Theorem isleag 24883

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 3054	. . . . 5 |- ( G e. TarskiG -> G e. _V )
3		fveq2 5865	. . . . . . . . . . . 12 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		isleag.p	. . . . . . . . . . . 12 |- P = ( Base ` G )
5	3, 4	syl6eqr 2503	. . . . . . . . . . 11 |- ( g = G -> ( Base ` g ) = P )
6	5	oveq1d 6305	. . . . . . . . . 10 |- ( g = G -> ( ( Base ` g ) ^m ( 0..^ 3 ) ) = ( P ^m ( 0..^ 3 ) ) )
7	6	eleq2d 2514	. . . . . . . . 9 |- ( g = G -> ( a e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) <-> a e. ( P ^m ( 0..^ 3 ) ) ) )
8	6	eleq2d 2514	. . . . . . . . 9 |- ( g = G -> ( b e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) <-> b e. ( P ^m ( 0..^ 3 ) ) ) )
9	7, 8	anbi12d 717	. . . . . . . 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 5865	. . . . . . . . . . 11 |- ( g = G -> (inA ` g ) = (inA ` G ) )
11	10	breqd 4413	. . . . . . . . . 10 |- ( g = G -> ( x (inA ` g ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> <-> x (inA ` G ) <" ( b ` 0 ) ( b ` 1 ) ( b ` 2 ) "> ) )
12		fveq2 5865	. . . . . . . . . . 11 |- ( g = G -> (cgrA ` g ) = (cgrA ` G ) )
13	12	breqd 4413	. . . . . . . . . 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 717	. . . . . . . . 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 3002	. . . . . . . 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 717	. . . . . . 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 4466	. . . . . 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 24882	. . . . . 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 6318	. . . . . . . 8 |- ( P ^m ( 0..^ 3 ) ) e. _V
20		xpexg 6593	. . . . . . . 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 678	. . . . . . 7 |- ( ( P ^m ( 0..^ 3 ) ) X. ( P ^m ( 0..^ 3 ) ) ) e. _V
22		opabssxp 4909	. . . . . . 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 4548	. . . . . 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 5948	. . . . 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 4413	. . 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 463	. . . . . . . . . 10 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> b = <" D E F "> )
28	27	fveq1d 5867	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( b ` 0 ) = ( <" D E F "> ` 0 ) )
29	27	fveq1d 5867	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( b ` 1 ) = ( <" D E F "> ` 1 ) )
30	27	fveq1d 5867	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( b ` 2 ) = ( <" D E F "> ` 2 ) )
31	28, 29, 30	s3eqd 12959	. . . . . . . 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 4414	. . . . . . 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 459	. . . . . . . . . 10 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> a = <" A B C "> )
34	33	fveq1d 5867	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( a ` 0 ) = ( <" A B C "> ` 0 ) )
35	33	fveq1d 5867	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( a ` 1 ) = ( <" A B C "> ` 1 ) )
36	33	fveq1d 5867	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> ( a ` 2 ) = ( <" A B C "> ` 2 ) )
37	34, 35, 36	s3eqd 12959	. . . . . . . 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 2452	. . . . . . . . 9 |- ( ( a = <" A B C "> /\ b = <" D E F "> ) -> x = x )
39	28, 29, 38	s3eqd 12959	. . . . . . . 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 4415	. . . . . . 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 717	. . . . . 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 2901	. . . . 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 2451	. . . . 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 4884	. . . 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 12985	. . . . . . . . 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 12986	. . . . . . . . 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 12987	. . . . . . . . 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 12959	. . . . . . 7 |- ( ph -> <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) ( <" D E F "> ` 2 ) "> = <" D E F "> )
56	55	breq2d 4414	. . . . . 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 12985	. . . . . . . . 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 12986	. . . . . . . . 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 12987	. . . . . . . . 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 12959	. . . . . . 7 |- ( ph -> <" ( <" A B C "> ` 0 ) ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) "> = <" A B C "> )
67		eqidd 2452	. . . . . . . 8 |- ( ph -> x = x )
68	48, 51, 67	s3eqd 12959	. . . . . . 7 |- ( ph -> <" ( <" D E F "> ` 0 ) ( <" D E F "> ` 1 ) x "> = <" D E x "> )
69	66, 68	breq12d 4415	. . . . . 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 717	. . . . 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 2901	. . . 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 710	. . 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 283	. 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 12966	. . . . . 6 |- ( ph -> <" A B C "> e. Word P )
75		s3len 12988	. . . . . . 7 |- ( # ` <" A B C "> ) = 3
76	75	a1i 11	. . . . . 6 |- ( ph -> ( # ` <" A B C "> ) = 3 )
77	74, 76	jca 535	. . . . 5 |- ( ph -> ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) )
78		fvex 5875	. . . . . . 7 |- ( Base ` G ) e. _V
79	4, 78	eqeltri 2525	. . . . . 6 |- P e. _V
80		3nn0 10887	. . . . . 6 |- 3 e. NN0
81		wrdmap 12698	. . . . . 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 678	. . . . 5 |- ( ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) <-> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) )
83	77, 82	sylib 200	. . . 4 |- ( ph -> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) )
84	46, 49, 52	s3cld 12966	. . . . . 6 |- ( ph -> <" D E F "> e. Word P )
85		s3len 12988	. . . . . . 7 |- ( # ` <" D E F "> ) = 3
86	85	a1i 11	. . . . . 6 |- ( ph -> ( # ` <" D E F "> ) = 3 )
87	84, 86	jca 535	. . . . 5 |- ( ph -> ( <" D E F "> e. Word P /\ ( # ` <" D E F "> ) = 3 ) )
88		wrdmap 12698	. . . . . 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 678	. . . . 5 |- ( ( <" D E F "> e. Word P /\ ( # ` <" D E F "> ) = 3 ) <-> <" D E F "> e. ( P ^m ( 0..^ 3 ) ) )
90	87, 89	sylib 200	. . . 4 |- ( ph -> <" D E F "> e. ( P ^m ( 0..^ 3 ) ) )
91	83, 90	jca 535	. . 3 |- ( ph -> ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" D E F "> e. ( P ^m ( 0..^ 3 ) ) ) )
92	91	biantrurd 511	. 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 260	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 188   /\ wa 371   = wceq 1444   e. wcel 1887   E.wrex 2738   _Vcvv 3045   class class class wbr 4402   {copab 4460   X. cxp 4832   ` cfv 5582  (class class class)co 6290   ^m cmap 7472   0cc0 9539   1c1 9540   2c2 10659   3c3 10660   NN0cn0 10869  ..^cfzo 11915   #chash 12515  Word cword 12656   <"cs3 12938   Basecbs 15121  TarskiGcstrkg 24478  cgrAccgra 24849  inAcinag 24876  ≤_∠cleag 24877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-concat 12666  df-s1 12667  df-s2 12944  df-s3 12945  df-leag 24882

This theorem is referenced by: (None)