Theorem isleag 25733

Description: Geometrical "less than" property for angles. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.)

Hypotheses

Ref	Expression
isleag.p	|- P = ( Base ` G )
isleag.g	|- ( ph -> G e. TarskiG )
isleag.a	|- ( ph -> A e. P )
isleag.b	|- ( ph -> B e. P )
isleag.c	|- ( ph -> C e. P )
isleag.d	|- ( ph -> D e. P )
isleag.e	|- ( ph -> E e. P )
isleag.f	|- ( ph -> F e. P )

Assertion

Ref	Expression
isleag	|- ( ph -> ( <" A B C "> (≤∠ ` G ) <" D E F "> <-> E. x e. P ( x (inA ` G ) <" D E F "> /\ <" A B C "> (cgrA ` G ) <" D E x "> ) ) )

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

Proof of Theorem isleag

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

Step	Hyp	Ref	Expression
1		isleag.g	. . . . 5 |- ( ph -> G e. TarskiG )
2		elex 3212	. . . . 5 |- ( G e. TarskiG -> G e. _V )
3		fveq2 6191	. . . . . . . . . . . 12 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		isleag.p	. . . . . . . . . . . 12 |- P = ( Base ` G )
5	3, 4	syl6eqr 2674	. . . . . . . . . . 11 |- ( g = G -> ( Base ` g ) = P )
6	5	oveq1d 6665	. . . . . . . . . 10 |- ( g = G -> ( ( Base ` g ) ^m ( 0..^ 3 ) ) = ( P ^m ( 0..^ 3 ) ) )
7	6	eleq2d 2687	. . . . . . . . 9 |- ( g = G -> ( a e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) <-> a e. ( P ^m ( 0..^ 3 ) ) ) )
8	6	eleq2d 2687	. . . . . . . . 9 |- ( g = G -> ( b e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) <-> b e. ( P ^m ( 0..^ 3 ) ) ) )
9	7, 8	anbi12d 747	. . . . . . . 8 |- ( g = G -> ( ( a e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) /\ b e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) ) <-> ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) ) )
10		fveq2 6191	. . . . . . . . . . 11 |- ( g = G -> (inA ` g ) = (inA ` G ) )
11	10	breqd 4664	. . . . . . . . . 10 |- ( g = G -> ( x (inA ` g ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> <-> x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> ) )
12		fveq2 6191	. . . . . . . . . . 11 |- ( g = G -> (cgrA ` g ) = (cgrA ` G ) )
13	12	breqd 4664	. . . . . . . . . 10 |- ( g = G -> ( <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` g ) <" ( b ` 0 ) ( b ` 1 ) x "> <-> <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) )
14	11, 13	anbi12d 747	. . . . . . . . 9 |- ( g = G -> ( ( x (inA ` g ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` g ) <" ( b ` 0 ) ( b ` 1 ) x "> ) <-> ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) )
15	5, 14	rexeqbidv 3153	. . . . . . . 8 |- ( g = G -> ( E. x e. ( Base ` g ) ( x (inA ` g ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` g ) <" ( b ` 0 ) ( b ` 1 ) x "> ) <-> E. x e. P ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) )
16	9, 15	anbi12d 747	. . . . . . 7 |- ( g = G -> ( ( ( a e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) /\ b e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) ) /\ E. x e. ( Base ` g ) ( x (inA ` g ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` g ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) <-> ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) ) )
17	16	opabbidv 4716	. . . . . 6 |- ( g = G -> { <. a , b >. | ( ( a e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) /\ b e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) ) /\ E. x e. ( Base ` g ) ( x (inA ` g ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` g ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) } = { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) } )
18		df-leag 25732	. . . . . 6 |- ≤∠ = ( g e. _V |-> { <. a , b >. | ( ( a e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) /\ b e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) ) /\ E. x e. ( Base ` g ) ( x (inA ` g ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` g ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) } )
19		ovex 6678	. . . . . . . 8 |- ( P ^m ( 0..^ 3 ) ) e. _V
20		xpexg 6960	. . . . . . . 8 |- ( ( ( P ^m ( 0..^ 3 ) ) e. _V /\ ( P ^m ( 0..^ 3 ) ) e. _V ) -> ( ( P ^m ( 0..^ 3 ) ) X. ( P ^m ( 0..^ 3 ) ) ) e. _V )
21	19, 19, 20	mp2an 708	. . . . . . 7 |- ( ( P ^m ( 0..^ 3 ) ) X. ( P ^m ( 0..^ 3 ) ) ) e. _V
22		opabssxp 5193	. . . . . . 7 |- { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) } C_ ( ( P ^m ( 0..^ 3 ) ) X. ( P ^m ( 0..^ 3 ) ) )
23	21, 22	ssexi 4803	. . . . . 6 |- { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) } e. _V
