Theorem tgcgr4 25426

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.

Step	Hyp	Ref	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 12548	. . . . 5 |- ( 0..^ 4 ) C_ NN0
6		nn0ssre 11296	. . . . 5 |- NN0 C_ RR
7	5, 6	sstri 3612	. . . 4 |- ( 0..^ 4 ) C_ RR
8	7	a1i 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 )
13	9, 10, 11, 12	s4cld 13618	. . . . 5 |- ( ph -> <" A B C D "> e. Word P )
14		wrdf 13310	. . . . 5 |- ( <" A B C D "> e. Word P -> <" A B C D "> : ( 0..^ ( # ` <" A B C D "> ) ) --> P )
15	13, 14	syl 17	. . . 4 |- ( ph -> <" A B C D "> : ( 0..^ ( # ` <" A B C D "> ) ) --> P )
16		s4len 13644	. . . . . 6 |- ( # ` <" A B C D "> ) = 4
17	16	oveq2i 6661	. . . . 5 |- ( 0..^ ( # ` <" A B C D "> ) ) = ( 0..^ 4 )
18	17	feq2i 6037	. . . 4 |- ( <" A B C D "> : ( 0..^ ( # ` <" A B C D "> ) ) --> P <-> <" A B C D "> : ( 0..^ 4 ) --> P )
19	15, 18	sylib 208	. . 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 )
24	20, 21, 22, 23	s4cld 13618	. . . . 5 |- ( ph -> <" W X Y Z "> e. Word P )
25		wrdf 13310	. . . . 5 |- ( <" W X Y Z "> e. Word P -> <" W X Y Z "> : ( 0..^ ( # ` <" W X Y Z "> ) ) --> P )
26	24, 25	syl 17	. . . 4 |- ( ph -> <" W X Y Z "> : ( 0..^ ( # ` <" W X Y Z "> ) ) --> P )
27		s4len 13644	. . . . . 6 |- ( # ` <" W X Y Z "> ) = 4
28	27	oveq2i 6661	. . . . 5 |- ( 0..^ ( # ` <" W X Y Z "> ) ) = ( 0..^ 4 )
29	28	feq2i 6037	. . . 4 |- ( <" W X Y Z "> : ( 0..^ ( # ` <" W X Y Z "> ) ) --> P <-> <" W X Y Z "> : ( 0..^ 4 ) --> P )
30	26, 29	sylib 208	. . 3 |- ( ph -> <" W X Y Z "> : ( 0..^ 4 ) --> P )
31	1, 2, 3, 4, 8, 19, 30	iscgrglt 25409	. 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 6051	. . . . . . . 8 |- ( <" A B C D "> : ( 0..^ 4 ) --> P -> dom <" A B C D "> = ( 0..^ 4 ) )
33	19, 32	syl 17	. . . . . . 7 |- ( ph -> dom <" A B C D "> = ( 0..^ 4 ) )
34		3p1e4 11153	. . . . . . . . 9 |- ( 3 + 1 ) = 4
35	34	oveq2i 6661	. . . . . . . 8 |- ( 0..^ ( 3 + 1 ) ) = ( 0..^ 4 )
36		3nn0 11310	. . . . . . . . . 10 |- 3 e. NN0
37		nn0uz 11722	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
38	36, 37	eleqtri 2699	. . . . . . . . 9 |- 3 e. ( ZZ>= ` 0 )
39		fzosplitsn 12576	. . . . . . . . 9 |- ( 3 e. ( ZZ>= ` 0 ) -> ( 0..^ ( 3 + 1 ) ) = ( ( 0..^ 3 ) u. { 3 } ) )
40	38, 39	ax-mp 5	. . . . . . . 8 |- ( 0..^ ( 3 + 1 ) ) = ( ( 0..^ 3 ) u. { 3 } )
41	35, 40	eqtr3i 2646	. . . . . . 7 |- ( 0..^ 4 ) = ( ( 0..^ 3 ) u. { 3 } )
42	33, 41	syl6eq 2672	. . . . . 6 |- ( ph -> dom <" A B C D "> = ( ( 0..^ 3 ) u. { 3 } ) )
43	42	raleqdv 3144	. . . . 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 4657	. . . . . . . 8 |- ( j = 3 -> ( i < j <-> i < 3 ) )
45		fveq2 6191	. . . . . . . . . 10 |- ( j = 3 -> ( <" A B C D "> ` j ) = ( <" A B C D "> ` 3 ) )
46	45	oveq2d 6666	. . . . . . . . 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 6191	. . . . . . . . . 10 |- ( j = 3 -> ( <" W X Y Z "> ` j ) = ( <" W X Y Z "> ` 3 ) )
48	47	oveq2d 6666	. . . . . . . . 9 |- ( j = 3 -> ( ( <" W X Y Z "> ` i ) .- ( <" W X Y Z "> ` j ) ) = ( ( <" W X Y Z "> ` i ) .- ( <" W X Y Z "> ` 3 ) ) )
