Theorem iscgra 25701

Description: Property for two angles ABC and DEF to be congruent. This is a modified version of the definition 11.3 of [Schwabhauser] p. 95. where the number of constructed points has been reduced to two. We chose this version rather than the textbook version because it is shorter and therefore simpler to work with. Theorem dfcgra2 25721 shows that those definitions are indeed equivalent. (Contributed by Thierry Arnoux, 31-Jul-2020.)

Hypotheses

Ref	Expression
iscgra.p	|- P = ( Base ` G )
iscgra.i	|- I = (Itv ` G )
iscgra.k	|- K = (hlG ` G )
iscgra.g	|- ( ph -> G e. TarskiG )
iscgra.a	|- ( ph -> A e. P )
iscgra.b	|- ( ph -> B e. P )
iscgra.c	|- ( ph -> C e. P )
iscgra.d	|- ( ph -> D e. P )
iscgra.e	|- ( ph -> E e. P )
iscgra.f	|- ( ph -> F e. P )

Assertion

Ref	Expression
iscgra	|- ( ph -> ( <" A B C "> (cgrA ` G ) <" D E F "> <-> E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) )

Distinct variable groups:   x, A, y   x, B, y   x, C, y   x, D, y   x, E, y   x, F, y   x, K, y   ph, x, y   x, G, y   x, I, y   x, P, y

