Theorem isinag 25729

Description: Property for point X to lie in the angle <" A B C "> Defnition 11.23 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.)

Hypotheses

Ref	Expression
isinag.p	|- P = ( Base ` G )
isinag.i	|- I = (Itv ` G )
isinag.k	|- K = (hlG ` G )
isinag.x	|- ( ph -> X e. P )
isinag.a	|- ( ph -> A e. P )
isinag.b	|- ( ph -> B e. P )
isinag.c	|- ( ph -> C e. P )
isinag.g	|- ( ph -> G e. V )

Assertion

Ref	Expression
isinag	|- ( ph -> ( X (inA ` G ) <" A B C "> <-> ( ( A =/= B /\ C =/= B /\ X =/= B ) /\ E. x e. P ( x e. ( A I C ) /\ ( x = B \/ x ( K ` B ) X ) ) ) ) )

Distinct variable groups:   x, A   x, B   x, C   x, G   x, P   x, X   ph, x

Allowed substitution hints:   I( x)   K( x)   V( x)

Proof of Theorem isinag

Dummy variables p t g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . . . 9 |- ( ( p = X /\ t = <" A B C "> ) -> t = <" A B C "> )
2	1	fveq1d 6193	. . . . . . . 8 |- ( ( p = X /\ t = <" A B C "> ) -> ( t ` 0 ) = ( <" A B C "> ` 0 ) )
3	1	fveq1d 6193	. . . . . . . 8 |- ( ( p = X /\ t = <" A B C "> ) -> ( t ` 1 ) = ( <" A B C "> ` 1 ) )
4	2, 3	neeq12d 2855	. . . . . . 7 |- ( ( p = X /\ t = <" A B C "> ) -> ( ( t ` 0 ) =/= ( t ` 1 ) <-> ( <" A B C "> ` 0 ) =/= ( <" A B C "> ` 1 ) ) )
5	1	fveq1d 6193	. . . . . . . 8 |- ( ( p = X /\ t = <" A B C "> ) -> ( t ` 2 ) = ( <" A B C "> ` 2 ) )
6	5, 3	neeq12d 2855	. . . . . . 7 |- ( ( p = X /\ t = <" A B C "> ) -> ( ( t ` 2 ) =/= ( t ` 1 ) <-> ( <" A B C "> ` 2 ) =/= ( <" A B C "> ` 1 ) ) )
7		simpl 473	. . . . . . . 8 |- ( ( p = X /\ t = <" A B C "> ) -> p = X )
8	7, 3	neeq12d 2855	. . . . . . 7 |- ( ( p = X /\ t = <" A B C "> ) -> ( p =/= ( t ` 1 ) <-> X =/= ( <" A B C "> ` 1 ) ) )
9	4, 6, 8	3anbi123d 1399	. . . . . 6 |- ( ( p = X /\ t = <" A B C "> ) -> ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) <-> ( ( <" A B C "> ` 0 ) =/= ( <" A B C "> ` 1 ) /\ ( <" A B C "> ` 2 ) =/= ( <" A B C "> ` 1 ) /\ X =/= ( <" A B C "> ` 1 ) ) ) )
10		eqidd 2623	. . . . . . . . 9 |- ( ( p = X /\ t = <" A B C "> ) -> x = x )
11	2, 5	oveq12d 6668	. . . . . . . . 9 |- ( ( p = X /\ t = <" A B C "> ) -> ( ( t ` 0 ) I ( t ` 2 ) ) = ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) )
12	10, 11	eleq12d 2695	. . . . . . . 8 |- ( ( p = X /\ t = <" A B C "> ) -> ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) <-> x e. ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) ) )
13	10, 3	eqeq12d 2637	. . . . . . . . 9 |- ( ( p = X /\ t = <" A B C "> ) -> ( x = ( t ` 1 ) <-> x = ( <" A B C "> ` 1 ) ) )
14	3	fveq2d 6195	. . . . . . . . . 10 |- ( ( p = X /\ t = <" A B C "> ) -> ( K ` ( t ` 1 ) ) = ( K ` ( <" A B C "> ` 1 ) ) )
15	10, 14, 7	breq123d 4667	. . . . . . . . 9 |- ( ( p = X /\ t = <" A B C "> ) -> ( x ( K ` ( t ` 1 ) ) p <-> x ( K ` ( <" A B C "> ` 1 ) ) X ) )
