Theorem isleag 24962

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 3040	. . . . 5 |- ( G e. TarskiG -> G e. _V )
3		fveq2 5879	. . . . . . . . . . . 12 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		isleag.p	. . . . . . . . . . . 12 |- P = ( Base ` G )
5	3, 4	syl6eqr 2523	. . . . . . . . . . 11 |- ( g = G -> ( Base ` g ) = P )
6	5	oveq1d 6323	. . . . . . . . . 10 |- ( g = G -> ( ( Base ` g ) ^m ( 0..^ 3 ) ) = ( P ^m ( 0..^ 3 ) ) )
7	6	eleq2d 2534	. . . . . . . . 9 |- ( g = G -> ( a e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) <-> a e. ( P ^m ( 0..^ 3 ) ) ) )
8	6	eleq2d 2534	. . . . . . . . 9 |- ( g = G -> ( b e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) <-> b e. ( P ^m ( 0..^ 3 ) ) ) )
9	7, 8	anbi12d 725	. . . . . . . 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 5879	. . . . . . . . . . 11 |- ( g = G -> (inA ` g ) = (inA ` G ) )
11	10	breqd 4406	. . . . . . . . . 10 |- ( g = G -> ( x (inA ` g ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> <-> x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> ) )
12		fveq2 5879	. . . . . . . . . . 11 |- ( g = G -> (cgrA ` g ) = (cgrA ` G ) )
13	12	breqd 4406	. . . . . . . . . 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 725	. . . . . . . . 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 2988	. . . . . . . 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 725	. . . . . . 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 4459	. . . . . 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 24961	. . . . . 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 6336	. . . . . . . 8 |- ( P ^m ( 0..^ 3 ) ) e. _V
20		xpexg 6612	. . . . . . . 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 686	. . . . . . 7 |- ( ( P ^m ( 0..^ 3 ) ) X. ( P ^m ( 0..^ 3 ) ) ) e. _V
22		opabssxp 4914	. . . . . . 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 4541	. . . . . 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 5963	. . . . 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 4406	. . 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 468	. . . . . . . . . 10 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> b = <" D E F "> )
28	27	fveq1d 5881	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( b ` 0 ) = ( <" D E F "> ` 0 ) )
29	27	fveq1d 5881	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( b ` 1 ) = ( <" D E F "> ` 1 ) )
30	27	fveq1d 5881	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( b ` 2 ) = ( <" D E F "> ` 2 ) )
31	28, 29, 30	s3eqd 13019	. . . . . . . 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 4407	. . . . . . 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 464	. . . . . . . . . 10 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> a = <" A B C "> )
34	33	fveq1d 5881	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( a ` 0 ) = ( <" A B C "> ` 0 ) )
35	33	fveq1d 5881	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( a ` 1 ) = ( <" A B C "> ` 1 ) )
36	33	fveq1d 5881	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( a ` 2 ) = ( <" A B C "> ` 2 ) )
37	34, 35, 36	s3eqd 13019	. . . . . . . 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 2472	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> x = x )
39	28, 29, 38	s3eqd 13019	. . . . . . . 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 4408	. . . . . . 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 725	. . . . . 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 2892	. . . . 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 2471	. . . . 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 4889	. . . 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 13045	. . . . . . . . 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 13046	. . . . . . . . 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 13047	. . . . . . . . 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 13019	. . . . . . 7 |- ( ph -> <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> = <" D E F "> )
56	55	breq2d 4407	. . . . . 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 13045	. . . . . . . . 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 13046	. . . . . . . . 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 13047	. . . . . . . . 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 13019	. . . . . . 7 |- ( ph -> <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> = <" A B C "> )
67		eqidd 2472	. . . . . . . 8 |- ( ph -> x = x )
68	48, 51, 67	s3eqd 13019	. . . . . . 7 |- ( ph -> <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> = <" D E x "> )
69	66, 68	breq12d 4408	. . . . . 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 725	. . . . 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 2892	. . . 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 718	. . 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 287	. 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 13026	. . . . . 6 |- ( ph -> <" A B C "> e. Word P )
75		s3len 13048	. . . . . . 7 |- ( # ` <" A B C "> ) = 3
76	75	a1i 11	. . . . . 6 |- ( ph -> ( # ` <" A B C "> ) = 3 )
77	74, 76	jca 541	. . . . 5 |- ( ph -> ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) )
78		fvex 5889	. . . . . . 7 |- ( Base ` G ) e. _V
79	4, 78	eqeltri 2545	. . . . . 6 |- P e. _V
80		3nn0 10911	. . . . . 6 |- 3 e. NN0
81		wrdmap 12749	. . . . . 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 686	. . . . 5 |- ( ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) <-> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) )
83	77, 82	sylib 201	. . . 4 |- ( ph -> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) )
84	46, 49, 52	s3cld 13026	. . . . . 6 |- ( ph -> <" D E F "> e. Word P )
85		s3len 13048	. . . . . . 7 |- ( # ` <" D E F "> ) = 3
86	85	a1i 11	. . . . . 6 |- ( ph -> ( # ` <" D E F "> ) = 3 )
87	84, 86	jca 541	. . . . 5 |- ( ph -> ( <" D E F "> e. Word P /\ ( # ` <" D E F "> ) = 3 ) )
88		wrdmap 12749	. . . . . 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 686	. . . . 5 |- ( ( <" D E F "> e. Word P /\ ( # ` <" D E F "> ) = 3 ) <-> <" D E F "> e. ( P ^m ( 0..^ 3 ) ) )
90	87, 89	sylib 201	. . . 4 |- ( ph -> <" D E F "> e. ( P ^m ( 0..^ 3 ) ) )
91	83, 90	jca 541	. . 3 |- ( ph -> ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) )
92	91	biantrurd 516	. 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 264	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 189   /\ wa 376   = wceq 1452   e. wcel 1904   E.wrex 2757   _Vcvv 3031   class class class wbr 4395   {copab 4453   X. cxp 4837   ` cfv 5589  (class class class)co 6308   ^m cmap 7490   0cc0 9557   1c1 9558   2c2 10681   3c3 10682   NN0cn0 10893  ..^cfzo 11942   #chash 12553  Word cword 12703   <"cs3 12997   Basecbs 15199  TarskiGcstrkg 24557  cgrAccgra 24928  inAcinag 24955  ≤_∠cleag 24956

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-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-map 7492  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-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-leag 24961

This theorem is referenced by: (None)