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Theorem tgcgr4 24574
Description: Two quadrilaterals to be congruent to each other if one triangle formed by their vertices is, and the additional points are equidistant too. (Contributed by Thierry Arnoux, 8-Oct-2020.)
Hypotheses
Ref Expression
tgcgrxfr.p  |-  P  =  ( Base `  G
)
tgcgrxfr.m  |-  .-  =  ( dist `  G )
tgcgrxfr.i  |-  I  =  (Itv `  G )
tgcgrxfr.r  |-  .~  =  (cgrG `  G )
tgcgrxfr.g  |-  ( ph  ->  G  e. TarskiG )
tgcgr4.a  |-  ( ph  ->  A  e.  P )
tgcgr4.b  |-  ( ph  ->  B  e.  P )
tgcgr4.c  |-  ( ph  ->  C  e.  P )
tgcgr4.d  |-  ( ph  ->  D  e.  P )
tgcgr4.w  |-  ( ph  ->  W  e.  P )
tgcgr4.x  |-  ( ph  ->  X  e.  P )
tgcgr4.y  |-  ( ph  ->  Y  e.  P )
tgcgr4.z  |-  ( ph  ->  Z  e.  P )
Assertion
Ref Expression
tgcgr4  |-  ( ph  ->  ( <" A B C D ">  .~ 
<" W X Y Z ">  <->  ( <" A B C ">  .~  <" W X Y ">  /\  (
( A  .-  D
)  =  ( W 
.-  Z )  /\  ( B  .-  D )  =  ( X  .-  Z )  /\  ( C  .-  D )  =  ( Y  .-  Z
) ) ) ) )

Proof of Theorem tgcgr4
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgcgrxfr.p . . 3  |-  P  =  ( Base `  G
)
2 tgcgrxfr.m . . 3  |-  .-  =  ( dist `  G )
3 tgcgrxfr.r . . 3  |-  .~  =  (cgrG `  G )
4 tgcgrxfr.g . . 3  |-  ( ph  ->  G  e. TarskiG )
5 fzo0ssnn0 12000 . . . . 5  |-  ( 0..^ 4 )  C_  NN0
6 nn0ssre 10880 . . . . 5  |-  NN0  C_  RR
75, 6sstri 3473 . . . 4  |-  ( 0..^ 4 )  C_  RR
87a1i 11 . . 3  |-  ( ph  ->  ( 0..^ 4 ) 
C_  RR )
9 tgcgr4.a . . . . . 6  |-  ( ph  ->  A  e.  P )
10 tgcgr4.b . . . . . 6  |-  ( ph  ->  B  e.  P )
11 tgcgr4.c . . . . . 6  |-  ( ph  ->  C  e.  P )
12 tgcgr4.d . . . . . 6  |-  ( ph  ->  D  e.  P )
139, 10, 11, 12s4cld 12969 . . . . 5  |-  ( ph  ->  <" A B C D ">  e. Word  P )
14 wrdf 12680 . . . . 5  |-  ( <" A B C D ">  e. Word  P  ->  <" A B C D "> : ( 0..^ ( # `  <" A B C D "> ) ) --> P )
1513, 14syl 17 . . . 4  |-  ( ph  ->  <" A B C D "> : ( 0..^ ( # `  <" A B C D "> ) ) --> P )
16 s4len 12994 . . . . . 6  |-  ( # `  <" A B C D "> )  =  4
1716oveq2i 6316 . . . . 5  |-  ( 0..^ ( # `  <" A B C D "> ) )  =  ( 0..^ 4 )
1817feq2i 5739 . . . 4  |-  ( <" A B C D "> :
( 0..^ ( # `  <" A B C D "> ) ) --> P  <->  <" A B C D "> : ( 0..^ 4 ) --> P )
1915, 18sylib 199 . . 3  |-  ( ph  ->  <" A B C D "> : ( 0..^ 4 ) --> P )
20 tgcgr4.w . . . . . 6  |-  ( ph  ->  W  e.  P )
21 tgcgr4.x . . . . . 6  |-  ( ph  ->  X  e.  P )
22 tgcgr4.y . . . . . 6  |-  ( ph  ->  Y  e.  P )
23 tgcgr4.z . . . . . 6  |-  ( ph  ->  Z  e.  P )
2420, 21, 22, 23s4cld 12969 . . . . 5  |-  ( ph  ->  <" W X Y Z ">  e. Word  P )
25 wrdf 12680 . . . . 5  |-  ( <" W X Y Z ">  e. Word  P  ->  <" W X Y Z "> : ( 0..^ ( # `  <" W X Y Z "> ) ) --> P )
2624, 25syl 17 . . . 4  |-  ( ph  ->  <" W X Y Z "> : ( 0..^ ( # `  <" W X Y Z "> ) ) --> P )
27 s4len 12994 . . . . . 6  |-  ( # `  <" W X Y Z "> )  =  4
2827oveq2i 6316 . . . . 5  |-  ( 0..^ ( # `  <" W X Y Z "> ) )  =  ( 0..^ 4 )
2928feq2i 5739 . . . 4  |-  ( <" W X Y Z "> :
( 0..^ ( # `  <" W X Y Z "> ) ) --> P  <->  <" W X Y Z "> : ( 0..^ 4 ) --> P )
3026, 29sylib 199 . . 3  |-  ( ph  ->  <" W X Y Z "> : ( 0..^ 4 ) --> P )
