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Theorem isleag 24883
Description: Geometrical "less than" property for angles. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.)
Hypotheses
Ref Expression
isleag.p  |-  P  =  ( Base `  G
)
isleag.g  |-  ( ph  ->  G  e. TarskiG )
isleag.a  |-  ( ph  ->  A  e.  P )
isleag.b  |-  ( ph  ->  B  e.  P )
isleag.c  |-  ( ph  ->  C  e.  P )
isleag.d  |-  ( ph  ->  D  e.  P )
isleag.e  |-  ( ph  ->  E  e.  P )
isleag.f  |-  ( ph  ->  F  e.  P )
Assertion
Ref Expression
isleag  |-  ( ph  ->  ( <" A B C "> ( `  G
) <" D E F ">  <->  E. x  e.  P  ( x
(inA `  G ) <" D E F ">  /\  <" A B C "> (cgrA `  G ) <" D E x "> ) ) )
Distinct variable groups:    x, A    x, B    x, C    x, D    x, E    x, F    x, G    x, P    ph, x

Proof of Theorem isleag
Dummy variables  a 
b  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isleag.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
2 elex 3054 . . . . 5  |-  ( G  e. TarskiG  ->  G  e.  _V )
3 fveq2 5865 . . . . . . . . . . . 12  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
4 isleag.p . . . . . . . . . . . 12  |-  P  =  ( Base `  G
)
53, 4syl6eqr 2503 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( Base `  g )  =  P )
65oveq1d 6305 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( Base `  g )  ^m  ( 0..^ 3 ) )  =  ( P  ^m  ( 0..^ 3 ) ) )
76eleq2d 2514 . . . . . . . . 9  |-  ( g  =  G  ->  (
a  e.  ( (
Base `  g )  ^m  ( 0..^ 3 ) )  <->  a  e.  ( P  ^m  ( 0..^ 3 ) ) ) )
86eleq2d 2514 . . . . . . . . 9  |-  ( g  =  G  ->  (
b  e.  ( (
Base `  g )  ^m  ( 0..^ 3 ) )  <->  b  e.  ( P  ^m  ( 0..^ 3 ) ) ) )
97, 8anbi12d 717 . . . . . . . 8  |-  ( g  =  G  ->  (
( a  e.  ( ( Base `  g
)  ^m  ( 0..^ 3 ) )  /\  b  e.  ( ( Base `  g )  ^m  ( 0..^ 3 ) ) )  <->  ( a  e.  ( P  ^m  (
0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) ) ) )
10 fveq2 5865 . . . . . . . . . . 11  |-  ( g  =  G  ->  (inA `  g )  =  (inA
`  G ) )
1110breqd 4413 . . . . . . . . . 10  |-  ( g  =  G  ->  (
x (inA `  g
) <" ( b `
 0 ) ( b `  1 ) ( b `  2
) ">  <->  x (inA `  G ) <" (
b `  0 )
( b `  1
) ( b ` 
2 ) "> ) )
12 fveq2 5865 . . . . . . . . . . 11  |-  ( g  =  G  ->  (cgrA `  g )  =  (cgrA `  G ) )
1312breqd 4413 . . . . . . . . . 10  |-  ( g  =  G  ->  ( <" ( a ` 
0 ) ( a `
 1 ) ( a `  2 ) "> (cgrA `  g ) <" (
b `  0 )
( b `  1
) x ">  <->  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
)
1411, 13anbi12d 717 . . . . . . . . 9  |-  ( g  =  G  ->  (
( x (inA `  g ) <" (
b `  0 )
( b `  1
) ( b ` 
2 ) ">  /\ 
<" ( a ` 
0 ) ( a `
 1 ) ( a `  2 ) "> (cgrA `  g ) <" (
b `  0 )
( b `  1
) x "> ) 
<->  ( x (inA `  G ) <" (
b `  0 )
( b `  1
) ( b ` 
2 ) ">  /\ 
<" ( a ` 
0 ) ( a `
 1 ) ( a `  2 ) "> (cgrA `  G ) <" (
b `  0 )
( b `  1
) x "> ) ) )
155, 14rexeqbidv 3002 . . . . . . . 8  |-  ( g  =  G  ->  ( E. x  e.  ( Base `  g ) ( x (inA `  g
) <" ( b `
 0 ) ( b `  1 ) ( b `  2
) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  g
) <" ( b `
 0 ) ( b `  1 ) x "> )  <->  E. x  e.  P  ( x (inA `  G
) <" ( b `
 0 ) ( b `  1 ) ( b `  2
) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
) )
169, 15anbi12d 717 . . . . . . 7  |-  ( g  =  G  ->  (
( ( a  e.  ( ( Base `  g
)  ^m  ( 0..^ 3 ) )  /\  b  e.  ( ( Base `  g )  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  ( Base `  g
) ( x (inA
`  g ) <" ( b ` 
0 ) ( b `