49	46, 48	eqeq12d 2637	. . . . . . . 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 ) ) ) )
50	44, 49	imbi12d 334	. . . . . . 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 ) ) ) ) )
51	50	ralunsn 4422	. . . . . 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 ) ) ) ) ) )
52	36, 51	ax-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 ) ) ) ) )
53	43, 52	syl6bb 276	. . . 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 ) ) ) ) ) )
54	53	ralbidv 2986	. . 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 ) ) ) ) ) )
55	42	raleqdv 3144	. . . 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 12548	. . . . . . . . . . . . . . . 16 |- ( 0..^ 3 ) C_ NN0
57	56, 6	sstri 3612	. . . . . . . . . . . . . . 15 |- ( 0..^ 3 ) C_ RR
58		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( i = 3 /\ j e. ( 0..^ 3 ) ) -> j e. ( 0..^ 3 ) )
59	57, 58	sseldi 3601	. . . . . . . . . . . . . 14 |- ( ( i = 3 /\ j e. ( 0..^ 3 ) ) -> j e. RR )
60		simpl 473	. . . . . . . . . . . . . . 15 |- ( ( i = 3 /\ j e. ( 0..^ 3 ) ) -> i = 3 )
61	6, 36	sselii 3600	. . . . . . . . . . . . . . 15 |- 3 e. RR
62	60, 61	syl6eqel 2709	. . . . . . . . . . . . . 14 |- ( ( i = 3 /\ j e. ( 0..^ 3 ) ) -> i e. RR )
63		elfzolt2 12479	. . . . . . . . . . . . . . . 16 |- ( j e. ( 0..^ 3 ) -> j < 3 )
64	63	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( i = 3 /\ j e. ( 0..^ 3 ) ) -> j < 3 )
65	64, 60	breqtrrd 4681	. . . . . . . . . . . . . 14 |- ( ( i = 3 /\ j e. ( 0..^ 3 ) ) -> j < i )
66		ltnsym 10135	. . . . . . . . . . . . . . 15 |- ( ( j e. RR /\ i e. RR ) -> ( j < i -> -. i < j ) )
67	66	imp 445	. . . . . . . . . . . . . 14 |- ( ( ( j e. RR /\ i e. RR ) /\ j < i ) -> -. i < j )
68	59, 62, 65, 67	syl21anc 1325	. . . . . . . . . . . . 13 |- ( ( i = 3 /\ j e. ( 0..^ 3 ) ) -> -. i < j )
69	68	pm2.21d 118	. . . . . . . . . . . 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 1494	. . . . . . . . . . . 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. ) )
71	69, 70	sylib 208	. . . . . . . . . . 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. ) )
72	71	ralbidva 2985	. . . . . . . . . 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 11186	. . . . . . . . . . . . 13 |- 3 e. NN
74		lbfzo0 12507	. . . . . . . . . . . . 13 |- ( 0 e. ( 0..^ 3 ) <-> 3 e. NN )
75	73, 74	mpbir 221	. . . . . . . . . . . 12 |- 0 e. ( 0..^ 3 )
76	75	ne0ii 3923	. . . . . . . . . . 11 |- ( 0..^ 3 ) =/= (/)
77		r19.3rzv 4064	. . . . . . . . . . 11 |- ( ( 0..^ 3 ) =/= (/) -> ( T. <-> A. j e. ( 0..^ 3 ) T. ) )
78	76, 77	ax-mp 5	. . . . . . . . . 10 |- ( T. <-> A. j e. ( 0..^ 3 ) T. )
79	72, 78	syl6bbr 278	. . . . . . . . 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 4656	. . . . . . . . . . . 12 |- ( i = 3 -> ( i < 3 <-> 3 < 3 ) )
81	61	ltnri 10146	. . . . . . . . . . . . 13 |- -. 3 < 3
82	81	bifal 1497	. . . . . . . . . . . 12 |- ( 3 < 3 <-> F. )
83	80, 82	syl6bb 276	. . . . . . . . . . 11 |- ( i = 3 -> ( i < 3 <-> F. ) )
84	83	imbi1d 331	. . . . . . . . . 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 1498	. . . . . . . . . . 11 |- ( F. -> ( ( <" A B C D "> ` i ) .- ( <" A B C D "> ` 3 ) ) = ( ( <" W X Y Z "> ` i ) .- ( <" W X Y Z "> ` 3 ) ) )