311, 2, 3, 4, 8, 19, 30iscgrglt 24557 . 2  |-  ( ph  ->  ( <" A B C D ">  .~ 
<" W X Y Z ">  <->  A. i  e.  dom  <" A B C D "> A. j  e.  dom  <" A B C D "> ( i  <  j  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) ) ) )
32 fdm 5750 . . . . . . . 8  |-  ( <" A B C D "> :
( 0..^ 4 ) --> P  ->  dom  <" A B C D ">  =  ( 0..^ 4 ) )
3319, 32syl 17 . . . . . . 7  |-  ( ph  ->  dom  <" A B C D ">  =  ( 0..^ 4 ) )
34 3p1e4 10742 . . . . . . . . 9  |-  ( 3  +  1 )  =  4
3534oveq2i 6316 . . . . . . . 8  |-  ( 0..^ ( 3  +  1 ) )  =  ( 0..^ 4 )
36 3nn0 10894 . . . . . . . . . 10  |-  3  e.  NN0
37 nn0uz 11200 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
3836, 37eleqtri 2505 . . . . . . . . 9  |-  3  e.  ( ZZ>= `  0 )
39 fzosplitsn 12023 . . . . . . . . 9  |-  ( 3  e.  ( ZZ>= `  0
)  ->  ( 0..^ ( 3  +  1 ) )  =  ( ( 0..^ 3 )  u.  { 3 } ) )
4038, 39ax-mp 5 . . . . . . . 8  |-  ( 0..^ ( 3  +  1 ) )  =  ( ( 0..^ 3 )  u.  { 3 } )
4135, 40eqtr3i 2453 . . . . . . 7  |-  ( 0..^ 4 )  =  ( ( 0..^ 3 )  u.  { 3 } )
4233, 41syl6eq 2479 . . . . . 6  |-  ( ph  ->  dom  <" A B C D ">  =  ( ( 0..^ 3 )  u.  {
3 } ) )
4342raleqdv 3028 . . . . 5  |-  ( ph  ->  ( A. j  e. 
dom  <" A B C D "> ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <->  A. j  e.  ( ( 0..^ 3 )  u.  { 3 } ) ( i  <  j  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) ) ) )
44 breq2 4427 . . . . . . . 8  |-  ( j  =  3  ->  (
i  <  j  <->  i  <  3 ) )
45 fveq2 5881 . . . . . . . . . 10  |-  ( j  =  3  ->  ( <" A B C D "> `  j
)  =  ( <" A B C D "> `  3
) )
4645oveq2d 6321 . . . . . . . . 9  |-  ( j  =  3  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) ) )
47 fveq2 5881 . . . . . . . . . 10  |-  ( j  =  3  ->  ( <" W X Y Z "> `  j
)  =  ( <" W X Y Z "> `  3
) )
4847oveq2d 6321 . . . . . . . . 9  |-  ( j  =  3  ->  (
( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )
4946, 48eqeq12d 2444 . . . . . . . 8  |-  ( j  =  3  ->  (
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) )  <->  ( ( <" A B C D "> `  i
)  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )
5044, 49imbi12d 321 . . . . . . 7  |-  ( j  =  3  ->  (
( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <->  ( i  <  3  ->  ( ( <" A B C D "> `  i
)  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )
5150ralunsn 4207 . . . . . 6  |-  ( 3  e.  NN0  ->  ( A. j  e.  ( (
0..^ 3 )  u. 
{ 3 } ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <->  ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) ) )
5236, 51ax-mp 5 . . . . 5  |-  ( A. j  e.  ( (
0..^ 3 )  u. 
{ 3 } ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <->  ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )
5343, 52syl6bb 264 . . . 4  |-  ( ph  ->  ( A. j  e. 
dom  <" A B C D "> ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <->  ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) ) )
5453ralbidv 2861 . . 3  |-  ( ph  ->  ( A. i  e. 
dom  <" A B C D "> A. j  e.  dom  <" A B C D "> ( i  <  j  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <->  A. i  e.  dom  <" A B C D "> ( A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) ) )
5542raleqdv 3028 . . . 4  |-  ( ph  ->  ( A. i  e. 