 1 ) ( b `  2 ) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  g
) <" ( b `
 0 ) ( b `  1 ) x "> )
)  <->  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  ( x (inA
`  G ) <" ( b ` 
0 ) ( b `
 1 ) ( b `  2 ) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
) ) )
1716opabbidv 4466 . . . . . 6  |-  ( g  =  G  ->  { <. a ,  b >.  |  ( ( a  e.  ( ( Base `  g
)  ^m  ( 0..^ 3 ) )  /\  b  e.  ( ( Base `  g )  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  ( Base `  g
) ( x (inA
`  g ) <" ( b ` 
0 ) ( b `
 1 ) ( b `  2 ) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  g
) <" ( b `
 0 ) ( b `  1 ) x "> )
) }  =  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  ( x (inA
`  G ) <" ( b ` 
0 ) ( b `
 1 ) ( b `  2 ) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
) } )
18 df-leag 24882 . . . . . 6  |-  =  ( g  e.  _V  |->  { <. a ,  b >.  |  ( ( a  e.  ( ( Base `  g
)  ^m  ( 0..^ 3 ) )  /\  b  e.  ( ( Base `  g )  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  ( Base `  g
) ( x (inA
`  g ) <" ( b ` 
0 ) ( b `
 1 ) ( b `  2 ) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  g
) <" ( b `
 0 ) ( b `  1 ) x "> )
) } )
19 ovex 6318 . . . . . . . 8  |-  ( P  ^m  ( 0..^ 3 ) )  e.  _V
20 xpexg 6593 . . . . . . . 8  |-  ( ( ( P  ^m  (
0..^ 3 ) )  e.  _V  /\  ( P  ^m  ( 0..^ 3 ) )  e.  _V )  ->  ( ( P  ^m  ( 0..^ 3 ) )  X.  ( P  ^m  ( 0..^ 3 ) ) )  e. 
_V )
2119, 19, 20mp2an 678 . . . . . . 7  |-  ( ( P  ^m  ( 0..^ 3 ) )  X.  ( P  ^m  (
0..^ 3 ) ) )  e.  _V
22 opabssxp 4909 . . . . . . 7  |-  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  (
x (inA `  G
) <" ( b `
 0 ) ( b `  1 ) ( b `  2
) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
) }  C_  (
( P  ^m  (
0..^ 3 ) )  X.  ( P  ^m  ( 0..^ 3 ) ) )
2321, 22ssexi 4548 . . . . . 6  |-  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  (
x (inA `  G
) <" ( b `
 0 ) ( b `  1 ) ( b `  2
) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
) }  e.  _V
2417, 18, 23fvmpt 5948 . . . . 5  |-  ( G  e.  _V  ->  ( `  G
)  =  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  (
x (inA `  G
) <" ( b `
 0 ) ( b `  1 ) ( b `  2
) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
) } )
251, 2, 243syl 18 . . . 4  |-  ( ph  ->  (
`  G )  =  { <. a ,  b
>.  |  ( (
a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  (
0..^ 3 ) ) )  /\  E. x  e.  P  ( x
(inA `  G ) <" ( b ` 
0 ) ( b `
 1 ) ( b `  2 ) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
) } )
2625breqd 4413 . . 3  |-  ( ph  ->  ( <" A B C "> ( `  G
) <" D E F ">  <->  <" A B C "> { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  (
x (inA `  G
) <" ( b `
 0 ) ( b `  1 ) ( b `  2
) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
) } <" D E F "> )
)
27 simpr 463 . . . . . . . . . 10  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  b  =  <" D E F "> )
2827fveq1d 5867 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( b `  0
)  =  ( <" D E F "> `  0
) )
2927fveq1d 5867 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( b `  1
)  =  ( <" D E F "> `  1
) )
3027fveq1d 5867 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( b `  2
)  =  ( <" D E F "> `  2
) )
3128, 29, 30s3eqd 12959 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  <" ( b `
 0 ) ( b `  1 ) ( b `  2
) ">  =  <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) ( <" D E F "> `  2
) "> )
