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Theorem isleag 24962
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 3040 . . . . 5  |-  ( G  e. TarskiG  ->  G  e.  _V )
3 fveq2 5879 . . . . . . . . . . . 12  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
4 isleag.p . . . . . . . . . . . 12  |-  P  =  ( Base `  G
)
53, 4syl6eqr 2523 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( Base `  g )  =  P )
65oveq1d 6323 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( Base `  g )  ^m  ( 0..^ 3 ) )  =  ( P  ^m  ( 0..^ 3 ) ) )
76eleq2d 2534 . . . . . . . . 9  |-  ( g  =  G  ->  (
a  e.  ( (
Base `  g )  ^m  ( 0..^ 3 ) )  <->  a  e.  ( P  ^m  ( 0..^ 3 ) ) ) )
86eleq2d 2534 . . . . . . . . 9  |-  ( g  =  G  ->  (
b  e.  ( (
Base `  g )  ^m  ( 0..^ 3 ) )  <->  b  e.  ( P  ^m  ( 0..^ 3 ) ) ) )
97, 8anbi12d 725 . . . . . . . 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 5879 . . . . . . . . . . 11  |-  ( g  =  G  ->  (inA `  g )  =  (inA
`  G ) )
1110breqd 4406 . . . . . . . . . 10  |-  ( g  =  G  ->  (
x (inA `  g
) <" ( b `
 0 ) ( b `  1 ) ( b `  2
) ">  <->  x (inA `  G ) <" (
b `  0 )
( b `  1
) ( b ` 
2 ) "> ) )
12 fveq2 5879 . . . . . . . . . . 11  |-  ( g  =  G  ->  (cgrA `  g )  =  (cgrA `  G ) )
1312breqd 4406 . . . . . . . . . 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 725 . . . . . . . . 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 2988 . . . . . . . 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 725 . . . . . . 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 4459 . . . . . 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 24961 . . . . . 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 6336 . . . . . . . 8  |-  ( P  ^m  ( 0..^ 3 ) )  e.  _V
20 xpexg 6612 . . . . . . . 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 686 . . . . . . 7  |-  ( ( P  ^m  ( 0..^ 3 ) )  X.  ( P  ^m  (
0..^ 3 ) ) )  e.  _V
22 opabssxp 4914 . . . . . . 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 4541 . . . . . 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 5963 . . . . 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 4406 . . 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 468 . . . . . . . . . 10  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  b  =  <" D E F "> )
2827fveq1d 5881 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( b `  0
)  =  ( <" D E F "> `  0
) )
2927fveq1d 5881 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( b `  1
)  =  ( <" D E F "> `  1
) )
3027fveq1d 5881 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( b `  2
)  =  ( <" D E F "> `  2
) )
3128, 29, 30s3eqd 13019 . . . . . . . 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 4407 . . . . . . 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 464 . . . . . . . . . 10  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  a  =  <" A B C "> )
3433fveq1d 5881 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( a `  0
)  =  ( <" A B C "> `  0
) )
3533fveq1d 5881 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( a `  1
)  =  ( <" A B C "> `  1
) )
3633fveq1d 5881 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( a `  2
)  =  ( <" A B C "> `  2
) )
3734, 35, 36s3eqd 13019 . . . . . . . 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 2472 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  x  =  x )
3928, 29, 38s3eqd 13019 . . . . . . . 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 4408 . . . . . . 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 725 . . . . . 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 2892 . . . . 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 2471 . . . . 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 4889 . . . 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 13045 . . . . . . . . 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 13046 . . . . . . . . 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 13047 . . . . . . . . 9  |-  ( F  e.  P  ->  ( <" D E F "> `  2
)  =  F )
5452, 53syl 17 . . . . . . . 8  |-  ( ph  ->  ( <" D E F "> `  2
)  =  F )
5548, 51, 54s3eqd 13019 . . . . . . 7  |-  ( ph  ->  <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) ( <" D E F "> `  2
) ">  =  <" D E F "> )
5655breq2d 4407 . . . . . 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 13045 . . . . . . . . 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 13046 . . . . . . . . 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 13047 . . . . . . . . 9  |-  ( C  e.  P  ->  ( <" A B C "> `  2
)  =  C )
6563, 64syl 17 . . . . . . . 8  |-  ( ph  ->  ( <" A B C "> `  2
)  =  C )
6659, 62, 65s3eqd 13019 . . . . . . 7  |-  ( ph  ->  <" ( <" A B C "> `  0
) ( <" A B C "> `  1
) ( <" A B C "> `  2
) ">  =  <" A B C "> )
67 eqidd 2472 . . . . . . . 8  |-  ( ph  ->  x  =  x )
6848, 51, 67s3eqd 13019 . . . . . . 7  |-  ( ph  ->  <" ( <" D E F "> `  0
) ( <" D E F "> `  1
) x ">  =  <" D E x "> )
6966, 68breq12d 4408 . . . . . 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 725 . . . . 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 2892 . . . 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 718 . . 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 287 . 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 13026 . . . . . 6  |-  ( ph  ->  <" A B C ">  e. Word  P )
75 s3len 13048 . . . . . . 7  |-  ( # `  <" A B C "> )  =  3
7675a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  <" A B C "> )  =  3
)
7774, 76jca 541 . . . . 5  |-  ( ph  ->  ( <" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
) )
78 fvex 5889 . . . . . . 7  |-  ( Base `  G )  e.  _V
794, 78eqeltri 2545 . . . . . 6  |-  P  e. 
_V
80 3nn0 10911 . . . . . 6  |-  3  e.  NN0
81 wrdmap 12749 . . . . . 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 686 . . . . 5  |-  ( (
<" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
)  <->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )
8377, 82sylib 201 . . . 4  |-  ( ph  ->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )
8446, 49, 52s3cld 13026 . . . . . 6  |-  ( ph  ->  <" D E F ">  e. Word  P )
85 s3len 13048 . . . . . . 7  |-  ( # `  <" D E F "> )  =  3
8685a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  <" D E F "> )  =  3
)
8784, 86jca 541 . . . . 5  |-  ( ph  ->  ( <" D E F ">  e. Word  P  /\  ( # `  <" D E F "> )  =  3
) )
88 wrdmap 12749 . . . . . 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 686 . . . . 5  |-  ( (
<" D E F ">  e. Word  P  /\  ( # `  <" D E F "> )  =  3
)  <->  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9087, 89sylib 201 . . . 4  |-  ( ph  ->  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9183, 90jca 541 . . 3  |-  ( ph  ->  ( <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) ) )
9291biantrurd 516 . 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 264 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031   class class class wbr 4395   {copab 4453    X. cxp 4837   ` cfv 5589  (class class class)co 6308    ^m cmap 7490   0cc0 9557   1c1 9558   2c2 10681   3c3 10682   NN0cn0 10893  ..^cfzo 11942   #chash 12553  Word cword 12703   <"cs3 12997   Basecbs 15199  TarskiGcstrkg 24557  cgrAccgra 24928  inAcinag 24955  ≤cleag 24956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003  df-s3 13004  df-leag 24961
This theorem is referenced by: (None)
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