86	85	bitru 1496	. . . . . . . . . 10 |- ( ( F. -> ( ( <" A B C D "> ` i ) .- ( <" A B C D "> ` 3 ) ) = ( ( <" W X Y Z "> ` i ) .- ( <" W X Y Z "> ` 3 ) ) ) <-> T. )
87	84, 86	syl6bb 276	. . . . . . . . 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. ) )
88	79, 87	anbi12d 747	. . . . . . . 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 676	. . . . . . . 8 |- ( ( T. /\ T. ) <-> T. )
90	88, 89	syl6bb 276	. . . . . . 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. ) )
91	90	ralunsn 4422	. . . . . 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. ) ) )
92	36, 91	ax-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 466	. . . . 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 1501	. . . . 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 ) ) ) ) )
95	92, 93, 94	3bitri 286	. . . 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 ) ) ) ) )
96	55, 95	syl6bb 276	. . 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 ) ) ) ) ) )
97	54, 96	bitrd 268	. 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 3064	. . 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 ) ) ) ) )
99	9, 10, 11	s3cld 13617	. . . . . . . . 9 |- ( ph -> <" A B C "> e. Word P )
100		wrdf 13310	. . . . . . . . 9 |- ( <" A B C "> e. Word P -> <" A B C "> : ( 0..^ ( # ` <" A B C "> ) ) --> P )
101	99, 100	syl 17	. . . . . . . 8 |- ( ph -> <" A B C "> : ( 0..^ ( # ` <" A B C "> ) ) --> P )
102		s3len 13639	. . . . . . . . . 10 |- ( # ` <" A B C "> ) = 3
103	102	oveq2i 6661	. . . . . . . . 9 |- ( 0..^ ( # ` <" A B C "> ) ) = ( 0..^ 3 )
104	103	feq2i 6037	. . . . . . . 8 |- ( <" A B C "> : ( 0..^ ( # ` <" A B C "> ) ) --> P <-> <" A B C "> : ( 0..^ 3 ) --> P )
105	101, 104	sylib 208	. . . . . . 7 |- ( ph -> <" A B C "> : ( 0..^ 3 ) --> P )
106		fdm 6051	. . . . . . 7 |- ( <" A B C "> : ( 0..^ 3 ) --> P -> dom <" A B C "> = ( 0..^ 3 ) )
107	105, 106	syl 17	. . . . . 6 |- ( ph -> dom <" A B C "> = ( 0..^ 3 ) )
108		raleq 3138	. . . . . . 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 ) ) ) ) )
109	105, 106, 108	3syl 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 ) ) ) ) )
110	107, 109	raleqbidv 3152	. . . . 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 ) ) ) ) )
111	57	a1i 11	. . . . . 6 |- ( ph -> ( 0..^ 3 ) C_ RR )
112	20, 21, 22	s3cld 13617	. . . . . . . 8 |- ( ph -> <" W X Y "> e. Word P )
113		wrdf 13310	. . . . . . . 8 |- ( <" W X Y "> e. Word P -> <" W X Y "> : ( 0..^ ( # ` <" W X Y "> ) ) --> P )
114	112, 113	syl 17	. . . . . . 7 |- ( ph -> <" W X Y "> : ( 0..^ ( # ` <" W X Y "> ) ) --> P )
115		s3len 13639	. . . . . . . . 9 |- ( # ` <" W X Y "> ) = 3
116	115	oveq2i 6661	. . . . . . . 8 |- ( 0..^ ( # ` <" W X Y "> ) ) = ( 0..^ 3 )
117	116	feq2i 6037	. . . . . . 7 |- ( <" W X Y "> : ( 0..^ ( # ` <" W X Y "> ) ) --> P <-> <" W X Y "> : ( 0..^ 3 ) --> P )
118	114, 117	sylib 208	. . . . . 6 |- ( ph -> <" W X Y "> : ( 0..^ 3 ) --> P )
