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Theorem tgcgr4 24655
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 12023 . . . . 5  |-  ( 0..^ 4 )  C_  NN0
6 nn0ssre 10897 . . . . 5  |-  NN0  C_  RR
75, 6sstri 3427 . . . 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 13027 . . . . 5  |-  ( ph  ->  <" A B C D ">  e. Word  P )
14 wrdf 12723 . . . . 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 13053 . . . . . 6  |-  ( # `  <" A B C D "> )  =  4
1716oveq2i 6319 . . . . 5  |-  ( 0..^ ( # `  <" A B C D "> ) )  =  ( 0..^ 4 )
1817feq2i 5731 . . . 4  |-  ( <" A B C D "> :
( 0..^ ( # `  <" A B C D "> ) ) --> P  <->  <" A B C D "> : ( 0..^ 4 ) --> P )
1915, 18sylib 201 . . 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 13027 . . . . 5  |-  ( ph  ->  <" W X Y Z ">  e. Word  P )
25 wrdf 12723 . . . . 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 13053 . . . . . 6  |-  ( # `  <" W X Y Z "> )  =  4
2827oveq2i 6319 . . . . 5  |-  ( 0..^ ( # `  <" W X Y Z "> ) )  =  ( 0..^ 4 )
2928feq2i 5731 . . . 4  |-  ( <" W X Y Z "> :
( 0..^ ( # `  <" W X Y Z "> ) ) --> P  <->  <" W X Y Z "> : ( 0..^ 4 ) --> P )
3026, 29sylib 201 . . 3  |-  ( ph  ->  <" W X Y Z "> : ( 0..^ 4 ) --> P )
311, 2, 3, 4, 8, 19, 30iscgrglt 24638 . 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 5745 . . . . . . . 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 10758 . . . . . . . . 9  |-  ( 3  +  1 )  =  4
3534oveq2i 6319 . . . . . . . 8  |-  ( 0..^ ( 3  +  1 ) )  =  ( 0..^ 4 )
36 3nn0 10911 . . . . . . . . . 10  |-  3  e.  NN0
37 nn0uz 11217 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
3836, 37eleqtri 2547 . . . . . . . . 9  |-  3  e.  ( ZZ>= `  0 )
39 fzosplitsn 12048 . . . . . . . . 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 2495 . . . . . . 7  |-  ( 0..^ 4 )  =  ( ( 0..^ 3 )  u.  { 3 } )
4233, 41syl6eq 2521 . . . . . 6  |-  ( ph  ->  dom  <" A B C D ">  =  ( ( 0..^ 3 )  u.  {
3 } ) )
4342raleqdv 2979 . . . . 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 4399 . . . . . . . 8  |-  ( j  =  3  ->  (
i  <  j  <->  i  <  3 ) )
45 fveq2 5879 . . . . . . . . . 10  |-  ( j  =  3  ->  ( <" A B C D "> `  j
)  =  ( <" A B C D "> `  3
) )
4645oveq2d 6324 . . . . . . . . 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 5879 . . . . . . . . . 10  |-  ( j  =  3  ->  ( <" W X Y Z "> `  j
)  =  ( <" W X Y Z "> `  3
) )
4847oveq2d 6324 . . . . . . . . 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 2486 . . . . . . . 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 327 . . . . . . 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 4178 . . . . . 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 269 . . . 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 2829 . . 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 2979 . . . 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 12023 . . . . . . . . . . . . . . . 16  |-  ( 0..^ 3 )  C_  NN0
5756, 6sstri 3427 . . . . . . . . . . . . . . 15  |-  ( 0..^ 3 )  C_  RR
58 simpr 468 . . . . . . . . . . . . . . 15  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
j  e.  ( 0..^ 3 ) )
5957, 58sseldi 3416 . . . . . . . . . . . . . 14  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
j  e.  RR )
60 simpl 464 . . . . . . . . . . . . . . 15  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
i  =  3 )
616, 36sselii 3415 . . . . . . . . . . . . . . 15  |-  3  e.  RR
6260, 61syl6eqel 2557 . . . . . . . . . . . . . 14  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
i  e.  RR )
63 elfzolt2 11956 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( 0..^ 3 )  ->  j  <  3 )
6463adantl 473 . . . . . . . . . . . . . . 15  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
j  <  3 )
6564, 60breqtrrd 4422 . . . . . . . . . . . . . 14  |-  ( ( i  =  3  /\  j  e.  ( 0..^ 3 ) )  -> 
j  <  i )
66 ltnsym 9750 . . . . . . . . . . . . . . 15  |-  ( ( j  e.  RR  /\  i  e.  RR )  ->  ( j  <  i  ->  -.  i  <  j
) )