dom  <" A B C D "> ( A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  <->  A. i  e.  (
( 0..^ 3 )  u.  { 3 } ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) ) )
56 fzo0ssnn0 12000 . . . . . . . . . . . . . . . 16  |-  ( 0..^ 3 )  C_  NN0
5756, 6sstri 3473 . . . . . . . . . . . . . . 15  |-  ( 0..^ 3 )  C_  RR
58 simpr 462 . . . . . . . . . . . . . . 15  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
j  e.  ( 0..^ 3 ) )
5957, 58sseldi 3462 . . . . . . . . . . . . . 14  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
j  e.  RR )
60 simpl 458 . . . . . . . . . . . . . . 15  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
i  =  3 )
616, 36sselii 3461 . . . . . . . . . . . . . . 15  |-  3  e.  RR
6260, 61syl6eqel 2515 . . . . . . . . . . . . . 14  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
i  e.  RR )
63 elfzolt2 11936 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( 0..^ 3 )  ->  j  <  3 )
6463adantl 467 . . . . . . . . . . . . . . 15  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
j  <  3 )
6564, 60breqtrrd 4450 . . . . . . . . . . . . . 14  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
j  <  i )
66 ltnsym 9739 . . . . . . . . . . . . . . 15  |-  ( ( j  e.  RR  /\  i  e.  RR )  ->  ( j  <  i  ->  -.  i  <  j
) )
6766imp 430 . . . . . . . . . . . . . 14  |-  ( ( ( j  e.  RR  /\  i  e.  RR )  /\  j  <  i
)  ->  -.  i  <  j )
6859, 62, 65, 67syl21anc 1263 . . . . . . . . . . . . 13  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  ->  -.  i  <  j )
6968pm2.21d 109 . . . . . . . . . . . 12  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) ) )
70 tbtru 1447 . . . . . . . . . . . 12  |-  ( ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <->  ( (
i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <-> T.  )
)
7169, 70sylib 199 . . . . . . . . . . 11  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
( ( i  < 
j  ->  ( ( <" A B C D "> `  i
)  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <-> T.  )
)
7271ralbidva 2858 . . . . . . . . . 10  |-  ( i  =  3  ->  ( A. j  e.  (
0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <->  A. j  e.  ( 0..^ 3 ) T.  ) )
73 3nn 10775 . . . . . . . . . . . . 13  |-  3  e.  NN
74 lbfzo0 11962 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0..^ 3 )  <->  3  e.  NN )
7573, 74mpbir 212 . . . . . . . . . . . 12  |-  0  e.  ( 0..^ 3 )
7675ne0ii 3768 . . . . . . . . . . 11  |-  ( 0..^ 3 )  =/=  (/)
77 r19.3rzv 3892 . . . . . . . . . . 11  |-  ( ( 0..^ 3 )  =/=  (/)  ->  ( T.  <->  A. j  e.  ( 0..^ 3 ) T.  ) )
7876, 77ax-mp 5 . . . . . . . . . 10  |-  ( T.  <->  A. j  e.  (
0..^ 3 ) T.  )
7972, 78syl6bbr 266 . . . . . . . . 9  |-  ( i  =  3  ->  ( A. j  e.  (
0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <-> T.  )
)
80 breq1 4426 . . . . . . . . . . . 12  |-  ( i  =  3  ->  (
i  <  3  <->  3  <  3 ) )
8161ltnri 9750 . . . . . . . . . . . . 13  |-  -.  3  <  3
8281bifal 1450 . . . . . . . . . . . 12  |-  ( 3  <  3  <-> F.  )
8380, 82syl6bb 264 . . . . . . . . . . 11  |-  ( i  =  3  ->  (
i  <  3  <-> F.  )
)
8483imbi1d 318 . . . . . . . . . 10  |-  ( i  =  3  ->  (
( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )  <->  ( F.  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )
85 falim 1451 . . . . . . . . . . 11  |-  ( F. 
->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )
8685bitru 1449 . . . . . . . . . 10  |-  ( ( F.  ->  ( ( <" A B C D "> `  i
)  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )  <-> T.  )
8784, 86syl6bb 264 . . . . . . . . 9  |-  ( i  =  3  ->  (
( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )  <-> T.  )
)
8879, 87anbi12d 715 . . . . . . . 8  |-  ( i  =  3  ->  (
( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  <-> 
( T.  /\ T.  ) ) )
89 anidm 648 . . . . . . . 8  |-  ( ( T.  /\ T.  )  <-> T.  )
9088, 89syl6bb 264 . . . . . . 7  |-  ( i  =  3  ->  (
( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  <-> T.  ) )
9190ralunsn 4207 . . . . . 6  |-  ( 3  e.  NN0  ->  ( A. i  e.  ( (
0..^ 3 )  u. 
{ 3 } ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  <-> 
( A. i  e.  ( 0..^ 3 ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  /\ T.  ) ) )
9236, 91ax-mp 5 . . . . 5  |-  ( A. i  e.  ( (
0..^ 3 )  u. 
{ 3 } ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  <-> 
( A. i  e.  ( 0..^ 3 ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  /\ T.  ) )
93 ancom 451 . . . . 5  |-  ( ( A. i  e.  ( 0..^ 3 ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  /\ T.  )  <->  ( T.  /\  A. i  e.  ( 0..^ 3 ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) ) )
94 truan 1454 . . . . 5  |-  ( ( T.  /\  A. i  e.  ( 0..^ 3 ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )  <->  A. i  e.  ( 0..^ 3 ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )
9592, 93, 943bitri 274 . . . 4  |-  ( A. i  e.  ( (
0..^ 3 )  u. 