24	17, 18, 23	fvmpt 6282	. . . . 5 |- ( G e. _V -> (≤∠ ` G ) = { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) } )
25	1, 2, 24	3syl 18	. . . 4 |- ( ph -> (≤∠ ` G ) = { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) } )
26	25	breqd 4664	. . 3 |- ( ph -> ( <" A B C "> (≤∠ ` G ) <" D E F "> <-> <" A B C "> { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) } <" D E F "> ) )
27		simpr 477	. . . . . . . . . 10 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> b = <" D E F "> )
28	27	fveq1d 6193	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( b ` 0 ) = ( <" D E F "> ` 0 ) )
29	27	fveq1d 6193	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( b ` 1 ) = ( <" D E F "> ` 1 ) )
30	27	fveq1d 6193	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( b ` 2 ) = ( <" D E F "> ` 2 ) )
31	28, 29, 30	s3eqd 13609	. . . . . . . 8 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> = <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> )
32	31	breq2d 4665	. . . . . . 7 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> <-> x (inA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> ) )
33		simpl 473	. . . . . . . . . 10 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> a = <" A B C "> )
34	33	fveq1d 6193	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( a ` 0 ) = ( <" A B C "> ` 0 ) )
35	33	fveq1d 6193	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( a ` 1 ) = ( <" A B C "> ` 1 ) )
36	33	fveq1d 6193	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( a ` 2 ) = ( <" A B C "> ` 2 ) )
37	34, 35, 36	s3eqd 13609	. . . . . . . 8 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> = <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> )
38		eqidd 2623	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> x = x )
39	28, 29, 38	s3eqd 13609	. . . . . . . 8 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> <" ( b ` 0 ) ( b ` 1 ) x "> = <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> )
40	37, 39	breq12d 4666	. . . . . . 7 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> <-> <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> (cgrA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> ) )
41	32, 40	anbi12d 747	. . . . . 6 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) <-> ( x (inA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> /\ <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> (cgrA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> ) ) )
42	41	rexbidv 3052	. . . . 5 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( E. x e. P ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) <-> E. x e. P ( x (inA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> /\ <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> (cgrA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> ) ) )
43		eqid 2622	. . . . 5 |- { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) } = { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) }
44	42, 43	brab2a 5194	. . . 4 |- ( <" A B C "> { <. a , b >. | ( ( a e. ( P ^m ( 0..^ 3 ) ) /\ b e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) } <" D E F "> <-> ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> /\ <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> (cgrA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> ) ) )
45	44	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 ( x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> /\ <" ( a ` 0 ) ( a ` 1 ) ( a ` 2 ) "> (cgrA ` G ) <" ( b ` 0 ) ( b ` 1 ) x "> ) ) } <" D E F "> <-> ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> /\ <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> (cgrA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> ) ) ) )
46		isleag.d	. . . . . . . . 9 |- ( ph -> D e. P )
47		s3fv0 13636	. . . . . . . . 9 |- ( D e. P -> ( <" D E F "> ` 0 ) = D )
48	46, 47	syl 17	. . . . . . . 8 |- ( ph -> ( <" D E F "> ` 0 ) = D )
49		isleag.e	. . . . . . . . 9 |- ( ph -> E e. P )
50		s3fv1 13637	. . . . . . . . 9 |- ( E e. P -> ( <" D E F "> ` 1 ) = E )
51	49, 50	syl 17	. . . . . . . 8 |- ( ph -> ( <" D E F "> ` 1 ) = E )
52		isleag.f	. . . . . . . . 9 |- ( ph -> F e. P )
53		s3fv2 13638	. . . . . . . . 9 |- ( F e. P -> ( <" D E F "> ` 2 ) = F )
54	52, 53	syl 17	. . . . . . . 8 |- ( ph -> ( <" D E F "> ` 2 ) = F )
55	48, 51, 54	s3eqd 13609	. . . . . . 7 |- ( ph -> <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> = <" D E F "> )