6766imp 436 . . . . . . . . . . . . . 14  |-  ( ( ( j  e.  RR  /\  i  e.  RR )  /\  j  <  i
)  ->  -.  i  <  j )
6859, 62, 65, 67syl21anc 1291 . . . . . . . . . . . . 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 1462 . . . . . . . . . . . 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 201 . . . . . . . . . . 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 2828 . . . . . . . . . 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 10791 . . . . . . . . . . . . 13  |-  3  e.  NN
74 lbfzo0 11983 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0..^ 3 )  <->  3  e.  NN )
7573, 74mpbir 214 . . . . . . . . . . . 12  |-  0  e.  ( 0..^ 3 )
7675ne0ii 3729 . . . . . . . . . . 11  |-  ( 0..^ 3 )  =/=  (/)
77 r19.3rzv 3853 . . . . . . . . . . 11  |-  ( ( 0..^ 3 )  =/=  (/)  ->  ( T.  <->  A. j  e.  ( 0..^ 3 ) T.  ) )
7876, 77ax-mp 5 . . . . . . . . . 10  |-  ( T.  <->  A. j  e.  (
0..^ 3 ) T.  )
7972, 78syl6bbr 271 . . . . . . . . 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 4398 . . . . . . . . . . . 12  |-  ( i  =  3  ->  (
i  <  3  <->  3  <  3 ) )
8161ltnri 9761 . . . . . . . . . . . . 13  |-  -.  3  <  3
8281bifal 1465 . . . . . . . . . . . 12  |-  ( 3  <  3  <-> F.  )
8380, 82syl6bb 269 . . . . . . . . . . 11  |-  ( i  =  3  ->  (
i  <  3  <-> F.  )
)
8483imbi1d 324 . . . . . . . . . 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 1466 . . . . . . . . . . 11  |-  ( F. 
->  ( ( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( ( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) ) )
8685bitru 1464 . . . . . . . . . 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 269 . . . . . . . . 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 725 . . . . . . . 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 656 . . . . . . . 8  |-  ( ( T.  /\ T.  )  <-> T.  )
9088, 89syl6bb 269 . . . . . . 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 4178 . . . . . 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 457 . . . . 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 1469 . . . . 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 279 . . . 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 269 . . 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 261 . 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 2904 . . 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 13026 . . . . . . . . 9  |-  ( ph  ->  <" A B C ">  e. Word  P )
100 wrdf 12723 . . . . . . . . 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 13048 . . . . . . . . . 10  |-  ( # `  <" A B C "> )  =  3
103102oveq2i 6319 . . . . . . . . 9  |-  ( 0..^ ( # `  <" A B C "> ) )  =  ( 0..^ 3 )
104103feq2i 5731 . . . . . . . 8  |-  ( <" A B C "> : ( 0..^ ( # `  <" A B C "> ) ) --> P  <->  <" A B C "> :
( 0..^ 3 ) --> P )
105101, 104sylib 201 . . . . . . 7  |-  ( ph  ->  <" A B C "> :
( 0..^ 3 ) --> P )
106 fdm 5745 . . . . . . 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 2973 . . . . . . 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 2987 . . . . 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 13026 . . . . . . . 8  |-  ( ph  ->  <" W X Y ">  e. Word  P )
113 wrdf 12723 . . . . . . . 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 13048 . . . . . . . . 9  |-  ( # `  <" W X Y "> )  =  3
116115oveq2i 6319 . . . . . . . 8  |-  ( 0..^ ( # `  <" W X Y "> ) )  =  ( 0..^ 3 )
117116feq2i 5731 . . . . . . 7  |-  ( <" W X Y "> : ( 0..^ ( # `  <" W X Y "> ) ) --> P  <->  <" W X Y "> :
( 0..^ 3 ) --> P )
118114, 117sylib 201 . . . . . 6  |-  ( ph  ->  <" W X Y "> :
( 0..^ 3 ) --> P )
1191, 2, 3, 4, 111, 105, 118iscgrglt 24638 . . . . 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 13005 . . . . . . . . . . 11  |-  <" A B C D ">  =  ( <" A B C "> ++  <" D "> )
121120fveq1i 5880 . . . . . . . . . 10  |-  ( <" A B C D "> `  i
)  =  ( (
<" A B C "> ++  <" D "> ) `  i
)
1229adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  A  e.  P )
12310adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  B  e.  P )