{ 3 } ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  <->  A. i  e.  (
0..^ 3 ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )
9655, 95syl6bb 264 . . 3  |-  ( ph  ->  ( A. i  e. 
dom  <" A B C D "> ( A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  <->  A. i  e.  (
0..^ 3 ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) ) )
9754, 96bitrd 256 . 2  |-  ( ph  ->  ( A. i  e. 
dom  <" A B C D "> A. j  e.  dom  <" A B C D "> ( i  <  j  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <->  A. i  e.  ( 0..^ 3 ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) ) )
98 r19.26 2952 . . 3  |-  ( A. i  e.  ( 0..^ 3 ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  <-> 
( A. i  e.  ( 0..^ 3 ) A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  A. i  e.  ( 0..^ 3 ) ( i  <  3  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )
999, 10, 11s3cld 12968 . . . . . . . . 9  |-  ( ph  ->  <" A B C ">  e. Word  P )
100 wrdf 12680 . . . . . . . . 9  |-  ( <" A B C ">  e. Word  P  ->  <" A B C "> :
( 0..^ ( # `  <" A B C "> )
) --> P )
10199, 100syl 17 . . . . . . . 8  |-  ( ph  ->  <" A B C "> :
( 0..^ ( # `  <" A B C "> )
) --> P )
102 s3len 12989 . . . . . . . . . 10  |-  ( # `  <" A B C "> )  =  3
103102oveq2i 6316 . . . . . . . . 9  |-  ( 0..^ ( # `  <" A B C "> ) )  =  ( 0..^ 3 )
104103feq2i 5739 . . . . . . . 8  |-  ( <" A B C "> : ( 0..^ ( # `  <" A B C "> ) ) --> P  <->  <" A B C "> :
( 0..^ 3 ) --> P )
105101, 104sylib 199 . . . . . . 7  |-  ( ph  ->  <" A B C "> :
( 0..^ 3 ) --> P )
106 fdm 5750 . . . . . . 7  |-  ( <" A B C "> : ( 0..^ 3 ) --> P  ->  dom  <" A B C ">  =  ( 0..^ 3 ) )
107105, 106syl 17 . . . . . 6  |-  ( ph  ->  dom  <" A B C ">  =  ( 0..^ 3 ) )
108 raleq 3022 . . . . . . 7  |-  ( dom 
<" A B C ">  =  ( 0..^ 3 )  -> 
( A. j  e. 
dom  <" A B C "> (
i  <  j  ->  ( ( <" A B C "> `  i
)  .-  ( <" A B C "> `  j ) )  =  ( ( <" W X Y "> `  i
)  .-  ( <" W X Y "> `  j ) ) )  <->  A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C "> `  i
)  .-  ( <" A B C "> `  j ) )  =  ( ( <" W X Y "> `  i
)  .-  ( <" W X Y "> `  j ) ) ) ) )
109105, 106, 1083syl 18 . . . . . 6  |-  ( ph  ->  ( A. j  e. 
dom  <" A B C "> (
i  <  j  ->  ( ( <" A B C "> `  i
)  .-  ( <" A B C "> `  j ) )  =  ( ( <" W X Y "> `  i
)  .-  ( <" W X Y "> `  j ) ) )  <->  A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C "> `  i
)  .-  ( <" A B C "> `  j ) )  =  ( ( <" W X Y "> `  i
)  .-  ( <" W X Y "> `  j ) ) ) ) )
110107, 109raleqbidv 3036 . . . . 5  |-  ( ph  ->  ( A. i  e. 
dom  <" A B C "> A. j  e.  dom  <" A B C "> (
i  <  j  ->  ( ( <" A B C "> `  i
)  .-  ( <" A B C "> `  j ) )  =  ( ( <" W X Y "> `  i
)  .-  ( <" W X Y "> `  j ) ) )  <->  A. i  e.  ( 0..^ 3 ) A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  (
( <" A B C "> `  i
)  .-  ( <" A B C "> `  j ) )  =  ( ( <" W X Y "> `  i
)  .-  ( <" W X Y "> `  j ) ) ) ) )
11157a1i 11 . . . . . 6  |-  ( ph  ->  ( 0..^ 3 ) 
C_  RR )
11220, 21, 22s3cld 12968 . . . . . . . 8  |-  ( ph  ->  <" W X Y ">  e. Word  P )
113 wrdf 12680 . . . . . . . 8  |-  ( <" W X Y ">  e. Word  P  ->  <" W X Y "> :
( 0..^ ( # `  <" W X Y "> )
) --> P )
114112, 113syl 17 . . . . . . 7  |-  ( ph  ->  <" W X Y "> :
( 0..^ ( # `  <" W X Y "> )
) --> P )
115 s3len 12989 . . . . . . . . 9  |-  ( # `  <" W X Y "> )  =  3
116115oveq2i 6316 . . . . . . . 8  |-  ( 0..^ ( # `  <" W X Y "> ) )  =  ( 0..^ 3 )
117116feq2i 5739 . . . . . . 7  |-  ( <" W X Y "> : ( 0..^ ( # `  <" W X Y "> ) ) --> P  <->  <" W X Y "> :
( 0..^ 3 ) --> P )
118114, 117sylib 199 . . . . . 6  |-  ( ph  ->  <" W X Y "> :
( 0..^ 3 ) --> P )
1191, 2, 3, 4, 111, 105, 118iscgrglt 24557 . . . . 5  |-  ( ph  ->  ( <" A B C ">  .~  <" W X Y "> 
<-> 
A. i  e.  dom  <" A B C "> A. j  e.  dom  <" A B C "> (
i  <  j  ->  ( ( <" A B C "> `  i
)  .-  ( <" A B C "> `  j ) )  =  ( ( <" W X Y "> `  i
)  .-  ( <" W X Y "> `  j ) ) ) ) )
120 df-s4 12948 . . . . . . . . . . 11  |-  <" A B C D ">  =  ( <" A B C "> ++  <" D "> )
121120fveq1i 5882 . . . . . . . . . 10  |-  ( <" A B C D "> `  i
)  =  ( (
<" A B C "> ++  <" D "> ) `  i
)
