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Theorem iccllyscon 28568
Description: A closed interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
Assertion
Ref Expression
iccllyscon  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e. Locally SCon )

Proof of Theorem iccllyscon
Dummy variables  a 
b  u  v  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 756 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  x  e.  ( topGen `  ran  (,) )
)
2 inss1 3703 . . . . . 6  |-  ( x  i^i  ( A [,] B ) )  C_  x
3 simprr 757 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  y  e.  ( x  i^i  ( A [,] B ) ) )
42, 3sseldi 3487 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  y  e.  x )
5 tg2 19339 . . . . 5  |-  ( ( x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  x )  ->  E. z  e.  ran  (,) ( y  e.  z  /\  z  C_  x ) )
61, 4, 5syl2anc 661 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  E. z  e.  ran  (,) ( y  e.  z  /\  z  C_  x ) )
7 ioof 11631 . . . . . . . 8  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
8 ffn 5721 . . . . . . . 8  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
9 ovelrn 6436 . . . . . . . 8  |-  ( (,) 
Fn  ( RR*  X.  RR* )  ->  ( z  e. 
ran  (,)  <->  E. a  e.  RR*  E. b  e.  RR*  z  =  ( a (,) b ) ) )
107, 8, 9mp2b 10 . . . . . . 7  |-  ( z  e.  ran  (,)  <->  E. a  e.  RR*  E. b  e. 
RR*  z  =  ( a (,) b ) )
11 inss1 3703 . . . . . . . . . . . 12  |-  ( z  i^i  ( A [,] B ) )  C_  z
12 simprrr 766 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  -> 
z  C_  x )
1311, 12syl5ss 3500 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  -> 
( z  i^i  ( A [,] B ) ) 
C_  x )
14 simprrl 765 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  -> 
y  e.  z )
15 simprl 756 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  -> 
z  =  ( a (,) b ) )
1615ineq1d 3684 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  -> 
( z  i^i  ( A [,] B ) )  =  ( ( a (,) b )  i^i  ( A [,] B
) ) )
1716oveq2d 6297 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  -> 
( ( topGen `  ran  (,) )t  ( z  i^i  ( A [,] B ) ) )  =  ( (
topGen `  ran  (,) )t  (
( a (,) b
)  i^i  ( A [,] B ) ) ) )
18 iooscon 28565 . . . . . . . . . . . . . . . 16  |-  ( (
topGen `  ran  (,) )t  (
a (,) b ) )  e. SCon
19 ioossre 11595 . . . . . . . . . . . . . . . . 17  |-  ( a (,) b )  C_  RR
20 eqid 2443 . . . . . . . . . . . . . . . . . . 19  |-  ( (
topGen `  ran  (,) )t  (
a (,) b ) )  =  ( (
topGen `  ran  (,) )t  (
a (,) b ) )
2120rescon 28564 . . . . . . . . . . . . . . . . . 18  |-  ( ( a (,) b ) 
C_  RR  ->  ( ( ( topGen `  ran  (,) )t  (
a (,) b ) )  e. SCon  <->  ( ( topGen `
 ran  (,) )t  (
a (,) b ) )  e.  Con )
)
22 reconn 21206 . . . . . . . . . . . . . . . . . 18  |-  ( ( a (,) b ) 
C_  RR  ->  ( ( ( topGen `  ran  (,) )t  (
a (,) b ) )  e.  Con  <->  A. u  e.  ( a (,) b