12411adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  C  e.  P )
125122, 123, 124s3cld 13026 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  <" A B C ">  e. Word  P )
12612adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  D  e.  P )
127126s1cld 12795 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  <" D ">  e. Word  P )
128 simprl 772 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  i  e.  ( 0..^ 3 ) )
129128, 103syl6eleqr 2560 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  i  e.  ( 0..^ ( # `  <" A B C "> ) ) )
130 ccatval1 12773 . . . . . . . . . . 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 1292 . . . . . . . . . 10  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( ( <" A B C "> ++  <" D "> ) `  i
)  =  ( <" A B C "> `  i
) )
132121, 131syl5eq 2517 . . . . . . . . 9  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( <" A B C D "> `  i
)  =  ( <" A B C "> `  i
) )
133120fveq1i 5880 . . . . . . . . . 10  |-  ( <" A B C D "> `  j
)  =  ( (
<" A B C "> ++  <" D "> ) `  j
)
134 simprr 774 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  j  e.  ( 0..^ 3 ) )
135134, 103syl6eleqr 2560 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  j  e.  ( 0..^ ( # `  <" A B C "> ) ) )
136 ccatval1 12773 . . . . . . . . . . 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 1292 . . . . . . . . . 10  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( ( <" A B C "> ++  <" D "> ) `  j
)  =  ( <" A B C "> `  j
) )
138133, 137syl5eq 2517 . . . . . . . . 9  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( <" A B C D "> `  j
)  =  ( <" A B C "> `  j
) )
139132, 138oveq12d 6326 . . . . . . . 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 13005 . . . . . . . . . . 11  |-  <" W X Y Z ">  =  ( <" W X Y "> ++  <" Z "> )
141140fveq1i 5880 . . . . . . . . . 10  |-  ( <" W X Y Z "> `  i
)  =  ( (
<" W X Y "> ++  <" Z "> ) `  i
)
14220adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  W  e.  P )
14321adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  X  e.  P )
14422adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  Y  e.  P )
145142, 143, 144s3cld 13026 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  <" W X Y ">  e. Word  P )
14623adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  Z  e.  P )
147146s1cld 12795 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  <" Z ">  e. Word  P )
148128, 116syl6eleqr 2560 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  i  e.  ( 0..^ ( # `  <" W X Y "> ) ) )
149 ccatval1 12773 . . . . . . . . . . 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 1292 . . . . . . . . . 10  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( ( <" W X Y "> ++  <" Z "> ) `  i
)  =  ( <" W X Y "> `  i
) )
151141, 150syl5eq 2517 . . . . . . . . 9  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( <" W X Y Z "> `  i
)  =  ( <" W X Y "> `  i
) )
152140fveq1i 5880 . . . . . . . . . 10  |-  ( <" W X Y Z "> `  j
)  =  ( (
<" W X Y "> ++  <" Z "> ) `  j
)
153134, 116syl6eleqr 2560 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  j  e.  ( 0..^ ( # `  <" W X Y "> ) ) )
154 ccatval1 12773 . . . . . . . . . . 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 1292 . . . . . . . . . 10  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( ( <" W X Y "> ++  <" Z "> ) `  j
)  =  ( <" W X Y "> `  j
) )
156152, 155syl5eq 2517 . . . . . . . . 9  |-  ( (
ph  /\  ( i  e.  ( 0..^ 3 )  /\  j  e.  ( 0..^ 3 ) ) )  ->  ( <" W X Y Z "> `  j
)  =  ( <" W X Y "> `  j
) )
157151, 156oveq12d 6326 . . . . . . . 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 2486 . . . . . . 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 323 . . . . . 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 2831 . . . . 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 294 . . . 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 12028 . . . . . 6  |-  ( 0..^ 3 )  =  {
0 ,  1 ,  2 }