Proof of Theorem iscgra

Dummy variables a b g k p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . . . 8 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> a = <" A B C "> )
2		eqidd 2623	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> x = x )
3		simpr 477	. . . . . . . . . 10 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> b = <" D E F "> )
4	3	fveq1d 6193	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( b ` 1 ) = ( <" D E F "> ` 1 ) )
5		eqidd 2623	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> y = y )
6	2, 4, 5	s3eqd 13609	. . . . . . . 8 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> <" x ( b ` 1 ) y "> = <" x ( <" D E F "> ` 1 ) y "> )
7	1, 6	breq12d 4666	. . . . . . 7 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( a (cgrG ` G ) <" x ( b ` 1 ) y "> <-> <" A B C "> (cgrG ` G ) <" x ( <" D E F "> ` 1 ) y "> ) )
8	4	fveq2d 6195	. . . . . . . 8 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( K ` ( b ` 1 ) ) = ( K ` ( <" D E F "> ` 1 ) ) )
9	3	fveq1d 6193	. . . . . . . 8 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( b ` 0 ) = ( <" D E F "> ` 0 ) )
10	2, 8, 9	breq123d 4667	. . . . . . 7 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( x ( K ` ( b ` 1 ) ) ( b ` 0 ) <-> x ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 0 ) ) )
11	3	fveq1d 6193	. . . . . . . 8 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( b ` 2 ) = ( <" D E F "> ` 2 ) )
12	5, 8, 11	breq123d 4667	. . . . . . 7 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( y ( K ` ( b ` 1 ) ) ( b ` 2 ) <-> y ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 2 ) ) )
13	7, 10, 12	3anbi123d 1399	. . . . . 6 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) <-> ( <" A B C "> (cgrG ` G ) <" x ( <" D E F "> ` 1 ) y "> /\ x ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 0 ) /\ y ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 2 ) ) ) )
14	13	2rexbidv 3057	. . . . 5 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) <-> E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x ( <" D E F "> ` 1 ) y "> /\ x ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 0 ) /\ y ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 2 ) ) ) )
15		eqid 2622	. . . . 5 |- { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } = { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) }
16	14, 15	brab2a 5194	. . . 4 |- ( <" A B C "> { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } <" D E F "> <-> ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x ( <" D E F "> ` 1 ) y "> /\ x ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 0 ) /\ y ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 2 ) ) ) )
17	16	a1i 11	. . 3 |- ( ph -> ( <" A B C "> { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } <" D E F "> <-> ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x ( <" D E F "> ` 1 ) y "> /\ x ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 0 ) /\ y ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 2 ) ) ) ) )
18		eqidd 2623	. . . . . . . 8 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> x = x )
19		iscgra.e	. . . . . . . . . 10 |- ( ph -> E e. P )
20		s3fv1 13637	. . . . . . . . . 10 |- ( E e. P -> ( <" D E F "> ` 1 ) = E )
21	19, 20	syl 17	. . . . . . . . 9 |- ( ph -> ( <" D E F "> ` 1 ) = E )
22	21	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( <" D E F "> ` 1 ) = E )
23		eqidd 2623	. . . . . . . 8 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> y = y )
24	18, 22, 23	s3eqd 13609	. . . . . . 7 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> <" x ( <" D E F "> ` 1 ) y "> = <" x E y "> )
25	24	breq2d 4665	. . . . . 6 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( <" A B C "> (cgrG ` G ) <" x ( <" D E F "> ` 1 ) y "> <-> <" A B C "> (cgrG ` G ) <" x E y "> ) )
26	22	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( K ` ( <" D E F "> ` 1 ) ) = ( K ` E ) )
27		iscgra.d	. . . . . . . . 9 |- ( ph -> D e. P )
28		s3fv0 13636	. . . . . . . . 9 |- ( D e. P -> ( <" D E F "> ` 0 ) = D )
29	27, 28	syl 17	. . . . . . . 8 |- ( ph -> ( <" D E F "> ` 0 ) = D )
30	29	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( <" D E F "> ` 0 ) = D )
31	18, 26, 30	breq123d 4667	. . . . . 6 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( x ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 0 ) <-> x ( K ` E ) D ) )
32		iscgra.f	. . . . . . . . 9 |- ( ph -> F e. P )
33		s3fv2 13638	. . . . . . . . 9 |- ( F e. P -> ( <" D E F "> ` 2 ) = F )
34	32, 33	syl 17	. . . . . . . 8 |- ( ph -> ( <" D E F "> ` 2 ) = F )
35	34	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( <" D E F "> ` 2 ) = F )
36	23, 26, 35	breq123d 4667	. . . . . 6 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( y ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 2 ) <-> y ( K ` E ) F ) )
37	25, 31, 36	3anbi123d 1399	. . . . 5 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( ( <" A B C "> (cgrG ` G ) <" x ( <" D E F "> ` 1 ) y "> /\ x ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 0 ) /\ y ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 2 ) ) <-> ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) )
