  e.  ~P x )
3525, 34elind 3619 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  (
r  X.  s )  e.  ( ( R 
tX  S )  i^i 
~P x ) )
36 1st2nd2 6758 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( u  X.  v )  ->  y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
378, 36syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
3837adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
39 simprl2 1040 . . . . . . . . . . . . . . 15  |-  ( ( ( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) )  ->  ( 1st `  y )  e.  r )
40 simprr2 1043 . . . . . . . . . . . . . . 15  |-  ( ( ( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) )  ->  ( 2nd `  y )  e.  s )
41 opelxpi 4962 . . . . . . . . . . . . . . 15  |-  ( ( ( 1st `  y
)  e.  r  /\  ( 2nd `  y )  e.  s )  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  ( r  X.  s ) )
4239, 40, 41syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ( ( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) )  ->  <. ( 1st `  y ) ,  ( 2nd `  y )
>.  e.  ( r  X.  s ) )
4342adantl 464 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  e.  ( r  X.  s ) )
4438, 43eqeltrd 2484 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  y  e.  ( r  X.  s
) )
45 txrest 20240 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( r  e.  R  /\  s  e.  S
) )  ->  (
( R  tX  S
)t  ( r  X.  s
) )  =  ( ( Rt  r )  tX  ( St  s ) ) )
4620, 21, 22, 23, 45syl22anc 1227 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  (
( R  tX  S
)t  ( r  X.  s
) )  =  ( ( Rt  r )  tX  ( St  s ) ) )
47 simprl3 1041 . . . . . . . . . . . . . . 15  |-  ( ( ( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) )  ->  ( Rt  r
)  e.  A )
48 simprr3 1044 . . . . . . . . . . . . . . 15  |-  ( ( ( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) )  ->  ( St  s
)  e.  A )
49 txlly.1 . . . . . . . . . . . . . . . 16  |-  ( ( j  e.  A  /\  k  e.  A )  ->  ( j  tX  k
)  e.  A )
5049caovcl 6390 . . . . . . . . . . . . . . 15  |-  ( ( ( Rt  r )  e.  A  /\  ( St  s )  e.  A )  ->  ( ( Rt  r )  tX  ( St  s ) )  e.  A
)
5147, 48, 50syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ( ( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) )  ->  ( ( Rt  r )  tX  ( St  s ) )  e.  A )
5251adantl 464 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  (
( Rt  r )  tX  ( St  s ) )  e.  A )
5346, 52eqeltrd 2484 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  (
( R  tX  S
)t  ( r  X.  s
) )  e.  A
)
54 eleq2 2469 . . . . . . . . . . . . . 14  |-  ( z  =  ( r  X.  s )  ->  (
y  e.  z  <->  y  e.  ( r  X.  s
) ) )
55 oveq2 6226 . . . . . . . . . . . . . . 15  |-  ( z  =  ( r  X.  s )  ->  (
( R  tX  S
)t  z )  =  ( ( R  tX  S
)t  ( r  X.  s
) ) )
5655eleq1d 2465 . . . . . . . . . . . . . 14  |-  ( z  =  ( r  X.  s )  ->  (
( ( R  tX  S )t  z )  e.  A  <->  ( ( R 
tX  S )t  ( r  X.  s ) )  e.  A ) )
5754, 56anbi12d 708 . . . . . . . . . . . . 13  |-  ( z  =  ( r  X.  s )  ->  (
( y  e.  z  /\  ( ( R 
tX  S )t  z )  e.  A )  <->  ( y  e.  ( r  X.  s
)  /\  ( ( R  tX  S )t  ( r  X.  s ) )  e.  A ) ) )
5857rspcev 3152 . . . . . . . . . . . 12  |-  ( ( ( r  X.  s
)  e.  ( ( R  tX  S )  i^i  ~P x )  /\  ( y  e.  ( r  X.  s
)  /\  ( ( R  tX  S )t  ( r  X.  s ) )  e.  A ) )  ->  E. z  e.  ( ( R  tX  S
)  i^i  ~P x
) ( y  e.  z  /\  ( ( R  tX  S )t  z )  e.  A ) )
5935, 44, 53, 58syl12anc 1224 . . . . . . . . . . 11  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
( r  e.  R  /\  s  e.  S
)  /\  ( (
r  C_  u  /\  ( 1st `  y )  e.  r  /\  ( Rt  r )  e.  A
