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Theorem iccllyscon 29547
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 3659 . . . . . 6  |-  ( x  i^i  ( A [,] B ) )  C_  x
3 simprr 758 . . . . . 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 3440 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  y  e.  x )
5 tg2 19758 . . . . 5  |-  ( ( x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  x )  ->  E. z  e.  ran  (,) ( y  e.  z  /\  z  C_  x ) )
61, 4, 5syl2anc 659 . . . 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 11676 . . . . . . . 8  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
8 ffn 5714 . . . . . . . 8  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
9 ovelrn 6432 . . . . . . . 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 3659 . . . . . . . . . . . 12  |-  ( z  i^i  ( A [,] B ) )  C_  z
12 simprrr 767 . . . . . . . . . . . 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 3453 . . . . . . . . . . 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 766 . . . . . . . . . . 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 3640 . . . . . . . . . . . . 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 6294 . . . . . . . . . . . 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 29544 . . . . . . . . . . . . . . . 16  |-  ( (
topGen `  ran  (,) )t  (
a (,) b ) )  e. SCon
19 ioossre 11640 . . . . . . . . . . . . . . . . 17  |-  ( a (,) b )  C_  RR
20 eqid 2402 . . . . . . . . . . . . . . . . . . 19  |-  ( (
topGen `  ran  (,) )t  (
a (,) b ) )  =  ( (
topGen `  ran  (,) )t  (
a (,) b ) )
2120rescon 29543 . . . . . . . . . . . . . . . . . 18  |-  ( ( a (,) b ) 
C_  RR  ->  ( ( ( topGen `  ran  (,) )t  (
a (,) b ) )  e. SCon  <->  ( ( topGen `
 ran  (,) )t  (
a (,) b ) )  e.  Con )
)
22 reconn 21625 . . . . . . . . . . . . . . . . . 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 3659 . . . . . . . . . . . . . . . 16  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  ( a (,) b
)
27 ssralv 3503 . . . . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . . . 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 3503 . . . . . . . . . . . . . . . . 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 42 . . . . . . . . . . . . . . . 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 13 . . . . . . . . . . . . . 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 3660 . . . . . . . . . . . . . . 15  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  ( A [,] B )
34 iccconn 21627 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e.  Con )
35 iccssre 11660 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
36 reconn 21625 . . . . . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . 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 3503 . . . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . . 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 3503 . . . . . . . . . . . . . . . 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 42 . . . . . . . . . . . . . . 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 62 . . . . . . . . . . . . . 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 3661 . . . . . . . . . . . . . . . 16  |-  ( ( ( u [,] v
)  C_  ( a (,) b )  /\  (
u [,] v ) 
C_  ( A [,] B ) )  <->  ( u [,] v )  C_  (
( a (,) b
)  i^i  ( A [,] B ) ) )
46452ralbii 2836 . . . . . . . . . . . . . . 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 2935 . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . 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 3451 . . . . . . . . . . . . . 14  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  RR
51 eqid 2402 . . . . . . . . . . . . . . . 16  |-  ( (
topGen `  ran  (,) )t  (
( a (,) b
)  i^i  ( A [,] B ) ) )  =  ( ( topGen ` 
ran  (,) )t  ( ( a (,) b )  i^i  ( A [,] B
) ) )
5251rescon 29543 . . . . . . . . . . . . . . 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 21625 . . . . . . . . . . . . . . 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 2490 . . . . . . . . . . 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 603 . . . . . . . . 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 2899 . . . . . . . 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 2899 . . . . . . 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 2876 . . . . 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 21559 . . . . . 6  |-  ran  (,)  e. 
TopBases
65 bastg 19759 . . . . . 6  |-  ( ran 
(,)  e.  TopBases  ->  ran  (,)  C_  ( topGen `  ran  (,) )
)
66 ssrexv 3504 . . . . . 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 31 . . . 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 2825 . 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 21560 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
72 ovex 6306 . . 3  |-  ( A [,] B )  e. 
_V
73 subislly 20274 . . 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 670 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   _Vcvv 3059    i^i cin 3413    C_ wss 3414   ~Pcpw 3955    X. cxp 4821   ran crn 4824    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   RRcr 9521   RR*cxr 9657   (,)cioo 11582   [,]cicc 11585   ↾t crest 15035   topGenctg 15052   Topctop 19686   TopBasesctb 19690   Conccon 20204  Locally clly 20257  SConcscon 29517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-cn 20021  df-cnp 20022  df-con 20205  df-lly 20259  df-tx 20355  df-hmeo 20548  df-xms 21115  df-ms 21116  df-tms 21117  df-ii 21673  df-htpy 21762  df-phtpy 21763  df-phtpc 21784  df-pcon 29518  df-scon 29519
This theorem is referenced by:  iillyscon  29550
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