163 raleq 2973 . . . . . 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 10725 . . . . . . . . . 10  |-  0  <  3
166 breq1 4398 . . . . . . . . . 10  |-  ( i  =  0  ->  (
i  <  3  <->  0  <  3 ) )
167165, 166mpbiri 241 . . . . . . . . 9  |-  ( i  =  0  ->  i  <  3 )
168167adantl 473 . . . . . . . 8  |-  ( (
ph  /\  i  = 
0 )  ->  i  <  3 )
169 biimt 342 . . . . . . . 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 468 . . . . . . . . . . 11  |-  ( (
ph  /\  i  = 
0 )  ->  i  =  0 )
172171fveq2d 5883 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" A B C D "> `  i
)  =  ( <" A B C D "> `  0
) )
173 s4fv0 13049 . . . . . . . . . . . 12  |-  ( A  e.  P  ->  ( <" A B C D "> `  0
)  =  A )
1749, 173syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( <" A B C D "> `  0 )  =  A )
175174adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" A B C D "> `  0
)  =  A )
176172, 175eqtrd 2505 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" A B C D "> `  i
)  =  A )
177 s4fv3 13052 . . . . . . . . . . 11  |-  ( D  e.  P  ->  ( <" A B C D "> `  3
)  =  D )
17812, 177syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( <" A B C D "> `  3 )  =  D )
179178adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" A B C D "> `  3
)  =  D )
180176, 179oveq12d 6326 . . . . . . . 8  |-  ( (
ph  /\  i  = 
0 )  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( A  .-  D ) )
181171fveq2d 5883 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" W X Y Z "> `  i
)  =  ( <" W X Y Z "> `  0
) )
182 s4fv0 13049 . . . . . . . . . . . 12  |-  ( W  e.  P  ->  ( <" W X Y Z "> `  0
)  =  W )
18320, 182syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( <" W X Y Z "> `  0 )  =  W )
184183adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" W X Y Z "> `  0
)  =  W )
185181, 184eqtrd 2505 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" W X Y Z "> `  i
)  =  W )
186 s4fv3 13052 . . . . . . . . . . 11  |-  ( Z  e.  P  ->  ( <" W X Y Z "> `  3
)  =  Z )
18723, 186syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( <" W X Y Z "> `  3 )  =  Z )
188187adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
0 )  ->  ( <" W X Y Z "> `  3
)  =  Z )
189185, 188oveq12d 6326 . . . . . . . 8  |-  ( (
ph  /\  i  = 
0 )  ->  (
( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  =  ( W  .-  Z ) )
190180, 189eqeq12d 2486 . . . . . . 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 263 . . . . . 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 10801 . . . . . . . . . 10  |-  1  <  3
193 breq1 4398 . . . . . . . . . 10  |-  ( i  =  1  ->  (
i  <  3  <->  1  <  3 ) )
194192, 193mpbiri 241 . . . . . . . . 9  |-  ( i  =  1  ->  i  <  3 )
195194adantl 473 . . . . . . . 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 468 . . . . . . . . . . 11  |-  ( (
ph  /\  i  = 
1 )  ->  i  =  1 )
198197fveq2d 5883 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" A B C D "> `  i
)  =  ( <" A B C D "> `  1
) )
199 s4fv1 13050 . . . . . . . . . . . 12  |-  ( B  e.  P  ->  ( <" A B C D "> `  1
)  =  B )
20010, 199syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( <" A B C D "> `  1 )  =  B )
201200adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" A B C D "> `  1
)  =  B )
202198, 201eqtrd 2505 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" A B C D "> `  i
)  =  B )
203178adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" A B C D "> `  3
)  =  D )
204202, 203oveq12d 6326 . . . . . . . 8  |-  ( (
ph  /\  i  = 
1 )  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( B  .-  D ) )
205197fveq2d 5883 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" W X Y Z "> `  i
)  =  ( <" W X Y Z "> `  1
) )
206 s4fv1 13050 . . . . . . . . . . . 12  |-  ( X  e.  P  ->  ( <" W X Y Z "> `  1
)  =  X )
20721, 206syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( <" W X Y Z "> `  1 )  =  X )
208207adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" W X Y Z "> `  1
)  =  X )