) A. v  e.  ( a (,) b
) ( u [,] v )  C_  (
a (,) b ) ) )
2321, 22bitrd 253 . . . . . . . . . . . . . . . . 17  |-  ( ( a (,) b ) 
C_  RR  ->  ( ( ( topGen `  ran  (,) )t  (
a (,) b ) )  e. SCon  <->  A. u  e.  ( a (,) b
) A. v  e.  ( a (,) b
) ( u [,] v )  C_  (
a (,) b ) ) )
2419, 23ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( ( ( topGen `  ran  (,) )t  (
a (,) b ) )  e. SCon  <->  A. u  e.  ( a (,) b
) A. v  e.  ( a (,) b
) ( u [,] v )  C_  (
a (,) b ) )
2518, 24mpbi 208 . . . . . . . . . . . . . . 15  |-  A. u  e.  ( a (,) b
) A. v  e.  ( a (,) b
) ( u [,] v )  C_  (
a (,) b )
26 inss1 3703 . . . . . . . . . . . . . . . 16  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  ( a (,) b
)
27 ssralv 3549 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a (,) b
)  i^i  ( A [,] B ) )  C_  ( a (,) b
)  ->  ( A. v  e.  ( a (,) b ) ( u [,] v )  C_  ( a (,) b
)  ->  A. v  e.  ( ( a (,) b )  i^i  ( A [,] B ) ) ( u [,] v
)  C_  ( a (,) b ) ) )
2827ralimdv 2853 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a (,) b
)  i^i  ( A [,] B ) )  C_  ( a (,) b
)  ->  ( A. u  e.  ( a (,) b ) A. v  e.  ( a (,) b
) ( u [,] v )  C_  (
a (,) b )  ->  A. u  e.  ( a (,) b ) A. v  e.  ( ( a (,) b
)  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( a (,) b ) ) )
29 ssralv 3549 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a (,) b
)  i^i  ( A [,] B ) )  C_  ( a (,) b
)  ->  ( A. u  e.  ( a (,) b ) A. v  e.  ( ( a (,) b )  i^i  ( A [,] B ) ) ( u [,] v
)  C_  ( a (,) b )  ->  A. u  e.  ( ( a (,) b )  i^i  ( A [,] B ) ) A. v  e.  ( ( a (,) b
)  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( a (,) b ) ) )
3028, 29syld 44 . . . . . . . . . . . . . . . 16  |-  ( ( ( a (,) b
)  i^i  ( A [,] B ) )  C_  ( a (,) b
)  ->  ( A. u  e.  ( a (,) b ) A. v  e.  ( a (,) b
) ( u [,] v )  C_  (
a (,) b )  ->  A. u  e.  ( ( a (,) b
)  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( a (,) b ) ) )
3126, 30ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( A. u  e.  ( a (,) b ) A. v  e.  ( a (,) b
) ( u [,] v )  C_  (
a (,) b )  ->  A. u  e.  ( ( a (,) b
)  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( a (,) b ) )
3225, 31mp1i 12 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  ->  A. u  e.  (
( a (,) b
)  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( a (,) b ) )
33 inss2 3704 . . . . . . . . . . . . . . 15  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  ( A [,] B )
34 iccconn 21208 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e.  Con )
35 iccssre 11615 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
36 reconn 21206 . . . . . . . . . . . . . . . . . 18  |-  ( ( A [,] B ) 
C_  RR  ->  ( ( ( topGen `  ran  (,) )t  ( A [,] B ) )  e.  Con  <->  A. u  e.  ( A [,] B
) A. v  e.  ( A [,] B
) ( u [,] v )  C_  ( A [,] B ) ) )
3735, 36syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )  e. 