1229adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  A  e.  P )
12310adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  B  e.  P )
12411adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  C  e.  P )
125122, 123, 124s3cld 12968 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  <" A B C ">  e. Word  P )
12612adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  D  e.  P )
127126s1cld 12746 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  <" D ">  e. Word  P )
128 simprl 762 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  i  e.  ( 0..^ 3 ) )
129128, 103syl6eleqr 2518 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  i  e.  ( 0..^ ( # `  <" A B C "> ) ) )
130 ccatval1 12726 . . . . . . . . . . 11  |-  ( (
<" A B C ">  e. Word  P  /\  <" D ">  e. Word  P  /\  i  e.  ( 0..^ ( # `  <" A B C "> )
) )  ->  (
( <" A B C "> ++  <" D "> ) `  i
)  =  ( <" A B C "> `  i
) )
131125, 127, 129, 130syl3anc 1264 . . . . . . . . . 10  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( ( <" A B C "> ++  <" D "> ) `  i
)  =  ( <" A B C "> `  i
) )
132121, 131syl5eq 2475 . . . . . . . . 9  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( <" A B C D "> `  i
)  =  ( <" A B C "> `  i
) )
133120fveq1i 5882 . . . . . . . . . 10  |-  ( <" A B C D "> `  j
)  =  ( (
<" A B C "> ++  <" D "> ) `  j
)
134 simprr 764 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  j  e.  ( 0..^ 3 ) )
135134, 103syl6eleqr 2518 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  j  e.  ( 0..^ ( # `  <" A B C "> ) ) )
136 ccatval1 12726 . . . . . . . . . . 11  |-  ( (
<" A B C ">  e. Word  P  /\  <" D ">  e. Word  P  /\  j  e.  ( 0..^ ( # `  <" A B C "> )
) )  ->  (
( <" A B C "> ++  <" D "> ) `  j
)  =  ( <" A B C "> `  j
) )
137125, 127, 135, 136syl3anc 1264 . . . . . . . . . 10  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( ( <" A B C "> ++  <" D "> ) `  j
)  =  ( <" A B C "> `  j
) )
138133, 137syl5eq 2475 . . . . . . . . 9  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( <" A B C D "> `  j
)  =  ( <" A B C "> `  j
) )
139132, 138oveq12d 6323 . . . . . . . 8  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( ( <" A B C D "> `  i
)  .-  ( <" A B C D "> `  j
) )  =  ( ( <" A B C "> `  i
)  .-  ( <" A B C "> `  j ) ) )
140 df-s4 12948 . . . . . . . . . . 11  |-  <" W X Y Z ">  =  ( <" W X Y "> ++  <" Z "> )
141140fveq1i 5882 . . . . . . . . . 10  |-  ( <" W X Y Z "> `  i
)  =  ( (
<" W X Y "> ++  <" Z "> ) `  i
)
14220adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  W  e.  P )
14321adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  X  e.  P )
14422adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  Y  e.  P )
145142, 143, 144s3cld 12968 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  <" W X Y ">  e. Word  P )
14623adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  Z  e.  P )
147146s1cld 12746 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  <" Z ">  e. Word  P )
148128, 116syl6eleqr 2518 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  i  e.  ( 0..^ ( # `  <" W X Y "> ) ) )
149 ccatval1 12726 . . . . . . . . . . 11  |-  ( (
<" W X Y ">  e. Word  P  /\  <" Z ">  e. Word  P  /\  i  e.  ( 0..^ ( # `  <" W X Y "> )
) )  ->  (
( <" W X Y "> ++  <" Z "> ) `  i
)  =  ( <" W X Y "> `  i
) )
150145, 147, 148, 149syl3anc 1264 . . . . . . . . . 10  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( ( <" W X Y "> ++  <" Z "> ) `  i
)  =  ( <" W X Y "> `  i
) )
151141, 150syl5eq 2475 . . . . . . . . 9  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( <" W X Y Z "> `  i
)  =  ( <" W X Y "> `  i
) )
152140fveq1i 5882 . . . . . . . . . 10  |-  ( <" W X Y Z "> `  j
)  =  ( (
<" W X Y "> ++  <" Z "> ) `  j
)
153134, 116syl6eleqr 2518 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  j  e.  ( 0..^ ( # `  <" W X Y "> ) ) )
154 ccatval1 12726 . . . . . . . . . . 11  |-  ( (
<" W X Y ">  e. Word  P  /\  <" Z ">  e. Word  P  /\  j  e.  ( 0..^ ( # `  <" W X Y "> )
) )  ->  (
( <" W X Y "> ++  <" Z "> ) `  j
)  =  ( <" W X Y "> `  j
) )
155145, 147, 153, 154syl3anc 1264 . . . . . . . . . 10  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( ( <" W X Y "> ++  <" Z "> ) `  j