3231breq2d 4414 . . . . . . 7  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( x (inA `  G ) <" (
b `  0 )
( b `  1
) ( b ` 
2 ) ">  <->  x
(inA `  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) ( <" D E F "> `  2
) "> )
)
33 simpl 459 . . . . . . . . . 10  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  a  =  <" A B C "> )
3433fveq1d 5867 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( a `  0
)  =  ( <" A B C "> `  0
) )
3533fveq1d 5867 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( a `  1
)  =  ( <" A B C "> `  1
) )
3633fveq1d 5867 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( a `  2
)  =  ( <" A B C "> `  2
) )
3734, 35, 36s3eqd 12959 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  <" ( a `
 0 ) ( a `  1 ) ( a `  2
) ">  =  <" ( <" A B C "> `  0
) ( <" A B C "> `  1
) ( <" A B C "> `  2
) "> )
38 eqidd 2452 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  x  =  x )
3928, 29, 38s3eqd 12959 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  <" ( b `
 0 ) ( b `  1 ) x ">  =  <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) x "> )
4037, 39breq12d 4415 . . . . . . 7  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( <" (
a `  0 )
( a `  1
) ( a ` 
2 ) "> (cgrA `  G ) <" ( b ` 
0 ) ( b `
 1 ) x ">  <->  <" ( <" A B C "> `  0
) ( <" A B C "> `  1
) ( <" A B C "> `  2
) "> (cgrA `  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) x "> ) )
4132, 40anbi12d 717 . . . . . 6  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( ( x (inA
`  G ) <" ( b ` 
0 ) ( b `
 1 ) ( b `  2 ) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )  <->  ( x (inA `  G
) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) ( <" D E F "> `  2
) ">  /\  <" ( <" A B C "> `  0
) ( <" A B C "> `  1
) ( <" A B C "> `  2
) "> (cgrA `  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) x "> ) ) )
4241rexbidv 2901 . . . . 5  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( E. x  e.  P  ( x (inA
`  G ) <" ( b ` 
0 ) ( b `
 1 ) ( b `  2 ) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )  <->  E. x  e.  P  ( x (inA `  G
) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) ( <" D E F "> `  2
) ">  /\  <" ( <" A B C "> `  0
) ( <" A B C "> `  1
) ( <" A B C "> `  2
) "> (cgrA `  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) x "> ) ) )
43 eqid 2451 . . . . 5  |-  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  (
x (inA `  G
) <" ( b `
 0 ) ( b `  1 ) ( b `  2
) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
) }  =  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  ( x (inA
`  G ) <" ( b ` 
0 ) ( b `
 1 ) ( b `  2 ) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
) }
4442, 43brab2a 4884 . . . 4  |-  ( <" A B C "> { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  (
x (inA `  G
) <" ( b `
 0 ) ( b `  1 ) ( b `  2
) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
) } <" D E F ">  <->  ( ( <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  ( x (inA
`  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) ( <" D E F "> `  2
) ">  /\  <" ( <" A B C "> `  0
) ( <" A B C "> `  1
) ( <" A B C "> `  2
) "> (cgrA `  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) x "> ) ) )
4544a1i 11 . . 3  |-  ( ph  ->  ( <" A B C "> { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  (
x (inA `  G
) <" ( b `
 0 ) ( b `  1 ) ( b `  2
) ">  /\  <" ( a `  0
) ( a ` 
1 ) ( a `
 2 ) "> (cgrA `  G