Con 
<-> 
A. u  e.  ( A [,] B ) A. v  e.  ( A [,] B ) ( u [,] v
)  C_  ( A [,] B ) ) )
3834, 37mpbid 210 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A. u  e.  ( A [,] B ) A. v  e.  ( A [,] B ) ( u [,] v
)  C_  ( A [,] B ) )
3938ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  ->  A. u  e.  ( A [,] B ) A. v  e.  ( A [,] B ) ( u [,] v )  C_  ( A [,] B ) )
40 ssralv 3549 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a (,) b
)  i^i  ( A [,] B ) )  C_  ( A [,] B )  ->  ( A. v  e.  ( A [,] B
) ( u [,] v )  C_  ( A [,] B )  ->  A. v  e.  (
( a (,) b
)  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( A [,] B ) ) )
4140ralimdv 2853 . . . . . . . . . . . . . . . 16  |-  ( ( ( a (,) b
)  i^i  ( A [,] B ) )  C_  ( A [,] B )  ->  ( A. u  e.  ( A [,] B
) A. v  e.  ( A [,] B
) ( u [,] v )  C_  ( A [,] B )  ->  A. u  e.  ( A [,] B ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( A [,] B ) ) )
42 ssralv 3549 . . . . . . . . . . . . . . . 16  |-  ( ( ( a (,) b
)  i^i  ( A [,] B ) )  C_  ( A [,] B )  ->  ( A. u  e.  ( A [,] B
) A. v  e.  ( ( a (,) b )  i^i  ( A [,] B ) ) ( u [,] v
)  C_  ( A [,] B )  ->  A. u  e.  ( ( a (,) b )  i^i  ( A [,] B ) ) A. v  e.  ( ( a (,) b
)  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( A [,] B ) ) )
4341, 42syld 44 . . . . . . . . . . . . . . 15  |-  ( ( ( a (,) b
)  i^i  ( A [,] B ) )  C_  ( A [,] B )  ->  ( A. u  e.  ( A [,] B
) A. v  e.  ( A [,] B
) ( u [,] v )  C_  ( A [,] B )  ->  A. u  e.  (
( a (,) b
)  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( A [,] B ) ) )
4433, 39, 43mpsyl 63 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  ->  A. u  e.  (
( a (,) b
)  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( A [,] B ) )
45 ssin 3705 . . . . . . . . . . . . . . . 16  |-  ( ( ( u [,] v
)  C_  ( a (,) b )  /\  (
u [,] v ) 
C_  ( A [,] B ) )  <->  ( u [,] v )  C_  (
( a (,) b
)  i^i  ( A [,] B ) ) )
46452ralbii 2875 . . . . . . . . . . . . . . 15  |-  ( A. u  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( ( u [,] v
)  C_  ( a (,) b )  /\  (
u [,] v ) 
C_  ( A [,] B ) )  <->  A. u  e.  ( ( a (,) b )  i^i  ( A [,] B ) ) A. v  e.  ( ( a (,) b
)  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( ( a (,) b )  i^i  ( A [,] B
) ) )
47 r19.26-2 2971 . . . . . . . . . . . . . . 15  |-  ( A. u  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( ( u [,] v
)  C_  ( a (,) b )  /\  (
u [,] v ) 
C_  ( A [,] B ) )  <->  ( A. u  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( a (,) b )  /\  A. u  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( A [,] B ) ) )
4846, 47bitr3i 251 . . . . . . . . . . . . . 14  |-  ( A. u  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( ( a (,) b )  i^i  ( A [,] B
) )  <->  ( A. u  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( a (,) b )  /\  A. u  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( A [,] B ) ) )
4932, 44, 48sylanbrc 664 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  ->  A. u  e.  (
( a (,) b
)  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( ( a (,) b )  i^i  ( A [,] B
) ) )
5026, 19sstri 3498 . . . . . . . . . . . . . 14  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  RR
51 eqid 2443 . . . . . . . . . . . . . . . 16  |-  ( (