)  =  ( <" W X Y "> `  j
) )
156152, 155syl5eq 2475 . . . . . . . . 9  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( <" W X Y Z "> `  j
)  =  ( <" W X Y "> `  j
) )
157151, 156oveq12d 6323 . . . . . . . 8  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( ( <" W X Y Z "> `  i
)  .-  ( <" W X Y Z "> `  j
) )  =  ( ( <" W X Y "> `  i
)  .-  ( <" W X Y "> `  j ) ) )
158139, 157eqeq12d 2444 . . . . . . 7  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) )  <->  ( ( <" A B C "> `  i
)  .-  ( <" A B C "> `  j ) )  =  ( ( <" W X Y "> `  i
)  .-  ( <" W X Y "> `  j ) ) ) )
159158imbi2d 317 . . . . . 6  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( (
i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <->  ( i  <  j  ->  ( ( <" A B C "> `  i
)  .-  ( <" A B C "> `  j ) )  =  ( ( <" W X Y "> `  i
)  .-  ( <" W X Y "> `  j ) ) ) ) )
1601592ralbidva 2864 . . . . 5  |-  ( ph  ->  ( A. i  e.  ( 0..^ 3 ) A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <->  A. i  e.  ( 0..^ 3 ) A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C "> `  i
)  .-  ( <" A B C "> `  j ) )  =  ( ( <" W X Y "> `  i
)  .-  ( <" W X Y "> `  j ) ) ) ) )
161110, 119, 1603bitr4rd 289 . . . 4  |-  ( ph  ->  ( A. i  e.  ( 0..^ 3 ) A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  <->  <" A B C ">  .~  <" W X Y "> ) )
162 fzo0to3tp 12005 . . . . . 6  |-  ( 0..^ 3 )  =  {
0 ,  1 ,  2 }
163 raleq 3022 . . . . . 6  |-  ( ( 0..^ 3 )  =  { 0 ,  1 ,  2 }  ->  ( A. i  e.  ( 0..^ 3 ) ( i  <  3  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )  <->  A. i  e.  { 0 ,  1 ,  2 }  (
i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )
164162, 163mp1i 13 . . . . 5  |-  ( ph  ->  ( A. i  e.  ( 0..^ 3 ) ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )  <->  A. i  e.  { 0 ,  1 ,  2 }  (
i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )
165 3pos 10710 . . . . . . . . . 10  |-  0  <  3
166 breq1 4426 . . . . . . . . . 10  |-  ( i  =  0  ->  (
i  <  3  <->  0  <  3 ) )
167165, 166mpbiri 236 . . . . . . . . 9  |-  ( i  =  0  ->  i  <  3 )
168167adantl 467 . . . . . . . 8  |-  ( (
ph  /\  i  = 
0 )  ->  i  <  3 )
169 biimt 336 . . . . . . . 8  |-  ( i  <  3  ->  (
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  <->  ( i  <  3  ->  ( ( <" A B C D "> `  i
)  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )
170168, 169syl 17 . . . . . . 7  |-  ( (
ph  /\  i  = 
0 )  ->  (
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  <->  ( i  <  3  ->  ( ( <" A B C D "> `  i
)  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )
171 simpr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  i  = 
0 )  ->  i  =  0 )
172171fveq2d 5885 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" A B C D "> `  i
)  =  ( <" A B C D "> `  0
) )
173 s4fv0 12990 . . . . . . . . . . . 12  |-  ( A  e.  P  ->  ( <" A B C D "> `  0
)  =  A )
1749, 173syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( <" A B C D "> `  0 )  =  A )
175174adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" A B C D "> `  0
)  =  A )
176172, 175eqtrd 2463 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" A B C D "> `  i
)  =  A )
177 s4fv3 12993 . . . . . . . . . . 11  |-  ( D  e.  P  ->  ( <" A B C D "> `  3
)  =  D )
17812, 177syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( <" A B C D "> `  3 )  =  D )
179178adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" A B C D "> `  3
)  =  D )
180176, 179oveq12d 6323 . . . . . . . 8  |-  ( (
ph  /\  i  = 
0 )  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( A  .-  D ) )
181171fveq2d 5885 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" W X Y Z "> `  i
)  =  ( <" W X Y Z "> `  0
) )
182 s4fv0 12990 . . . . . . . . . . . 12  |-  ( W  e.  P  ->  ( <" W X Y Z "> `  0
)  =  W )
18320, 182syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( <" W X Y Z "> `  0 )  =  W )
184183adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" W X Y Z "> `  0
)  =  W )
185181, 184eqtrd 2463 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" W X Y Z "> `  i
