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Theorem iccllyscon 24929
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 733 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  x  e.  ( topGen `  ran  (,) )
)
2 inss1 3553 . . . . . 6  |-  ( x  i^i  ( A [,] B ) )  C_  x
3 simprr 734 . . . . . 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 3338 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  y  e.  x )
5 tg2 17022 . . . . 5  |-  ( ( x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  x )  ->  E. z  e.  ran  (,) ( y  e.  z  /\  z  C_  x ) )
61, 4, 5syl2anc 643 . . . 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 10994 . . . . . . . 8  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
8 ffn 5583 . . . . . . . 8  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
9 ovelrn 6214 . . . . . . . 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 3553 . . . . . . . . . . . 12  |-  ( z  i^i  ( A [,] B ) )  C_  z
12 simprrr 742 . . . . . . . . . . . 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 3351 . . . . . . . . . . 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 741 . . . . . . . . . . 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 733 . . . . . . . . . . . . . 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 3533 . . . . . . . . . . . . 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 6089 . . . . . . . . . . . 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 24926 . . . . . . . . . . . . . . . 16  |-  ( (
topGen `  ran  (,) )t  (
a (,) b ) )  e. SCon
19 ioossre 10964 . . . . . . . . . . . . . . . . 17  |-  ( a (,) b )  C_  RR
20 eqid 2435 . . . . . . . . . . . . . . . . . . 19  |-  ( (
topGen `  ran  (,) )t  (
a (,) b ) )  =  ( (
topGen `  ran  (,) )t  (
a (,) b ) )
2120rescon 24925 . . . . . . . . . . . . . . . . . 18  |-  ( ( a (,) b ) 
C_  RR  ->  ( ( ( topGen `  ran  (,) )t  (
a (,) b ) )  e. SCon  <->  ( ( topGen `
 ran  (,) )t  (
a (,) b ) )  e.  Con )
)
22 reconn 18851 . . . . . . . . . . . . . . . . . 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 245 . . . . . . . . . . . . . . . . 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 8 . . . . . . . . . . . . . . . 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 200 . . . . . . . . . . . . . . 15  |-  A. u  e.  ( a (,) b
) A. v  e.  ( a (,) b
) ( u [,] v )  C_  (
a (,) b )
26 inss1 3553 . . . . . . . . . . . . . . . 16  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  ( a (,) b
)
27 ssralv 3399 . . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . . 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 3399 . . . . . . . . . . . . . . . . 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 8 . . . . . . . . . . . . . . 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 3554 . . . . . . . . . . . . . . 15  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  ( A [,] B )
34 iccconn 18853 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e.  Con )
35 iccssre 10984 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
36 reconn 18851 . . . . . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A. u  e.  ( A [,] B ) A. v  e.  ( A [,] B ) ( u [,] v
)  C_  ( A [,] B ) )
3938ad2antrr 707 . . . . . . . . . . . . . . 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 3399 . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . 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 3399 . . . . . . . . . . . . . . . 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 61 . . . . . . . . . . . . . 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 3555 . . . . . . . . . . . . . . . 16  |-  ( ( ( u [,] v
)  C_  ( a (,) b )  /\  (
u [,] v ) 
C_  ( A [,] B ) )  <->  ( u [,] v )  C_  (
( a (,) b
)  i^i  ( A [,] B ) ) )
46452ralbii 2723 . . . . . . . . . . . . . . 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 2831 . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . 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 646 . . . . . . . . . . . . 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 3349 . . . . . . . . . . . . . 14  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  RR
51 eqid 2435 . . . . . . . . . . . . . . . 16  |-  ( (
topGen `  ran  (,) )t  (
( a (,) b
)  i^i  ( A [,] B ) ) )  =  ( ( topGen ` 
ran  (,) )t  ( ( a (,) b )  i^i  ( A [,] B
) ) )
5251rescon 24925 . . . . . . . . . . . . . . 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 18851 . . . . . . . . . . . . . . 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 245 . . . . . . . . . . . . . 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 8 . . . . . . . . . . . . 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 204 . . . . . . . . . . . 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 2509 . . . . . . . . . . 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 1134 . . . . . . . . . 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 589 . . . . . . . . 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 2825 . . . . . . . 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 2825 . . . . . . 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 209 . . . . . 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 2808 . . . . 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 18786 . . . . . 6  |-  ran  (,)  e. 
TopBases
65 bastg 17023 . . . . . 6  |-  ( ran 
(,)  e.  TopBases  ->  ran  (,)  C_  ( topGen `  ran  (,) )
)
66 ssrexv 3400 . . . . . 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 2790 . 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 18787 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
72 ovex 6098 . . 3  |-  ( A [,] B )  e. 
_V
73 subislly 17536 . . 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 654 . 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 204 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e. Locally SCon )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   ~Pcpw 3791    X. cxp 4868   ran crn 4871    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   RRcr 8981   RR*cxr 9111   (,)cioo 10908   [,]cicc 10911   ↾t crest 13640   topGenctg 13657   Topctop 16950   TopBasesctb 16954   Conccon 17466  Locally clly 17519  SConcscon 24899
This theorem is referenced by:  iillyscon  24932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-cn 17283  df-cnp 17284  df-con 17467  df-lly 17521  df-tx 17586  df-hmeo 17779  df-xms 18342  df-ms 18343  df-tms 18344  df-ii 18899  df-htpy 18987  df-phtpy 18988  df-phtpc 19009  df-pcon 24900  df-scon 24901
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