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Theorem iccllyscon 27069
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 750 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  x  e.  ( topGen `  ran  (,) )
)
2 inss1 3567 . . . . . 6  |-  ( x  i^i  ( A [,] B ) )  C_  x
3 simprr 751 . . . . . 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 3351 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( x  e.  ( topGen `  ran  (,) )  /\  y  e.  (
x  i^i  ( A [,] B ) ) ) )  ->  y  e.  x )
5 tg2 18529 . . . . 5  |-  ( ( x  e.  ( topGen ` 
ran  (,) )  /\  y  e.  x )  ->  E. z  e.  ran  (,) ( y  e.  z  /\  z  C_  x ) )
61, 4, 5syl2anc 656 . . . 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 11383 . . . . . . . 8  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
8 ffn 5556 . . . . . . . 8  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
9 ovelrn 6238 . . . . . . . 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 3567 . . . . . . . . . . . 12  |-  ( z  i^i  ( A [,] B ) )  C_  z
12 simprrr 759 . . . . . . . . . . . 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 3364 . . . . . . . . . . 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 758 . . . . . . . . . . 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 750 . . . . . . . . . . . . . 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 3548 . . . . . . . . . . . . 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 6106 . . . . . . . . . . . 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 27066 . . . . . . . . . . . . . . . 16  |-  ( (
topGen `  ran  (,) )t  (
a (,) b ) )  e. SCon
19 ioossre 11353 . . . . . . . . . . . . . . . . 17  |-  ( a (,) b )  C_  RR
20 eqid 2441 . . . . . . . . . . . . . . . . . . 19  |-  ( (
topGen `  ran  (,) )t  (
a (,) b ) )  =  ( (
topGen `  ran  (,) )t  (
a (,) b ) )
2120rescon 27065 . . . . . . . . . . . . . . . . . 18  |-  ( ( a (,) b ) 
C_  RR  ->  ( ( ( topGen `  ran  (,) )t  (
a (,) b ) )  e. SCon  <->  ( ( topGen `
 ran  (,) )t  (
a (,) b ) )  e.  Con )
)
22 reconn 20364 . . . . . . . . . . . . . . . . . 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 3567 . . . . . . . . . . . . . . . 16  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  ( a (,) b
)
27 ssralv 3413 . . . . . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . . . . 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 3413 . . . . . . . . . . . . . . . . 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 3568 . . . . . . . . . . . . . . 15  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  ( A [,] B )
34 iccconn 20366 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e.  Con )
35 iccssre 11373 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
36 reconn 20364 . . . . . . . . . . . . . . . . . 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 720 . . . . . . . . . . . . . . 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 3413 . . . . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . . . 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 3413 . . . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . . . . . 16  |-  ( ( ( u [,] v
)  C_  ( a (,) b )  /\  (
u [,] v ) 
C_  ( A [,] B ) )  <->  ( u [,] v )  C_  (
( a (,) b
)  i^i  ( A [,] B ) ) )
46452ralbii 2739 . . . . . . . . . . . . . . 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 2848 . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . 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 3362 . . . . . . . . . . . . . 14  |-  ( ( a (,) b )  i^i  ( A [,] B ) )  C_  RR
51 eqid 2441 . . . . . . . . . . . . . . . 16  |-  ( (
topGen `  ran  (,) )t  (
( a (,) b
)  i^i  ( A [,] B ) ) )  =  ( ( topGen ` 
ran  (,) )t  ( ( a (,) b )  i^i  ( A [,] B
) ) )
5251rescon 27065 . . . . . . . . . . . . . . 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 20364 . . . . . . . . . . . . . . 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 2515 . . . . . . . . . . 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 1163 . . . . . . . . . 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 602 . . . . . . . . 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 2842 . . . . . . . 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 2842 . . . . . . 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 2824 . . . . 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 20298 . . . . . 6  |-  ran  (,)  e. 
TopBases
65 bastg 18530 . . . . . 6  |-  ( ran 
(,)  e.  TopBases  ->  ran  (,)  C_  ( topGen `  ran  (,) )
)
66 ssrexv 3414 . . . . . 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 2806 . 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 20299 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
72 ovex 6115 . . 3  |-  ( A [,] B )  e. 
_V
73 subislly 19044 . . 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 667 . 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 960    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   _Vcvv 2970    i^i cin 3324    C_ wss 3325   ~Pcpw 3857    X. cxp 4834   ran crn 4837    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   RRcr 9277   RR*cxr 9413   (,)cioo 11296   [,]cicc 11299   ↾t crest 14355   topGenctg 14372   Topctop 18457   TopBasesctb 18461   Conccon 18974  Locally clly 19027  SConcscon 27039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-cn 18790  df-cnp 18791  df-con 18975  df-lly 19029  df-tx 19094  df-hmeo 19287  df-xms 19854  df-ms 19855  df-tms 19856  df-ii 20412  df-htpy 20501  df-phtpy 20502  df-phtpc 20523  df-pcon 27040  df-scon 27041
This theorem is referenced by:  iillyscon  27072
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