56	55	breq2d 4665	. . . . . 6 |- ( ph -> ( x (inA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> <-> x (inA ` G ) <" D E F "> ) )
57		isleag.a	. . . . . . . . 9 |- ( ph -> A e. P )
58		s3fv0 13636	. . . . . . . . 9 |- ( A e. P -> ( <" A B C "> ` 0 ) = A )
59	57, 58	syl 17	. . . . . . . 8 |- ( ph -> ( <" A B C "> ` 0 ) = A )
60		isleag.b	. . . . . . . . 9 |- ( ph -> B e. P )
61		s3fv1 13637	. . . . . . . . 9 |- ( B e. P -> ( <" A B C "> ` 1 ) = B )
62	60, 61	syl 17	. . . . . . . 8 |- ( ph -> ( <" A B C "> ` 1 ) = B )
63		isleag.c	. . . . . . . . 9 |- ( ph -> C e. P )
64		s3fv2 13638	. . . . . . . . 9 |- ( C e. P -> ( <" A B C "> ` 2 ) = C )
65	63, 64	syl 17	. . . . . . . 8 |- ( ph -> ( <" A B C "> ` 2 ) = C )
66	59, 62, 65	s3eqd 13609	. . . . . . 7 |- ( ph -> <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> = <" A B C "> )
67		eqidd 2623	. . . . . . . 8 |- ( ph -> x = x )
68	48, 51, 67	s3eqd 13609	. . . . . . 7 |- ( ph -> <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> = <" D E x "> )
69	66, 68	breq12d 4666	. . . . . 6 |- ( ph -> ( <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> (cgrA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> <-> <" A B C "> (cgrA ` G ) <" D E x "> ) )
70	56, 69	anbi12d 747	. . . . 5 |- ( ph -> ( ( x (inA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> /\ <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> (cgrA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> ) <-> ( x (inA ` G ) <" D E F "> /\ <" A B C "> (cgrA ` G ) <" D E x "> ) ) )
71	70	rexbidv 3052	. . . 4 |- ( ph -> ( E. x e. P ( x (inA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> /\ <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> (cgrA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> ) <-> E. x e. P ( x (inA ` G ) <" D E F "> /\ <" A B C "> (cgrA ` G ) <" D E x "> ) ) )
72	71	anbi2d 740	. . 3 |- ( ph -> ( ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> /\ <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> (cgrA ` G ) <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> ) ) <-> ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" D E F "> /\ <" A B C "> (cgrA ` G ) <" D E x "> ) ) ) )
73	26, 45, 72	3bitrd 294	. 2 |- ( ph -> ( <" A B C "> (≤∠ ` G ) <" D E F "> <-> ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" D E F "> /\ <" A B C "> (cgrA ` G ) <" D E x "> ) ) ) )
74	57, 60, 63	s3cld 13617	. . . . . 6 |- ( ph -> <" A B C "> e. Word P )
75		s3len 13639	. . . . . . 7 |- ( # ` <" A B C "> ) = 3
76	75	a1i 11	. . . . . 6 |- ( ph -> ( # ` <" A B C "> ) = 3 )
77	74, 76	jca 554	. . . . 5 |- ( ph -> ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) )
78		fvex 6201	. . . . . . 7 |- ( Base ` G ) e. _V
79	4, 78	eqeltri 2697	. . . . . 6 |- P e. _V
80		3nn0 11310	. . . . . 6 |- 3 e. NN0
81		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 ) ) ) )
82	79, 80, 81	mp2an 708	. . . . 5 |- ( ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) <-> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) )
83	77, 82	sylib 208	. . . 4 |- ( ph -> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) )
84	46, 49, 52	s3cld 13617	. . . . . 6 |- ( ph -> <" D E F "> e. Word P )
85		s3len 13639	. . . . . . 7 |- ( # ` <" D E F "> ) = 3
86	85	a1i 11	. . . . . 6 |- ( ph -> ( # ` <" D E F "> ) = 3 )
87	84, 86	jca 554	. . . . 5 |- ( ph -> ( <" D E F "> e. Word P /\ ( # ` <" D E F "> ) = 3 ) )
88		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 ) ) ) )
89	79, 80, 88	mp2an 708	. . . . 5 |- ( ( <" D E F "> e. Word P /\ ( # ` <" D E F "> ) = 3 ) <-> <" D E F "> e. ( P ^m ( 0..^ 3 ) ) )
90	87, 89	sylib 208	. . . 4 |- ( ph -> <" D E F "> e. ( P ^m ( 0..^ 3 ) ) )
91	83, 90	jca 554	. . 3 |- ( ph -> ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) )
92	91	biantrurd 529	. 2 |- ( ph -> ( E. x e. P ( x (inA ` G ) <" D E F "> /\ <" A B C "> (cgrA ` G ) <" D E x "> ) <-> ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) /\ E. x e. P ( x (inA ` G ) <" D E F "> /\ <" A B C "> (cgrA ` G ) <" D E x "> ) ) ) )
93	73, 92	bitr4d 271	1 |- ( ph -> ( <" A B C "> (≤∠ ` G ) <" D E F "> <-> E. x e. P ( x (inA ` G ) <" D E F "> /\ <" A B C "> (cgrA ` G ) <" D E x "> ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   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  cgrAccgra 25699  inAcinag 25726  ≤_∠cleag 25727

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-leag 25732

This theorem is referenced by: (None)