119	1, 2, 3, 4, 111, 105, 118	iscgrglt 25409	. . . . 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 13595	. . . . . . . . . . 11 |- <" A B C D "> = ( <" A B C "> ++ <" D "> )
121	120	fveq1i 6192	. . . . . . . . . 10 |- ( <" A B C D "> ` i ) = ( ( <" A B C "> ++ <" D "> ) ` i )
122	9	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> A e. P )
123	10	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> B e. P )
124	11	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> C e. P )
125	122, 123, 124	s3cld 13617	. . . . . . . . . . 11 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> <" A B C "> e. Word P )
126	12	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> D e. P )
127	126	s1cld 13383	. . . . . . . . . . 11 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> <" D "> e. Word P )
128		simprl 794	. . . . . . . . . . . 12 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> i e. ( 0..^ 3 ) )
129	128, 103	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> i e. ( 0..^ ( # ` <" A B C "> ) ) )
130		ccatval1 13361	. . . . . . . . . . 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 ) )
131	125, 127, 129, 130	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> ( ( <" A B C "> ++ <" D "> ) ` i ) = ( <" A B C "> ` i ) )
132	121, 131	syl5eq 2668	. . . . . . . . 9 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> ( <" A B C D "> ` i ) = ( <" A B C "> ` i ) )
133	120	fveq1i 6192	. . . . . . . . . 10 |- ( <" A B C D "> ` j ) = ( ( <" A B C "> ++ <" D "> ) ` j )
134		simprr 796	. . . . . . . . . . . 12 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> j e. ( 0..^ 3 ) )
135	134, 103	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> j e. ( 0..^ ( # ` <" A B C "> ) ) )
136		ccatval1 13361	. . . . . . . . . . 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 ) )
137	125, 127, 135, 136	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> ( ( <" A B C "> ++ <" D "> ) ` j ) = ( <" A B C "> ` j ) )
138	133, 137	syl5eq 2668	. . . . . . . . 9 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> ( <" A B C D "> ` j ) = ( <" A B C "> ` j ) )
139	132, 138	oveq12d 6668	. . . . . . . 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 13595	. . . . . . . . . . 11 |- <" W X Y Z "> = ( <" W X Y "> ++ <" Z "> )
141	140	fveq1i 6192	. . . . . . . . . 10 |- ( <" W X Y Z "> ` i ) = ( ( <" W X Y "> ++ <" Z "> ) ` i )
142	20	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> W e. P )
143	21	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> X e. P )
144	22	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> Y e. P )
145	142, 143, 144	s3cld 13617	. . . . . . . . . . 11 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> <" W X Y "> e. Word P )
146	23	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> Z e. P )
147	146	s1cld 13383	. . . . . . . . . . 11 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> <" Z "> e. Word P )
148	128, 116	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> i e. ( 0..^ ( # ` <" W X Y "> ) ) )
149		ccatval1 13361	. . . . . . . . . . 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 ) )