38	37	2rexbidva 3056	. . . 4 |- ( ph -> ( E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x ( <" D E F "> ` 1 ) y "> /\ x ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 0 ) /\ y ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 2 ) ) <-> E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) )
39	38	anbi2d 740	. . 3 |- ( ph -> ( ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x ( <" D E F "> ` 1 ) y "> /\ x ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 0 ) /\ y ( K ` ( <" D E F "> ` 1 ) ) ( <" D E F "> ` 2 ) ) ) <-> ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) ) )
40	17, 39	bitrd 268	. 2 |- ( ph -> ( <" A B C "> { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } <" D E F "> <-> ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) ) )
41		iscgra.g	. . . 4 |- ( ph -> G e. TarskiG )
42		elex 3212	. . . 4 |- ( G e. TarskiG -> G e. _V )
43		iscgra.p	. . . . . . . 8 |- P = ( Base ` G )
44		iscgra.k	. . . . . . . 8 |- K = (hlG ` G )
45		simpl 473	. . . . . . . . . . . . 13 |- ( ( p = P /\ k = K ) -> p = P )
46	45	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( p = P /\ k = K ) -> P = p )
47	46	oveq1d 6665	. . . . . . . . . . 11 |- ( ( p = P /\ k = K ) -> ( P ^m ( 0..^ 3 ) ) = ( p ^m ( 0..^ 3 ) ) )
48	47	eleq2d 2687	. . . . . . . . . 10 |- ( ( p = P /\ k = K ) -> ( a e. ( P ^m ( 0..^ 3 ) ) <-> a e. ( p ^m ( 0..^ 3 ) ) ) )
49	47	eleq2d 2687	. . . . . . . . . 10 |- ( ( p = P /\ k = K ) -> ( b e. ( P ^m ( 0..^ 3 ) ) <-> b e. ( p ^m ( 0..^ 3 ) ) ) )
50	48, 49	anbi12d 747	. . . . . . . . 9 |- ( ( p = P /\ k = K ) -> ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) <-> ( a e. ( p ^m ( 0..^ 3 ) ) /\ b e. ( p ^m ( 0..^ 3 ) ) ) ) )
51		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( p = P /\ k = K ) -> k = K )
52		eqidd 2623	. . . . . . . . . . . . . . 15 |- ( ( p = P /\ k = K ) -> ( b ` 1 ) = ( b ` 1 ) )
53	51, 52	fveq12d 6197	. . . . . . . . . . . . . 14 |- ( ( p = P /\ k = K ) -> ( k ` ( b ` 1 ) ) = ( K ` ( b ` 1 ) ) )
54	53	breqd 4664	. . . . . . . . . . . . 13 |- ( ( p = P /\ k = K ) -> ( x ( k ` ( b ` 1 ) ) ( b ` 0 ) <-> x ( K ` ( b ` 1 ) ) ( b ` 0 ) ) )
55	53	breqd 4664	. . . . . . . . . . . . 13 |- ( ( p = P /\ k = K ) -> ( y ( k ` ( b ` 1 ) ) ( b ` 2 ) <-> y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) )
56	54, 55	3anbi23d 1402	. . . . . . . . . . . 12 |- ( ( p = P /\ k = K ) -> ( ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( k ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( k ` ( b ` 1 ) ) ( b ` 2 ) ) <-> ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) )
57	56	bicomd 213	. . . . . . . . . . 11 |- ( ( p = P /\ k = K ) -> ( ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) <-> ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( k ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( k ` ( b ` 1 ) ) ( b ` 2 ) ) ) )
58	46, 57	rexeqbidv 3153	. . . . . . . . . 10 |- ( ( p = P /\ k = K ) -> ( E. y e. P ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) <-> E. y e. p ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( k ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( k ` ( b ` 1 ) ) ( b ` 2 ) ) ) )
59	46, 58	rexeqbidv 3153	. . . . . . . . 9 |- ( ( p = P /\ k = K ) -> ( E. x e. P E. y e. P ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) <-> E. x e. p E. y e. p ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( k ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( k ` ( b ` 1 ) ) ( b ` 2 ) ) ) )
60	50, 59	anbi12d 747	. . . . . . . 8 |- ( ( p = P /\ k = K ) -> ( ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) <-> ( ( a e. ( p ^m ( 0..^ 3 ) ) /\ b e. ( p ^m ( 0..^ 3 ) ) ) /\ E. x e. p E. y e. p ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( k ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( k ` ( b ` 1 ) ) ( b ` 2 ) ) ) ) )
61	43, 44, 60	sbcie2s 15916	. . . . . . 7 |- ( g = G -> ( [. ( Base ` g ) / p ]. [. (hlG ` g ) / k ]. ( ( a e. ( p ^m ( 0..^ 3 ) ) /\ b e. ( p ^m ( 0..^ 3 ) ) ) /\ E. x e. p E. y e. p ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( k ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( k ` ( b ` 1 ) ) ( b ` 2 ) ) ) <-> ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) ) )
62	61	opabbidv 4716	. . . . . 6 |- ( g = G -> { <. a , b >. | [. ( Base ` g ) / p ]. [. (hlG ` g ) / k ]. ( ( a e. ( p ^m ( 0..^ 3 ) ) /\ b e. ( p ^m ( 0..^ 3 ) ) ) /\ E. x e. p E. y e. p ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( k ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( k ` ( b ` 1 ) ) ( b ` 2 ) ) ) } = { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } )
63		fveq2 6191	. . . . . . . . . . . 12 |- ( g = G -> (cgrG ` g ) = (cgrG ` G ) )
64	63	breqd 4664	. . . . . . . . . . 11 |- ( g = G -> ( a (cgrG ` g ) <" x ( b ` 1 ) y "> <-> a (cgrG ` G ) <" x ( b ` 1 ) y "> ) )
65	64	3anbi1d 1403	. . . . . . . . . 10 |- ( g = G -> ( ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) <-> ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) )