topGen `  ran  (,) )t  (
( a (,) b
)  i^i  ( A [,] B ) ) )  =  ( ( topGen ` 
ran  (,) )t  ( ( a (,) b )  i^i  ( A [,] B
) ) )
5251rescon 28564 . . . . . . . . . . . . . . 15  |-  ( ( ( a (,) b
)  i^i  ( A [,] B ) )  C_  RR  ->  ( ( (
topGen `  ran  (,) )t  (
( a (,) b
)  i^i  ( A [,] B ) ) )  e. SCon 
<->  ( ( topGen `  ran  (,) )t  ( ( a (,) b )  i^i  ( A [,] B ) ) )  e.  Con )
)
53 reconn 21206 . . . . . . . . . . . . . . 15  |-  ( ( ( a (,) b
)  i^i  ( A [,] B ) )  C_  RR  ->  ( ( (
topGen `  ran  (,) )t  (
( a (,) b
)  i^i  ( A [,] B ) ) )  e.  Con  <->  A. u  e.  ( ( a (,) b )  i^i  ( A [,] B ) ) A. v  e.  ( ( a (,) b
)  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( ( a (,) b )  i^i  ( A [,] B
) ) ) )
5452, 53bitrd 253 . . . . . . . . . . . . . 14  |-  ( ( ( a (,) b
)  i^i  ( A [,] B ) )  C_  RR  ->  ( ( (
topGen `  ran  (,) )t  (
( a (,) b
)  i^i  ( A [,] B ) ) )  e. SCon 
<-> 
A. u  e.  ( ( a (,) b
)  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( ( a (,) b )  i^i  ( A [,] B
) ) ) )
5550, 54ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( ( topGen `  ran  (,) )t  (
( a (,) b
)  i^i  ( A [,] B ) ) )  e. SCon 
<-> 
A. u  e.  ( ( a (,) b
)  i^i  ( A [,] B ) ) A. v  e.  ( (
a (,) b )  i^i  ( A [,] B ) ) ( u [,] v ) 
C_  ( ( a (,) b )  i^i  ( A [,] B
) ) )
5649, 55sylibr 212 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  -> 
( ( topGen `  ran  (,) )t  ( ( a (,) b )  i^i  ( A [,] B ) ) )  e. SCon )
5717, 56eqeltrd 2531 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  -> 
( ( topGen `  ran  (,) )t  ( z  i^i  ( A [,] B ) ) )  e. SCon )
5813, 14, 573jca 1177 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  ( x  i^i  ( A [,] B ) ) ) )  /\  (
z  =  ( a (,) b )  /\  ( y  e.  z  /\  z  C_  x
) ) )  -> 
( ( z  i^i  ( A [,] B
) )  C_  x  /\  y  e.  z  /\  ( ( topGen `  ran  (,) )t  ( z  i^i  ( A [,] B ) ) )  e. SCon ) )
5958exp32 605 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  ( z  =  ( a (,) b )  ->  (
( y  e.  z  /\  z  C_  x
)  ->  ( (
z  i^i  ( A [,] B ) )  C_  x  /\  y  e.  z  /\  ( ( topGen ` 
ran  (,) )t  ( z  i^i  ( A [,] B
) ) )  e. SCon
) ) ) )
6059rexlimdvw 2938 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  ( E. b  e.  RR*  z  =  ( a (,) b
)  ->  ( (
y  e.  z  /\  z  C_  x )  -> 
( ( z  i^i  ( A [,] B
) )  C_  x  /\  y  e.  z  /\  ( ( topGen `  ran  (,) )t  ( z  i^i  ( A [,] B ) ) )  e. SCon ) ) ) )
6160rexlimdvw 2938 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  ( E. a  e.  RR*  E. b  e.  RR*  z  =  ( a (,) b )  ->  ( ( y  e.  z  /\  z  C_  x )  ->  (
( z  i^i  ( A [,] B ) ) 
C_  x  /\  y  e.  z  /\  (
( topGen `  ran  (,) )t  (
z  i^i  ( A [,] B ) ) )  e. SCon ) ) ) )
6210, 61syl5bi 217 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  ( z  e.  ran  (,)  ->  ( ( y  e.  z  /\  z  C_  x )  -> 
( ( z  i^i  ( A [,] B
) )  C_  x  /\  y  e.  z  /\  ( ( topGen `  ran  (,) )t  ( z  i^i  ( A [,] B ) ) )  e. SCon ) ) ) )
6362reximdvai 2915 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  ( E. z  e.  ran  (,) (
y  e.  z  /\  z  C_  x )  ->  E. z  e.  ran  (,) ( ( z  i^i  ( A [,] B
) )  C_  x  /\  y  e.  z  /\  ( ( topGen `  ran  (,) )t  ( z  i^i  ( A [,] B ) ) )  e. SCon ) ) )
64 retopbas 21140 . . . . . 6  |-  ran  (,)  e. 
