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Theorem txlly 19184
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 19051 . . 3  |-  ( R  e. Locally  A  ->  R  e. 
Top )
2 llytop 19051 . . 3  |-  ( S  e. Locally  A  ->  S  e. 
Top )
3 txtop 19117 . . 3  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( R  tX  S
)  e.  Top )
41, 2, 3syl2an 477 . 2  |-  ( ( R  e. Locally  A  /\  S  e. Locally  A )  -> 
( R  tX  S
)  e.  Top )
5 eltx 19116 . . . 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 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 )
) )  ->  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 6601 . . . . . . . . . 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 19053 . . . . . . . . 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 1218 . . . . . . . 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 754 . . . . . . . . 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 6602 . . . . . . . . . 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 19053 . . . . . . . . 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 1218 . . . . . . . 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 2883 . . . . . . . . 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 729 . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . 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 19150 . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . 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 1033 . . . . . . . . . . . . . . . 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 1036 . . . . . . . . . . . . . . . 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 4940 . . . . . . . . . . . . . . . 16  |-  ( ( r  C_  u  /\  s  C_  v )  -> 
( r  X.  s
)  C_  ( u  X.  v ) )
2926, 27, 28syl2anc 661 . . . . . . . . . . . . . . 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 3365 . . . . . . . . . . . . . 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 2970 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
3332elpw2 4451 . . . . . . . . . . . . . 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 3535 . . . . . . . . . . . 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 6608 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 1034 . . . . . . . . . . . . . . 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 1037 . . . . . . . . . . . . . . 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 4866 . . . . . . . . . . . . . . 15  |-  ( ( ( 1st `  y
)  e.  r  /\  ( 2nd `  y )  e.  s )  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  ( r  X.  s ) )
4239, 40, 41syl2anc 661 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . 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 2512 . . . . . . . . . . . 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 19179 . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . 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 1035 . . . . . . . . . . . . . . 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 1038 . . . . . . . . . . . . . . 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 6252 . . . . . . . . . . . . . . 15  |-  ( ( ( Rt  r )  e.  A  /\  ( St  s )  e.  A )  ->  ( ( Rt  r )  tX  ( St  s ) )  e.  A
)
5147, 48, 50syl2anc 661 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . 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 2512 . . . . . . . . . . . 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 2499 . . . . . . . . . . . . . 14  |-  ( z  =  ( r  X.  s )  ->  (
y  e.  z  <->  y  e.  ( r  X.  s
) ) )
55 oveq2 6094 . . . . . . . . . . . . . . 15  |-  ( z  =  ( r  X.  s )  ->  (
( R  tX  S
)t  z )  =  ( ( R  tX  S
)t  ( r  X.  s
) ) )
5655eleq1d 2504 . . . . . . . . . . . . . 14  |-  ( z  =  ( r  X.  s )  ->  (
( ( R  tX  S )t  z )  e.  A  <->  ( ( R 
tX  S )t  ( r  X.  s ) )  e.  A ) )
5754, 56anbi12d 710 . . . . . . . . . . . . 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 3068 . . . . . . . . . . . 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 1216 . . . . . . . . . . 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 615 . . . . . . . . . 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 2843 . . . . . . . . 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 679 . . . . . . 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 615 . . . . . 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 2843 . . . . 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 2790 . . . 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 2793 . 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 19047 . 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 664 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   E.wrex 2711    i^i cin 3322    C_ wss 3323   ~Pcpw 3855   <.cop 3878    X. cxp 4833   ` cfv 5413  (class class class)co 6086   1stc1st 6570   2ndc2nd 6571   ↾t crest 14351   Topctop 18473  Locally clly 19043    tX ctx 19108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-rest 14353  df-topgen 14374  df-top 18478  df-bases 18480  df-lly 19045  df-tx 19110
This theorem is referenced by: (None)
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