209205, 208eqtrd 2505 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" W X Y Z "> `  i
)  =  X )
210187adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
1 )  ->  ( <" W X Y Z "> `  3
)  =  Z )
211209, 210oveq12d 6326 . . . . . . . 8  |-  ( (
ph  /\  i  = 
1 )  ->  (
( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  =  ( X  .-  Z ) )
212204, 211eqeq12d 2486 . . . . . . 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 263 . . . . . 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 10800 . . . . . . . . . 10  |-  2  <  3
215 breq1 4398 . . . . . . . . . 10  |-  ( i  =  2  ->  (
i  <  3  <->  2  <  3 ) )
216214, 215mpbiri 241 . . . . . . . . 9  |-  ( i  =  2  ->  i  <  3 )
217216adantl 473 . . . . . . . 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 468 . . . . . . . . . . 11  |-  ( (
ph  /\  i  = 
2 )  ->  i  =  2 )
220219fveq2d 5883 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" A B C D "> `  i
)  =  ( <" A B C D "> `  2
) )
221 s4fv2 13051 . . . . . . . . . . . 12  |-  ( C  e.  P  ->  ( <" A B C D "> `  2
)  =  C )
22211, 221syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( <" A B C D "> `  2 )  =  C )
223222adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" A B C D "> `  2
)  =  C )
224220, 223eqtrd 2505 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" A B C D "> `  i
)  =  C )
225178adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" A B C D "> `  3
)  =  D )
226224, 225oveq12d 6326 . . . . . . . 8  |-  ( (
ph  /\  i  = 
2 )  ->  (
( <" A B C D "> `  i )  .-  ( <" A B C D "> `  3
) )  =  ( C  .-  D ) )
227219fveq2d 5883 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" W X Y Z "> `  i
)  =  ( <" W X Y Z "> `  2
) )
228 s4fv2 13051 . . . . . . . . . . . 12  |-  ( Y  e.  P  ->  ( <" W X Y Z "> `  2
)  =  Y )
22922, 228syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( <" W X Y Z "> `  2 )  =  Y )
230229adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" W X Y Z "> `  2
)  =  Y )
231227, 230eqtrd 2505 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" W X Y Z "> `  i
)  =  Y )
232187adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  i  = 
2 )  ->  ( <" W X Y Z "> `  3
)  =  Z )
233231, 232oveq12d 6326 . . . . . . . 8  |-  ( (
ph  /\  i  = 
2 )  ->  (
( <" W X Y Z "> `  i )  .-  ( <" W X Y Z "> `  3
) )  =  ( Y  .-  Z ) )
234226, 233eqeq12d 2486 . . . . . . 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 263 . . . . . 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 9662 . . . . . 6  |-  ( ph  ->  0  e.  RR )
237 1red 9676 . . . . . 6  |-  ( ph  ->  1  e.  RR )
238 2re 10701 . . . . . . 7  |-  2  e.  RR
239238a1i 11 . . . . . 6  |-  ( ph  ->  2  e.  RR )
240191, 213, 235, 236, 237, 239raltpd 4086 . . . . 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 261 . . . 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 725 . . 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 265 . 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 287 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452   T. wtru 1453   F. wfal 1457    e. wcel 1904    =/= wne 2641   A.wral 2756    u. cun 3388    C_ wss 3390   (/)c0 3722   {csn 3959   {ctp 3963   class class class wbr 4395   dom cdm 4839   -->wf 5585   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693   NNcn 10631   2c2 10681   3c3 10682   4c4 10683   NN0cn0 10893   ZZ>=cuz 11182  ..^cfzo 11942   #chash 12553  Word cword 12703   ++ cconcat 12705   <"cs1 12706   <"cs3 12997   <"cs4 12998   Basecbs 15199   distcds 15277  TarskiGcstrkg 24557  Itvcitv 24563  cgrGccgrg 24634
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-fal 1458  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-pm 7493  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-4 10692  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-s4 13005  df-trkgc 24575  df-trkgcb 24577  df-trkg 24580  df-cgrg 24635
This theorem is referenced by:  cgrg3col4  24963
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