66	65	rexbidv 3052	. . . . . . . . 9 |- ( g = G -> ( E. y e. P ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) <-> E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) )
67	66	rexbidv 3052	. . . . . . . 8 |- ( g = G -> ( E. x e. P E. y e. P ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) <-> E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) )
68	67	anbi2d 740	. . . . . . 7 |- ( g = G -> ( ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) <-> ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) ) )
69	68	opabbidv 4716	. . . . . 6 |- ( g = G -> { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } = { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } )
70	62, 69	eqtrd 2656	. . . . 5 |- ( g = G -> { <. a , b >. | [. ( Base ` g ) / p ]. [. (hlG ` g ) / k ]. ( ( a e. ( p ^m ( 0..^ 3 ) ) /\ b e. ( p ^m ( 0..^ 3 ) ) ) /\ E. x e. p E. y e. p ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( k ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( k ` ( b ` 1 ) ) ( b ` 2 ) ) ) } = { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } )
71		df-cgra 25700	. . . . 5 |- cgrA = ( g e. _V |-> { <. a , b >. | [. ( Base ` g ) / p ]. [. (hlG ` g ) / k ]. ( ( a e. ( p ^m ( 0..^ 3 ) ) /\ b e. ( p ^m ( 0..^ 3 ) ) ) /\ E. x e. p E. y e. p ( a (cgrG ` g ) <" x ( b ` 1 ) y "> /\ x ( k ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( k ` ( b ` 1 ) ) ( b ` 2 ) ) ) } )
72		ovex 6678	. . . . . . 7 |- ( P ^m ( 0..^ 3 ) ) e. _V
73	72, 72	xpex 6962	. . . . . 6 |- ( ( P ^m ( 0..^ 3 ) ) X. ( P ^m ( 0..^ 3 ) ) ) e. _V
74		opabssxp 5193	. . . . . 6 |- { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } C_ ( ( P ^m ( 0..^ 3 ) ) X. ( P ^m ( 0..^ 3 ) ) )
75	73, 74	ssexi 4803	. . . . 5 |- { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } e. _V
76	70, 71, 75	fvmpt 6282	. . . 4 |- ( G e. _V -> (cgrA ` G ) = { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } )
77	41, 42, 76	3syl 18	. . 3 |- ( ph -> (cgrA ` G ) = { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } )
78	77	breqd 4664	. 2 |- ( ph -> ( <" A B C "> (cgrA ` G ) <" D E F "> <-> <" A B C "> { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( a (cgrG ` G ) <" x ( b ` 1 ) y "> /\ x ( K ` ( b ` 1 ) ) ( b ` 0 ) /\ y ( K ` ( b ` 1 ) ) ( b ` 2 ) ) ) } <" D E F "> ) )
79		iscgra.a	. . . . . . 7 |- ( ph -> A e. P )
80		iscgra.b	. . . . . . 7 |- ( ph -> B e. P )
81		iscgra.c	. . . . . . 7 |- ( ph -> C e. P )
82	79, 80, 81	s3cld 13617	. . . . . 6 |- ( ph -> <" A B C "> e. Word P )
83		s3len 13639	. . . . . . 7 |- ( # ` <" A B C "> ) = 3
84	83	a1i 11	. . . . . 6 |- ( ph -> ( # ` <" A B C "> ) = 3 )
85	82, 84	jca 554	. . . . 5 |- ( ph -> ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) )
86		fvex 6201	. . . . . . 7 |- ( Base ` G ) e. _V
87	43, 86	eqeltri 2697	. . . . . 6 |- P e. _V
88		3nn0 11310	. . . . . 6 |- 3 e. NN0
89		wrdmap 13336	. . . . . 6 |- ( ( P e. _V /\ 3 e. NN0 ) -> ( ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) <-> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) ) )
90	87, 88, 89	mp2an 708	. . . . 5 |- ( ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) <-> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) )
91	85, 90	sylib 208	. . . 4 |- ( ph -> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) )
92	27, 19, 32	s3cld 13617	. . . . . 6 |- ( ph -> <" D E F "> e. Word P )
93		s3len 13639	. . . . . . 7 |- ( # ` <" D E F "> ) = 3
94	93	a1i 11	. . . . . 6 |- ( ph -> ( # ` <" D E F "> ) = 3 )
95	92, 94	jca 554	. . . . 5 |- ( ph -> ( <" D E F "> e. Word P /\ ( # ` <" D E F "> ) = 3 ) )
96		wrdmap 13336	. . . . . 6 |- ( ( P e. _V /\ 3 e. NN0 ) -> ( ( <" D E F "> e. Word P /\ ( # ` <" D E F "> ) = 3 ) <-> <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) )
97	87, 88, 96	mp2an 708	. . . . 5 |- ( ( <" D E F "> e. Word P /\ ( # ` <" D E F "> ) = 3 ) <-> <" D E F "> e. ( P ^m ( 0..^ 3 ) ) )
98	95, 97	sylib 208	. . . 4 |- ( ph -> <" D E F "> e. ( P ^m ( 0..^ 3 ) ) )
99	91, 98	jca 554	. . 3 |- ( ph -> ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) )
100	99	biantrurd 529	. 2 |- ( ph -> ( E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) <-> ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) ) )
101	40, 78, 100	3bitr4d 300	1 |- ( ph -> ( <" A B C "> (cgrA ` G ) <" D E F "> <-> E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   [.wsbc 3435   class class class wbr 4653   {copab 4712   X. cxp 5112   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   1c1 9937   2c2 11070   3c3 11071   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291   <"cs3 13587   Basecbs 15857  TarskiGcstrkg 25329  Itvcitv 25335  cgrGccgrg 25405  hlGchlg 25495  cgrAccgra 25699

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-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-map 7859  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-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-cgra 25700

This theorem is referenced by:  iscgra1  25702  iscgrad  25703  cgrane1  25704  cgrane2  25705  cgrane3  25706  cgrane4  25707  cgrahl1  25708  cgrahl2  25709  cgracgr  25710  cgraswap  25712  cgracom  25714  cgratr  25715  cgrabtwn  25717  cgrahl  25718  sacgr  25722