)  =  W )
186 s4fv3 12993 . . . . . . . . . . 11  |-  ( Z  e.  P  ->  ( <" W X Y Z "> `  3
)  =  Z )
18723, 186syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( <" W X Y Z "> `  3 )  =  Z )
188187adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" W X Y Z "> `  3
)  =  Z )
189185, 188oveq12d 6323 . . . . . . . 8  |-  ( (
ph  /\  i  = 
0 )  ->  (
( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  =  ( W  .-  Z ) )
190180, 189eqeq12d 2444 . . . . . . 7  |-  ( (
ph  /\  i  = 
0 )  ->  (
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  <->  ( A  .-  D )  =  ( W  .-  Z ) ) )
191170, 190bitr3d 258 . . . . . 6  |-  ( (
ph  /\  i  = 
0 )  ->  (
( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )  <->  ( A  .-  D )  =  ( W  .-  Z ) ) )
192 1lt3 10785 . . . . . . . . . 10  |-  1  <  3
193 breq1 4426 . . . . . . . . . 10  |-  ( i  =  1  ->  (
i  <  3  <->  1  <  3 ) )
194192, 193mpbiri 236 . . . . . . . . 9  |-  ( i  =  1  ->  i  <  3 )
195194adantl 467 . . . . . . . 8  |-  ( (
ph  /\  i  = 
1 )  ->  i  <  3 )
196195, 169syl 17 . . . . . . 7  |-  ( (
ph  /\  i  = 
1 )  ->  (
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  <->  ( i  <  3  ->  ( ( <" A B C D "> `  i
)  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )
197 simpr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  i  = 
1 )  ->  i  =  1 )
198197fveq2d 5885 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" A B C D "> `  i
)  =  ( <" A B C D "> `  1
) )
199 s4fv1 12991 . . . . . . . . . . . 12  |-  ( B  e.  P  ->  ( <" A B C D "> `  1
)  =  B )
20010, 199syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( <" A B C D "> `  1 )  =  B )
201200adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" A B C D "> `  1
)  =  B )
202198, 201eqtrd 2463 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" A B C D "> `  i
)  =  B )
203178adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" A B C D "> `  3
)  =  D )
204202, 203oveq12d 6323 . . . . . . . 8  |-  ( (
ph  /\  i  = 
1 )  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( B  .-  D ) )
205197fveq2d 5885 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" W X Y Z "> `  i
)  =  ( <" W X Y Z "> `  1
) )
206 s4fv1 12991 . . . . . . . . . . . 12  |-  ( X  e.  P  ->  ( <" W X Y Z "> `  1
)  =  X )
20721, 206syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( <" W X Y Z "> `  1 )  =  X )
208207adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" W X Y Z "> `  1
)  =  X )
209205, 208eqtrd 2463 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" W X Y Z "> `  i
)  =  X )
210187adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" W X Y Z "> `  3
)  =  Z )
211209, 210oveq12d 6323 . . . . . . . 8  |-  ( (
ph  /\  i  = 
1 )  ->  (
( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  =  ( X  .-  Z ) )
212204, 211eqeq12d 2444 . . . . . . 7  |-  ( (
ph  /\  i  = 
1 )  ->  (
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  <->  ( B  .-  D )  =  ( X  .-  Z ) ) )
213196, 212bitr3d 258 . . . . . 6  |-  ( (
ph  /\  i  = 
1 )  ->  (
( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )  <->  ( B  .-  D )  =  ( X  .-  Z ) ) )
214 2lt3 10784 . . . . . . . . . 10  |-  2  <  3
215 breq1 4426 . . . . . . . . . 10  |-  ( i  =  2  ->  (
i  <  3  <->  2  <  3 ) )
216214, 215mpbiri 236 . . . . . . . . 9  |-  ( i  =  2  ->  i  <  3 )
217216adantl 467 . . . . . . . 8  |-  ( (
ph  /\  i  = 
2 )  ->  i  <  3 )
218217, 169syl 17 . . . . . . 7  |-  ( (
ph  /\  i  = 
2 )  ->  (
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  <->  ( i  <  3  ->  ( ( <" A B C D "> `  i
)  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) ) )
219 simpr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  i  = 
2 )  ->  i  =  2 )
220219fveq2d 5885 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" A B C D "> `  i
)  =  ( <" A B C D "> `  2
) )
221 s4fv2 12992 . . . . . . . . . . . 12  |-  ( C  e.  P  ->  ( <" A B C D "> `  2
)  =  C )
22211, 221syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( <" A B C D "> `  2 )  =  C )
223222adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" A B C D "> `  2