150	145, 147, 148, 149	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> ( ( <" W X Y "> ++ <" Z "> ) ` i ) = ( <" W X Y "> ` i ) )
151	141, 150	syl5eq 2668	. . . . . . . . 9 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> ( <" W X Y Z "> ` i ) = ( <" W X Y "> ` i ) )
152	140	fveq1i 6192	. . . . . . . . . 10 |- ( <" W X Y Z "> ` j ) = ( ( <" W X Y "> ++ <" Z "> ) ` j )
153	134, 116	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> j e. ( 0..^ ( # ` <" W X Y "> ) ) )
154		ccatval1 13361	. . . . . . . . . . 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 ) )
155	145, 147, 153, 154	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> ( ( <" W X Y "> ++ <" Z "> ) ` j ) = ( <" W X Y "> ` j ) )
156	152, 155	syl5eq 2668	. . . . . . . . 9 |- ( ( ph /\ ( i e. ( 0..^ 3 ) /\ j e. ( 0..^ 3 ) ) ) -> ( <" W X Y Z "> ` j ) = ( <" W X Y "> ` j ) )
157	151, 156	oveq12d 6668	. . . . . . . 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 ) ) )
158	139, 157	eqeq12d 2637	. . . . . . 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 ) ) ) )
159	158	imbi2d 330	. . . . . 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 ) ) ) ) )
160	159	2ralbidva 2988	. . . . 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 ) ) ) ) )
161	110, 119, 160	3bitr4rd 301	. . . 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 12554	. . . . . 6 |- ( 0..^ 3 ) = { 0 , 1 , 2 }
163		raleq 3138	. . . . . 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 ) ) ) ) )
164	162, 163	mp1i 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 11114	. . . . . . . . . 10 |- 0 < 3
166		breq1 4656	. . . . . . . . . 10 |- ( i = 0 -> ( i < 3 <-> 0 < 3 ) )
167	165, 166	mpbiri 248	. . . . . . . . 9 |- ( i = 0 -> i < 3 )
168	167	adantl 482	. . . . . . . 8 |- ( ( ph /\ i = 0 ) -> i < 3 )
169		biimt 350	. . . . . . . 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 ) ) ) ) )
170	168, 169	syl 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 477	. . . . . . . . . . 11 |- ( ( ph /\ i = 0 ) -> i = 0 )
172	171	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ i = 0 ) -> ( <" A B C D "> ` i ) = ( <" A B C D "> ` 0 ) )
173		s4fv0 13640	. . . . . . . . . . . 12 |- ( A e. P -> ( <" A B C D "> ` 0 ) = A )
174	9, 173	syl 17	. . . . . . . . . . 11 |- ( ph -> ( <" A B C D "> ` 0 ) = A )
175	174	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ i = 0 ) -> ( <" A B C D "> ` 0 ) = A )
176	172, 175	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ i = 0 ) -> ( <" A B C D "> ` i ) = A )
177		s4fv3 13643	. . . . . . . . . . 11 |- ( D e. P -> ( <" A B C D "> ` 3 ) = D )
178	12, 177	syl 17	. . . . . . . . . 10 |- ( ph -> ( <" A B C D "> ` 3 ) = D )
179	178	adantr 481	. . . . . . . . 9 |- ( ( ph /\ i = 0 ) -> ( <" A B C D "> ` 3 ) = D )
180	176, 179	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ i = 0 ) -> ( ( <" A B C D "> ` i ) .- ( <" A B C D "> ` 3 ) ) = ( A .- D ) )
181	171	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ i = 0 ) -> ( <" W X Y Z "> ` i ) = ( <" W X Y Z "> ` 0 ) )
182		s4fv0 13640	. . . . . . . . . . . 12 |- ( W e. P -> ( <" W X Y Z "> ` 0 ) = W )