) <" ( b `
 0 ) ( b `  1 ) x "> )
) } <" D E F ">  <->  ( ( <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  ( x (inA
`  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) ( <" D E F "> `  2
) ">  /\  <" ( <" A B C "> `  0
) ( <" A B C "> `  1
) ( <" A B C "> `  2
) "> (cgrA `  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) x "> ) ) ) )
46 isleag.d . . . . . . . . 9  |-  ( ph  ->  D  e.  P )
47 s3fv0 12985 . . . . . . . . 9  |-  ( D  e.  P  ->  ( <" D E F "> `  0
)  =  D )
4846, 47syl 17 . . . . . . . 8  |-  ( ph  ->  ( <" D E F "> `  0
)  =  D )
49 isleag.e . . . . . . . . 9  |-  ( ph  ->  E  e.  P )
50 s3fv1 12986 . . . . . . . . 9  |-  ( E  e.  P  ->  ( <" D E F "> `  1
)  =  E )
5149, 50syl 17 . . . . . . . 8  |-  ( ph  ->  ( <" D E F "> `  1
)  =  E )
52 isleag.f . . . . . . . . 9  |-  ( ph  ->  F  e.  P )
53 s3fv2 12987 . . . . . . . . 9  |-  ( F  e.  P  ->  ( <" D E F "> `  2
)  =  F )
5452, 53syl 17 . . . . . . . 8  |-  ( ph  ->  ( <" D E F "> `  2
)  =  F )
5548, 51, 54s3eqd 12959 . . . . . . 7  |-  ( ph  ->  <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) ( <" D E F "> `  2
) ">  =  <" D E F "> )
5655breq2d 4414 . . . . . 6  |-  ( ph  ->  ( x (inA `  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) ( <" D E F "> `  2
) ">  <->  x (inA `  G ) <" D E F "> )
)
57 isleag.a . . . . . . . . 9  |-  ( ph  ->  A  e.  P )
58 s3fv0 12985 . . . . . . . . 9  |-  ( A  e.  P  ->  ( <" A B C "> `  0
)  =  A )
5957, 58syl 17 . . . . . . . 8  |-  ( ph  ->  ( <" A B C "> `  0
)  =  A )
60 isleag.b . . . . . . . . 9  |-  ( ph  ->  B  e.  P )
61 s3fv1 12986 . . . . . . . . 9  |-  ( B  e.  P  ->  ( <" A B C "> `  1
)  =  B )
6260, 61syl 17 . . . . . . . 8  |-  ( ph  ->  ( <" A B C "> `  1
)  =  B )
63 isleag.c . . . . . . . . 9  |-  ( ph  ->  C  e.  P )
64 s3fv2 12987 . . . . . . . . 9  |-  ( C  e.  P  ->  ( <" A B C "> `  2
)  =  C )
6563, 64syl 17 . . . . . . . 8  |-  ( ph  ->  ( <" A B C "> `  2
)  =  C )
6659, 62, 65s3eqd 12959 . . . . . . 7  |-  ( ph  ->  <" ( <" A B C "> `  0
) ( <" A B C "> `  1
) ( <" A B C "> `  2
) ">  =  <" A B C "> )
67 eqidd 2452 . . . . . . . 8  |-  ( ph  ->  x  =  x )
6848, 51, 67s3eqd 12959 . . . . . . 7  |-  ( ph  ->  <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) x ">  =  <" D E x "> )
6966, 68breq12d 4415 . . . . . 6  |-  ( ph  ->  ( <" ( <" A B C "> `  0
) ( <" A B C "> `  1
) ( <" A B C "> `  2
) "> (cgrA `  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) x ">  <->  <" A B C "> (cgrA `  G ) <" D E x "> ) )
7056, 69anbi12d 717 . . . . 5  |-  ( ph  ->  ( ( x (inA
`  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) ( <" D E F "> `  2
) ">  /\  <" ( <" A B C "> `  0
) ( <" A B C "> `  1
) ( <" A B C "> `  2
) "> (cgrA `  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) x "> ) 
<->  ( x (inA `  G ) <" D E F ">  /\  <" A B C "> (cgrA `  G ) <" D E x "> ) ) )
7170rexbidv 2901 . . . 4  |-  ( ph  ->  ( E. x  e.  P  ( x (inA
`  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) ( <" D E F "> `  2
) ">  /\  <" ( <" A B C "> `  0
) ( <" A B C "> `  1
) ( <" A B C "> `  2
) "> (cgrA `  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) x "> ) 
<->  E. x  e.  P  ( x (inA `  G ) <" D E F ">  /\  <" A B C "> (cgrA `  G ) <" D E x "> ) ) )
7271anbi2d 710 . . 3  |-  ( ph  ->  ( ( ( <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  ( x (inA