)  =  C )
224220, 223eqtrd 2463 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" A B C D "> `  i
)  =  C )
225178adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" A B C D "> `  3
)  =  D )
226224, 225oveq12d 6323 . . . . . . . 8  |-  ( (
ph  /\  i  = 
2 )  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( C  .-  D ) )
227219fveq2d 5885 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" W X Y Z "> `  i
)  =  ( <" W X Y Z "> `  2
) )
228 s4fv2 12992 . . . . . . . . . . . 12  |-  ( Y  e.  P  ->  ( <" W X Y Z "> `  2
)  =  Y )
22922, 228syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( <" W X Y Z "> `  2 )  =  Y )
230229adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" W X Y Z "> `  2
)  =  Y )
231227, 230eqtrd 2463 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" W X Y Z "> `  i
)  =  Y )
232187adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" W X Y Z "> `  3
)  =  Z )
233231, 232oveq12d 6323 . . . . . . . 8  |-  ( (
ph  /\  i  = 
2 )  ->  (
( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  =  ( Y  .-  Z ) )
234226, 233eqeq12d 2444 . . . . . . 7  |-  ( (
ph  /\  i  = 
2 )  ->  (
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  <->  ( C  .-  D )  =  ( Y  .-  Z ) ) )
235218, 234bitr3d 258 . . . . . 6  |-  ( (
ph  /\  i  = 
2 )  ->  (
( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )  <->  ( C  .-  D )  =  ( Y  .-  Z ) ) )
236 0red 9651 . . . . . 6  |-  ( ph  ->  0  e.  RR )
237 1red 9665 . . . . . 6  |-  ( ph  ->  1  e.  RR )
238 2re 10686 . . . . . . 7  |-  2  e.  RR
239238a1i 11 . . . . . 6  |-  ( ph  ->  2  e.  RR )
240191, 213, 235, 236, 237, 239raltpd 4123 . . . . 5  |-  ( ph  ->  ( A. i  e. 
{ 0 ,  1 ,  2 }  (
i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )  <->  ( ( A  .-  D )  =  ( W  .-  Z
)  /\  ( B  .-  D )  =  ( X  .-  Z )  /\  ( C  .-  D )  =  ( Y  .-  Z ) ) ) )
241164, 240bitrd 256 . . . 4  |-  ( ph  ->  ( A. i  e.  ( 0..^ 3 ) ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )  <->  ( ( A  .-  D )  =  ( W  .-  Z
)  /\  ( B  .-  D )  =  ( X  .-  Z )  /\  ( C  .-  D )  =  ( Y  .-  Z ) ) ) )
242161, 241anbi12d 715 . . 3  |-  ( ph  ->  ( ( A. i  e.  ( 0..^ 3 ) A. j  e.  ( 0..^ 3 ) ( i  <  j  -> 
( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  A. i  e.  ( 0..^ 3 ) ( i  <  3  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  <-> 
( <" A B C ">  .~  <" W X Y ">  /\  ( ( A 
.-  D )  =  ( W  .-  Z
)  /\  ( B  .-  D )  =  ( X  .-  Z )  /\  ( C  .-  D )  =  ( Y  .-  Z ) ) ) ) )
24398, 242syl5bb 260 . 2  |-  ( ph  ->  ( A. i  e.  ( 0..^ 3 ) ( A. j  e.  ( 0..^ 3 ) ( i  <  j  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  j
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  j
) ) )  /\  ( i  <  3  ->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) ) )  <-> 
( <" A B C ">  .~  <" W X Y ">  /\  ( ( A 
.-  D )  =  ( W  .-  Z
)  /\  ( B  .-  D )  =  ( X  .-  Z )  /\  ( C  .-  D )  =  ( Y  .-  Z ) ) ) ) )
24431, 97, 2433bitrd 282 1  |-  ( ph  ->  ( <" A B C D ">  .~ 
<" W X Y Z ">  <->  ( <" A B C ">  .~  <" W X Y ">  /\  (
( A  .-  D
)  =  ( W 
.-  Z )  /\  ( B  .-  D )  =  ( X  .-  Z )  /\  ( C  .-  D )  =  ( Y  .-  Z
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   T. wtru 1438   F. wfal 1442    e. wcel 1872    =/= wne 2614   A.wral 2771    u. cun 3434    C_ wss 3436   (/)c0 3761   {csn 3998   {ctp 4002   class class class wbr 4423   dom cdm 4853   -->wf 5597   ` cfv 5601  (class class class)co 6305   RRcr 9545   0cc0 9546   1c1 9547    + caddc 9549    < clt 9682   NNcn 10616   2c2 10666   3c3 10667   4c4 10668   NN0cn0 10876   ZZ>=cuz 11166  ..^cfzo 11922   #chash 12521  Word cword 12660   ++ cconcat 12662   <"cs1 12663   <"cs3 12940   <"cs4 12941   Basecbs 15120   distcds 15198  TarskiGcstrkg 24476  Itvcitv 24482  cgrGccgrg 24553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12522  df-word 12668  df-concat 12670  df-s1 12671  df-s2 12946  df-s3 12947  df-s4 12948  df-trkgc 24494  df-trkgcb 24496  df-trkg 24499  df-cgrg 24554
This theorem is referenced by:  cgrg3col4  24882
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