16	13, 15	orbi12d 746	. . . . . . . 8 |- ( ( p = X /\ t = <" A B C "> ) -> ( ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) <-> ( x = ( <" A B C "> ` 1 ) \/ x ( K ` ( <" A B C "> ` 1 ) ) X ) ) )
17	12, 16	anbi12d 747	. . . . . . 7 |- ( ( p = X /\ t = <" A B C "> ) -> ( ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) <-> ( x e. ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) /\ ( x = ( <" A B C "> ` 1 ) \/ x ( K ` ( <" A B C "> ` 1 ) ) X ) ) ) )
18	17	rexbidv 3052	. . . . . 6 |- ( ( p = X /\ t = <" A B C "> ) -> ( E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) <-> E. x e. P ( x e. ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) /\ ( x = ( <" A B C "> ` 1 ) \/ x ( K ` ( <" A B C "> ` 1 ) ) X ) ) ) )
19	9, 18	anbi12d 747	. . . . 5 |- ( ( p = X /\ t = <" A B C "> ) -> ( ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) <-> ( ( ( <" A B C "> ` 0 ) =/= ( <" A B C "> ` 1 ) /\ ( <" A B C "> ` 2 ) =/= ( <" A B C "> ` 1 ) /\ X =/= ( <" A B C "> ` 1 ) ) /\ E. x e. P ( x e. ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) /\ ( x = ( <" A B C "> ` 1 ) \/ x ( K ` ( <" A B C "> ` 1 ) ) X ) ) ) ) )
20		eqid 2622	. . . . 5 |- { <. p , t >. | ( ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) } = { <. p , t >. | ( ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) }
21	19, 20	brab2a 5194	. . . 4 |- ( X { <. p , t >. | ( ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) } <" A B C "> <-> ( ( X e. P /\ <" A B C "> e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( <" A B C "> ` 0 ) =/= ( <" A B C "> ` 1 ) /\ ( <" A B C "> ` 2 ) =/= ( <" A B C "> ` 1 ) /\ X =/= ( <" A B C "> ` 1 ) ) /\ E. x e. P ( x e. ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) /\ ( x = ( <" A B C "> ` 1 ) \/ x ( K ` ( <" A B C "> ` 1 ) ) X ) ) ) ) )
22	21	a1i 11	. . 3 |- ( ph -> ( X { <. p , t >. | ( ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) } <" A B C "> <-> ( ( X e. P /\ <" A B C "> e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( <" A B C "> ` 0 ) =/= ( <" A B C "> ` 1 ) /\ ( <" A B C "> ` 2 ) =/= ( <" A B C "> ` 1 ) /\ X =/= ( <" A B C "> ` 1 ) ) /\ E. x e. P ( x e. ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) /\ ( x = ( <" A B C "> ` 1 ) \/ x ( K ` ( <" A B C "> ` 1 ) ) X ) ) ) ) ) )
23		biidd 252	. . . 4 |- ( ph -> ( ( X e. P /\ <" A B C "> e. ( P ^m ( 0..^ 3 ) ) ) <-> ( X e. P /\ <" A B C "> e. ( P ^m ( 0..^ 3 ) ) ) ) )
24		isinag.a	. . . . . . . 8 |- ( ph -> A e. P )
25		s3fv0 13636	. . . . . . . 8 |- ( A e. P -> ( <" A B C "> ` 0 ) = A )
26	24, 25	syl 17	. . . . . . 7 |- ( ph -> ( <" A B C "> ` 0 ) = A )
27		isinag.b	. . . . . . . 8 |- ( ph -> B e. P )