TopBases
65 bastg 19340 . . . . . 6  |-  ( ran 
(,)  e.  TopBases  ->  ran  (,)  C_  ( topGen `  ran  (,) )
)
66 ssrexv 3550 . . . . . 6  |-  ( ran 
(,)  C_  ( topGen `  ran  (,) )  ->  ( E. z  e.  ran  (,) (
( z  i^i  ( A [,] B ) ) 
C_  x  /\  y  e.  z  /\  (
( topGen `  ran  (,) )t  (
z  i^i  ( A [,] B ) ) )  e. SCon )  ->  E. z  e.  ( topGen `  ran  (,) )
( ( z  i^i  ( A [,] B
) )  C_  x  /\  y  e.  z  /\  ( ( topGen `  ran  (,) )t  ( z  i^i  ( A [,] B ) ) )  e. SCon ) ) )
6764, 65, 66mp2b 10 . . . . 5  |-  ( E. z  e.  ran  (,) ( ( z  i^i  ( A [,] B
) )  C_  x  /\  y  e.  z  /\  ( ( topGen `  ran  (,) )t  ( z  i^i  ( A [,] B ) ) )  e. SCon )  ->  E. z  e.  ( topGen `
 ran  (,) )
( ( z  i^i  ( A [,] B
) )  C_  x  /\  y  e.  z  /\  ( ( topGen `  ran  (,) )t  ( z  i^i  ( A [,] B ) ) )  e. SCon ) )
6863, 67syl6 33 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  ( E. z  e.  ran  (,) (
y  e.  z  /\  z  C_  x )  ->  E. z  e.  ( topGen `
 ran  (,) )
( ( z  i^i  ( A [,] B
) )  C_  x  /\  y  e.  z  /\  ( ( topGen `  ran  (,) )t  ( z  i^i  ( A [,] B ) ) )  e. SCon ) ) )
696, 68mpd 15 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  E. z  e.  ( topGen `  ran  (,) )
( ( z  i^i  ( A [,] B
) )  C_  x  /\  y  e.  z  /\  ( ( topGen `  ran  (,) )t  ( z  i^i  ( A [,] B ) ) )  e. SCon ) )
7069ralrimivva 2864 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A. x  e.  (
topGen `  ran  (,) ) A. y  e.  (
x  i^i  ( A [,] B ) ) E. z  e.  ( topGen ` 
ran  (,) ) ( ( z  i^i  ( A [,] B ) ) 
C_  x  /\  y  e.  z  /\  (
( topGen `  ran  (,) )t  (
z  i^i  ( A [,] B ) ) )  e. SCon ) )
71 retop 21141 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
72 ovex 6309 . . 3  |-  ( A [,] B )  e. 
_V
73 subislly 19855 . . 3  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A [,] B )  e. 
_V )  ->  (
( ( topGen `  ran  (,) )t  ( A [,] B
) )  e. Locally SCon  <->  A. x  e.  ( topGen `  ran  (,) ) A. y  e.  (
x  i^i  ( A [,] B ) ) E. z  e.  ( topGen ` 
ran  (,) ) ( ( z  i^i  ( A [,] B ) ) 
C_  x  /\  y  e.  z  /\  (
( topGen `  ran  (,) )t  (
z  i^i  ( A [,] B ) ) )  e. SCon ) ) )
7471, 72, 73mp2an 672 . 2  |-  ( ( ( topGen `  ran  (,) )t  ( A [,] B ) )  e. Locally SCon 
<-> 
A. x  e.  (
topGen `  ran  (,) ) A. y  e.  (
x  i^i  ( A [,] B ) ) E. z  e.  ( topGen ` 
ran  (,) ) ( ( z  i^i  ( A [,] B ) ) 
C_  x  /\  y  e.  z  /\  (
( topGen `  ran  (,) )t  (
z  i^i  ( A [,] B ) ) )  e. SCon ) )
7570, 74sylibr 212 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e. Locally SCon )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   _Vcvv 3095    i^i cin 3460    C_ wss 3461   ~Pcpw 3997    X. cxp 4987   ran crn 4990    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   RRcr 9494   RR*cxr 9630   (,)cioo 11538   [,]cicc 11541   ↾t crest 14695   topGenctg 14712   Topctop 19267   TopBasesctb 19271   Conccon 19785  Locally clly 19838  SConcscon 28538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-seq 12087  df-exp 12146  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-xrs 14776  df-qtop 14781  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-mulg 15934  df-cntz 16229  df-cmn 16674  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-cn 19601  df-cnp 19602  df-con 19786  df-lly 19840  df-tx 19936  df-hmeo 20129  df-xms 20696  df-ms 20697  df-tms 20698  df-ii 21254  df-htpy 21343  df-phtpy 21344  df-phtpc 21365  df-pcon 28539  df-scon 28540
This theorem is referenced by:  iillyscon  28571
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