)  /\  ( s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A ) ) ) )  ->  E. z  e.  ( ( R  tX  S )  i^i  ~P x ) ( y  e.  z  /\  (
( R  tX  S
)t  z )  e.  A
) )
6059expr 613 . . . . . . . . . 10  |-  ( ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  /\  (
r  e.  R  /\  s  e.  S )
)  ->  ( (
( r  C_  u  /\  ( 1st `  y
)  e.  r  /\  ( Rt  r )  e.  A )  /\  (
s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A
) )  ->  E. z  e.  ( ( R  tX  S )  i^i  ~P x ) ( y  e.  z  /\  (
( R  tX  S
)t  z )  e.  A
) ) )
6160rexlimdvva 2895 . . . . . . . . 9  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  ( E. r  e.  R  E. s  e.  S  ( ( r  C_  u  /\  ( 1st `  y
)  e.  r  /\  ( Rt  r )  e.  A )  /\  (
s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A
) )  ->  E. z  e.  ( ( R  tX  S )  i^i  ~P x ) ( y  e.  z  /\  (
( R  tX  S
)t  z )  e.  A
) ) )
6219, 61syl5bir 218 . . . . . . . 8  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  (
( E. r  e.  R  ( r  C_  u  /\  ( 1st `  y
)  e.  r  /\  ( Rt  r )  e.  A )  /\  E. s  e.  S  (
s  C_  v  /\  ( 2nd `  y )  e.  s  /\  ( St  s )  e.  A
) )  ->  E. z  e.  ( ( R  tX  S )  i^i  ~P x ) ( y  e.  z  /\  (
( R  tX  S
)t  z )  e.  A
) ) )
6312, 18, 62mp2and 677 . . . . . . 7  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( ( u  e.  R  /\  v  e.  S )  /\  (
y  e.  ( u  X.  v )  /\  ( u  X.  v
)  C_  x )
) )  ->  E. z  e.  ( ( R  tX  S )  i^i  ~P x ) ( y  e.  z  /\  (
( R  tX  S
)t  z )  e.  A
) )
6463expr 613 . . . . . 6  |-  ( ( ( R  e. Locally  A  /\  S  e. Locally  A )  /\  ( u  e.  R  /\  v  e.  S
) )  ->  (
( y  e.  ( u  X.  v )  /\  ( u  X.  v )  C_  x
)  ->  E. z  e.  ( ( R  tX  S )  i^i  ~P x ) ( y  e.  z  /\  (
( R  tX  S
)t  z )  e.  A
) ) )
6564rexlimdvva 2895 . . . . 5  |-  ( ( R  e. Locally  A  /\  S  e. Locally  A )  -> 
( E. u  e.  R  E. v  e.  S  ( y  e.  ( u  X.  v
)  /\  ( u  X.  v )  C_  x
)  ->  E. z  e.  ( ( R  tX  S )  i^i  ~P x ) ( y  e.  z  /\  (
( R  tX  S
)t  z )  e.  A
) ) )
6665ralimdv 2806 . . . 4  |-  ( ( R  e. Locally  A  /\  S  e. Locally  A )  -> 
( A. y  e.  x  E. u  e.  R  E. v  e.  S  ( y  e.  ( u  X.  v
)  /\  ( u  X.  v )  C_  x
)  ->  A. y  e.  x  E. z  e.  ( ( R  tX  S )  i^i  ~P x ) ( y  e.  z  /\  (
( R  tX  S
)t  z )  e.  A
) ) )
675, 66sylbid 215 . . 3  |-  ( ( R  e. Locally  A  /\  S  e. Locally  A )  -> 
( x  e.  ( R  tX  S )  ->  A. y  e.  x  E. z  e.  (
( R  tX  S
)  i^i  ~P x
) ( y  e.  z  /\  ( ( R  tX  S )t  z )  e.  A ) ) )
6867ralrimiv 2808 . 2  |-  ( ( R  e. Locally  A  /\  S  e. Locally  A )  ->  A. x  e.  ( R  tX  S ) A. y  e.  x  E. z  e.  ( ( R  tX  S )  i^i 
~P x ) ( y  e.  z  /\  ( ( R  tX  S )t  z )  e.  A ) )
69 islly 20077 . 2  |-  ( ( R  tX  S )  e. Locally  A  <->  ( ( R 
tX  S )  e. 
Top  /\  A. x  e.  ( R  tX  S
) A. y  e.  x  E. z  e.  ( ( R  tX  S )  i^i  ~P x ) ( y  e.  z  /\  (
( R  tX  S
)t  z )  e.  A
) ) )
704, 68, 69sylanbrc 662 1  |-  ( ( R  e. Locally  A  /\  S  e. Locally  A )  -> 
( R  tX  S
)  e. Locally  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   A.wral 2746   E.wrex 2747    i^i cin 3405    C_ wss 3406   ~Pcpw 3944   <.cop 3967    X. cxp 4928   ` cfv 5513  (class class class)co 6218   1stc1st 6719   2ndc2nd 6720   ↾t crest 14851   Topctop 19502  Locally clly 20073    tX ctx 20169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-1st 6721  df-2nd 6722  df-rest 14853  df-topgen 14874  df-top 19507  df-bases 19509  df-lly 20075  df-tx 20171
This theorem is referenced by: (None)
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