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Theorem iccllyscon 23796
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 732 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  x  e.  ( topGen `  ran  (,) )
)
2 inss1 3402 . . . . . 6  |-  ( x  i^i  ( A [,] B ) )  C_  x
3 simprr 733 . . . . . 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 3191 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  y  e.  x )
5 tg2 16719 . . . . 5  |-  ( ( x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  x )  ->  E. z  e.  ran  (,) ( y  e.  z  /\  z  C_  x ) )
61, 4, 5syl2anc 642 . . . 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 10757 . . . . . . . 8  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
8 ffn 5405 . . . . . . . 8  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
9 ovelrn 6012 . . . . . . . 8  |-  ( (,) 
Fn  ( RR*  X.  RR* )  ->  ( z  e. 
ran  (,)  <->  E. a  e.  RR*  E. b  e.  RR*  z  =  ( a (,) b ) ) )
107, 8, 9mp2b 9 . . . . . . 7  |-  ( z  e.  ran  (,)  <->  E. a  e.  RR*  E. b  e. 
RR*  z  =  ( a (,) b ) )
11 inss1 3402 . . . . . . . . . . . 12  |-  ( z  i^i  ( A [,] B ) )  C_  z
12 simprrr 741 . . . . . . . . . . . 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 3203 . . . . . . . . . . 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 740 . . . . . . . . . . 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 732 . . . . . . . . . . . . . 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 3382 . . . . . . . . . . . . 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 5890 . . . . . . . . . . . 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 23793 . . . . . . . . . . . . . . . 16  |-  ( (
topGen `  ran  (,) )t  (
a (,) b ) )  e. SCon
19 ioossre 10728 . . . . . . . . . . . . . . . . 17  |-  ( a (,) b )  C_  RR
20 eqid 2296 . . . . . . . . . . . . . . . . . . 19  |-  ( (
topGen `  ran  (,) )t  (
a (,) b ) )  =  ( (
topGen `  ran  (,) )t  (
a (,) b ) )
2120rescon 23792 . . . . . . . . . . . . . . . . . 18  |-  ( ( a (,) b ) 
C_  RR  ->  ( ( ( topGen `  ran  (,) )t  (
a (,) b ) )  e. SCon  <->  ( ( topGen `
 ran  (,) )t  (
a (,) b ) )  e.  Con )
)
22 reconn 18349 . . . . . . . . . . . . . . . . . 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 244 . . . . . . . . . . . . . . . . 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 199 . . . . . . . . . . . . . . 15  |-  A. u  e.  ( a (,) b
) A. v  e.  ( a (,) b
) ( u [,] v )  C_  (
a (,) b )
26 inss1 3402 . . . . . . . . . . . . . . . 16  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  ( a (,) b
)
27 ssralv 3250 . . . . . . . . . . . . . . . . . 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 2635 . . . . . . . . . . . . . . . . 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 3250 . . . . . . . . . . . . . . . . 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 40 . . . . . . . . . . . . . . . 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 11 . . . . . . . . . . . . . 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 3403 . . . . . . . . . . . . . . 15  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  ( A [,] B )
34 iccconn 18351 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e.  Con )
35 iccssre 10747 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
36 reconn 18349 . . . . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A. u  e.  ( A [,] B ) A. v  e.  ( A [,] B ) ( u [,] v
)  C_  ( A [,] B ) )
3938ad2antrr 706 . . . . . . . . . . . . . . 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 3250 . . . . . . . . . . . . . . . . 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 2635 . . . . . . . . . . . . . . . 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 3250 . . . . . . . . . . . . . . . 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 40 . . . . . . . . . . . . . . 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 59 . . . . . . . . . . . . . 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 3404 . . . . . . . . . . . . . . . 16  |-  ( ( ( u [,] v
)  C_  ( a (,) b )  /\  (
u [,] v ) 
C_  ( A [,] B ) )  <->  ( u [,] v )  C_  (
( a (,) b
)  i^i  ( A [,] B ) ) )
46452ralbii 2582 . . . . . . . . . . . . . . 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 2689 . . . . . . . . . . . . . . 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 242 . . . . . . . . . . . . . 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 645 . . . . . . . . . . . . 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 3201 . . . . . . . . . . . . . 14  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  RR
51 eqid 2296 . . . . . . . . . . . . . . . 16  |-  ( (
topGen `  ran  (,) )t  (
( a (,) b
)  i^i  ( A [,] B ) ) )  =  ( ( topGen ` 
ran  (,) )t  ( ( a (,) b )  i^i  ( A [,] B
) ) )
5251rescon 23792 . . . . . . . . . . . . . . 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 18349 . . . . . . . . . . . . . . 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 244 . . . . . . . . . . . . . 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 203 . . . . . . . . . . . 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 2370 . . . . . . . . . . 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 1132 . . . . . . . . . 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 588 . . . . . . . . 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 2683 . . . . . . . 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 2683 . . . . . . 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 208 . . . . . 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 2666 . . . . 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 18285 . . . . . 6  |-  ran  (,)  e. 
TopBases
65 bastg 16720 . . . . . 6  |-  ( ran 
(,)  e.  TopBases  ->  ran  (,)  C_  ( topGen `  ran  (,) )
)
66 ssrexv 3251 . . . . . 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 9 . . . . 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 29 . . . 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 14 . . 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 2648 . 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 18286 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
72 ovex 5899 . . 3  |-  ( A [,] B )  e. 
_V
73 subislly 17223 . . 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 653 . 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 203 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638    X. cxp 4703   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   RR*cxr 8882   (,)cioo 10672   [,]cicc 10675   ↾t crest 13341   topGenctg 13358   Topctop 16647   TopBasesctb 16651   Conccon 17153  Locally clly 17206  SConcscon 23766
This theorem is referenced by:  iillyscon  23799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-cn 16973  df-cnp 16974  df-con 17154  df-lly 17208  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903  df-ii 18397  df-htpy 18484  df-phtpy 18485  df-phtpc 18506  df-pcon 23767  df-scon 23768
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