183	20, 182	syl 17	. . . . . . . . . . 11 |- ( ph -> ( <" W X Y Z "> ` 0 ) = W )
184	183	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ i = 0 ) -> ( <" W X Y Z "> ` 0 ) = W )
185	181, 184	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ i = 0 ) -> ( <" W X Y Z "> ` i ) = W )
186		s4fv3 13643	. . . . . . . . . . 11 |- ( Z e. P -> ( <" W X Y Z "> ` 3 ) = Z )
187	23, 186	syl 17	. . . . . . . . . 10 |- ( ph -> ( <" W X Y Z "> ` 3 ) = Z )
188	187	adantr 481	. . . . . . . . 9 |- ( ( ph /\ i = 0 ) -> ( <" W X Y Z "> ` 3 ) = Z )
189	185, 188	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ i = 0 ) -> ( ( <" W X Y Z "> ` i ) .- ( <" W X Y Z "> ` 3 ) ) = ( W .- Z ) )
190	180, 189	eqeq12d 2637	. . . . . . 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 ) ) )
191	170, 190	bitr3d 270	. . . . . 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 11196	. . . . . . . . . 10 |- 1 < 3
193		breq1 4656	. . . . . . . . . 10 |- ( i = 1 -> ( i < 3 <-> 1 < 3 ) )
194	192, 193	mpbiri 248	. . . . . . . . 9 |- ( i = 1 -> i < 3 )
195	194	adantl 482	. . . . . . . 8 |- ( ( ph /\ i = 1 ) -> i < 3 )
196	195, 169	syl 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 477	. . . . . . . . . . 11 |- ( ( ph /\ i = 1 ) -> i = 1 )
198	197	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ i = 1 ) -> ( <" A B C D "> ` i ) = ( <" A B C D "> ` 1 ) )
199		s4fv1 13641	. . . . . . . . . . . 12 |- ( B e. P -> ( <" A B C D "> ` 1 ) = B )
200	10, 199	syl 17	. . . . . . . . . . 11 |- ( ph -> ( <" A B C D "> ` 1 ) = B )
201	200	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ i = 1 ) -> ( <" A B C D "> ` 1 ) = B )
202	198, 201	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ i = 1 ) -> ( <" A B C D "> ` i ) = B )
203	178	adantr 481	. . . . . . . . 9 |- ( ( ph /\ i = 1 ) -> ( <" A B C D "> ` 3 ) = D )
204	202, 203	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ i = 1 ) -> ( ( <" A B C D "> ` i ) .- ( <" A B C D "> ` 3 ) ) = ( B .- D ) )
205	197	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ i = 1 ) -> ( <" W X Y Z "> ` i ) = ( <" W X Y Z "> ` 1 ) )
206		s4fv1 13641	. . . . . . . . . . . 12 |- ( X e. P -> ( <" W X Y Z "> ` 1 ) = X )
207	21, 206	syl 17	. . . . . . . . . . 11 |- ( ph -> ( <" W X Y Z "> ` 1 ) = X )
208	207	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ i = 1 ) -> ( <" W X Y Z "> ` 1 ) = X )
209	205, 208	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ i = 1 ) -> ( <" W X Y Z "> ` i ) = X )
210	187	adantr 481	. . . . . . . . 9 |- ( ( ph /\ i = 1 ) -> ( <" W X Y Z "> ` 3 ) = Z )
211	209, 210	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ i = 1 ) -> ( ( <" W X Y Z "> ` i ) .- ( <" W X Y Z "> ` 3 ) ) = ( X .- Z ) )
212	204, 211	eqeq12d 2637	. . . . . . 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 ) ) )
213	196, 212	bitr3d 270	. . . . . 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 11195	. . . . . . . . . 10 |- 2 < 3
215		breq1 4656	. . . . . . . . . 10 |- ( i = 2 -> ( i < 3 <-> 2 < 3 ) )
216	214, 215	mpbiri 248	. . . . . . . . 9 |- ( i = 2 -> i < 3 )