`  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) ( <" D E F "> `  2
) ">  /\  <" ( <" A B C "> `  0
) ( <" A B C "> `  1
) ( <" A B C "> `  2
) "> (cgrA `  G ) <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) x "> ) )  <->  ( ( <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  ( x (inA
`  G ) <" D E F ">  /\  <" A B C "> (cgrA `  G ) <" D E x "> ) ) ) )
7326, 45, 723bitrd 283 . 2  |-  ( ph  ->  ( <" A B C "> ( `  G
) <" D E F ">  <->  ( ( <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  ( x (inA
`  G ) <" D E F ">  /\  <" A B C "> (cgrA `  G ) <" D E x "> ) ) ) )
7457, 60, 63s3cld 12966 . . . . . 6  |-  ( ph  ->  <" A B C ">  e. Word  P )
75 s3len 12988 . . . . . . 7  |-  ( # `  <" A B C "> )  =  3
7675a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  <" A B C "> )  =  3
)
7774, 76jca 535 . . . . 5  |-  ( ph  ->  ( <" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
) )
78 fvex 5875 . . . . . . 7  |-  ( Base `  G )  e.  _V
794, 78eqeltri 2525 . . . . . 6  |-  P  e. 
_V
80 3nn0 10887 . . . . . 6  |-  3  e.  NN0
81 wrdmap 12698 . . . . . 6  |-  ( ( P  e.  _V  /\  3  e.  NN0 )  -> 
( ( <" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
)  <->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) ) )
8279, 80, 81mp2an 678 . . . . 5  |-  ( (
<" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
)  <->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )
8377, 82sylib 200 . . . 4  |-  ( ph  ->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )
8446, 49, 52s3cld 12966 . . . . . 6  |-  ( ph  ->  <" D E F ">  e. Word  P )
85 s3len 12988 . . . . . . 7  |-  ( # `  <" D E F "> )  =  3
8685a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  <" D E F "> )  =  3
)
8784, 86jca 535 . . . . 5  |-  ( ph  ->  ( <" D E F ">  e. Word  P  /\  ( # `  <" D E F "> )  =  3
) )
88 wrdmap 12698 . . . . . 6  |-  ( ( P  e.  _V  /\  3  e.  NN0 )  -> 
( ( <" D E F ">  e. Word  P  /\  ( # `  <" D E F "> )  =  3
)  <->  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) ) )
8979, 80, 88mp2an 678 . . . . 5  |-  ( (
<" D E F ">  e. Word  P  /\  ( # `  <" D E F "> )  =  3
)  <->  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9087, 89sylib 200 . . . 4  |-  ( ph  ->  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9183, 90jca 535 . . 3  |-  ( ph  ->  ( <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) ) )
9291biantrurd 511 . 2  |-  ( ph  ->  ( E. x  e.  P  ( x (inA
`  G ) <" D E F ">  /\  <" A B C "> (cgrA `  G ) <" D E x "> )  <->  ( ( <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  ( x (inA
`  G ) <" D E F ">  /\  <" A B C "> (cgrA `  G ) <" D E x "> ) ) ) )
9373, 92bitr4d 260 1  |-  ( ph  ->  ( <" A B C "> ( `  G
) <" D E F ">  <->  E. x  e.  P  ( x
(inA `  G ) <" D E F ">  /\  <" A B C "> (cgrA `  G ) <" D E x "> ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   E.wrex 2738   _Vcvv 3045   class class class wbr 4402   {copab 4460    X. cxp 4832   ` cfv 5582  (class class class)co 6290    ^m cmap 7472   0cc0 9539   1c1 9540   2c2 10659   3c3 10660   NN0cn0 10869  ..^cfzo 11915   #chash 12515  Word cword 12656   <"cs3 12938   Basecbs 15121  TarskiGcstrkg 24478  cgrAccgra 24849  inAcinag 24876  ≤cleag 24877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-concat 12666  df-s1 12667  df-s2 12944  df-s3 12945  df-leag 24882
This theorem is referenced by: (None)
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