217	216	adantl 482	. . . . . . . 8 |- ( ( ph /\ i = 2 ) -> i < 3 )
218	217, 169	syl 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 477	. . . . . . . . . . 11 |- ( ( ph /\ i = 2 ) -> i = 2 )
220	219	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ i = 2 ) -> ( <" A B C D "> ` i ) = ( <" A B C D "> ` 2 ) )
221		s4fv2 13642	. . . . . . . . . . . 12 |- ( C e. P -> ( <" A B C D "> ` 2 ) = C )
222	11, 221	syl 17	. . . . . . . . . . 11 |- ( ph -> ( <" A B C D "> ` 2 ) = C )
223	222	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ i = 2 ) -> ( <" A B C D "> ` 2 ) = C )
224	220, 223	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ i = 2 ) -> ( <" A B C D "> ` i ) = C )
225	178	adantr 481	. . . . . . . . 9 |- ( ( ph /\ i = 2 ) -> ( <" A B C D "> ` 3 ) = D )
226	224, 225	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ i = 2 ) -> ( ( <" A B C D "> ` i ) .- ( <" A B C D "> ` 3 ) ) = ( C .- D ) )
227	219	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ i = 2 ) -> ( <" W X Y Z "> ` i ) = ( <" W X Y Z "> ` 2 ) )
228		s4fv2 13642	. . . . . . . . . . . 12 |- ( Y e. P -> ( <" W X Y Z "> ` 2 ) = Y )
229	22, 228	syl 17	. . . . . . . . . . 11 |- ( ph -> ( <" W X Y Z "> ` 2 ) = Y )
230	229	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ i = 2 ) -> ( <" W X Y Z "> ` 2 ) = Y )
231	227, 230	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ i = 2 ) -> ( <" W X Y Z "> ` i ) = Y )
232	187	adantr 481	. . . . . . . . 9 |- ( ( ph /\ i = 2 ) -> ( <" W X Y Z "> ` 3 ) = Z )
233	231, 232	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ i = 2 ) -> ( ( <" W X Y Z "> ` i ) .- ( <" W X Y Z "> ` 3 ) ) = ( Y .- Z ) )
234	226, 233	eqeq12d 2637	. . . . . . 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 ) ) )
235	218, 234	bitr3d 270	. . . . . 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 10041	. . . . . 6 |- ( ph -> 0 e. RR )
237		1red 10055	. . . . . 6 |- ( ph -> 1 e. RR )
238		2re 11090	. . . . . . 7 |- 2 e. RR
239	238	a1i 11	. . . . . 6 |- ( ph -> 2 e. RR )
240	191, 213, 235, 236, 237, 239	raltpd 4315	. . . . 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 ) ) ) )
241	164, 240	bitrd 268	. . . 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 ) ) ) )
242	161, 241	anbi12d 747	. . 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 ) ) ) ) )
243	98, 242	syl5bb 272	. 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 ) ) ) ) )
244	31, 97, 243	3bitrd 294	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 ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   F. wfal 1488   e. wcel 1990   =/= wne 2794   A.wral 2912   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   {ctp 4181   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   NN0cn0 11292   ZZ>=cuz 11687  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294   <"cs3 13587   <"cs4 13588   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  cgrGccgrg 25405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-s4 13595  df-trkgc 25347  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406

This theorem is referenced by:  cgrg3col4  25734