28		s3fv1 13637	. . . . . . . 8 |- ( B e. P -> ( <" A B C "> ` 1 ) = B )
29	27, 28	syl 17	. . . . . . 7 |- ( ph -> ( <" A B C "> ` 1 ) = B )
30	26, 29	neeq12d 2855	. . . . . 6 |- ( ph -> ( ( <" A B C "> ` 0 ) =/= ( <" A B C "> ` 1 ) <-> A =/= B ) )
31		isinag.c	. . . . . . . 8 |- ( ph -> C e. P )
32		s3fv2 13638	. . . . . . . 8 |- ( C e. P -> ( <" A B C "> ` 2 ) = C )
33	31, 32	syl 17	. . . . . . 7 |- ( ph -> ( <" A B C "> ` 2 ) = C )
34	33, 29	neeq12d 2855	. . . . . 6 |- ( ph -> ( ( <" A B C "> ` 2 ) =/= ( <" A B C "> ` 1 ) <-> C =/= B ) )
35	29	neeq2d 2854	. . . . . 6 |- ( ph -> ( X =/= ( <" A B C "> ` 1 ) <-> X =/= B ) )
36	30, 34, 35	3anbi123d 1399	. . . . 5 |- ( ph -> ( ( ( <" A B C "> ` 0 ) =/= ( <" A B C "> ` 1 ) /\ ( <" A B C "> ` 2 ) =/= ( <" A B C "> ` 1 ) /\ X =/= ( <" A B C "> ` 1 ) ) <-> ( A =/= B /\ C =/= B /\ X =/= B ) ) )
37	26, 33	oveq12d 6668	. . . . . . . 8 |- ( ph -> ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) = ( A I C ) )
38	37	eleq2d 2687	. . . . . . 7 |- ( ph -> ( x e. ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) <-> x e. ( A I C ) ) )
39	29	eqeq2d 2632	. . . . . . . 8 |- ( ph -> ( x = ( <" A B C "> ` 1 ) <-> x = B ) )
40	29	fveq2d 6195	. . . . . . . . 9 |- ( ph -> ( K ` ( <" A B C "> ` 1 ) ) = ( K ` B ) )
41	40	breqd 4664	. . . . . . . 8 |- ( ph -> ( x ( K ` ( <" A B C "> ` 1 ) ) X <-> x ( K ` B ) X ) )
42	39, 41	orbi12d 746	. . . . . . 7 |- ( ph -> ( ( x = ( <" A B C "> ` 1 ) \/ x ( K ` ( <" A B C "> ` 1 ) ) X ) <-> ( x = B \/ x ( K ` B ) X ) ) )
43	38, 42	anbi12d 747	. . . . . 6 |- ( ph -> ( ( x e. ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) /\ ( x = ( <" A B C "> ` 1 ) \/ x ( K ` ( <" A B C "> ` 1 ) ) X ) ) <-> ( x e. ( A I C ) /\ ( x = B \/ x ( K ` B ) X ) ) ) )
44	43	rexbidv 3052	. . . . 5 |- ( ph -> ( E. x e. P ( x e. ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) /\ ( x = ( <" A B C "> ` 1 ) \/ x ( K ` ( <" A B C "> ` 1 ) ) X ) ) <-> E. x e. P ( x e. ( A I C ) /\ ( x = B \/ x ( K ` B ) X ) ) ) )
45	36, 44	anbi12d 747	. . . 4 |- ( ph -> ( ( ( ( <" A B C "> ` 0 ) =/= ( <" A B C "> ` 1 ) /\ ( <" A B C "> ` 2 ) =/= ( <" A B C "> ` 1 ) /\ X =/= ( <" A B C "> ` 1 ) ) /\ E. x e. P ( x e. ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) /\ ( x = ( <" A B C "> ` 1 ) \/ x ( K ` ( <" A B C "> ` 1 ) ) X ) ) ) <-> ( ( A =/= B /\ C =/= B /\ X =/= B ) /\ E. x e. P ( x e. ( A I C ) /\ ( x = B \/ x ( K ` B ) X ) ) ) ) )
46	23, 45	anbi12d 747	. . 3 |- ( ph -> ( ( ( X e. P /\ <" A B C "> e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( <" A B C "> ` 0 ) =/= ( <" A B C "> ` 1 ) /\ ( <" A B C "> ` 2 ) =/= ( <" A B C "> ` 1 ) /\ X =/= ( <" A B C "> ` 1 ) ) /\ E. x e. P ( x e. ( ( <" A B C "> ` 0 ) I ( <" A B C "> ` 2 ) ) /\ ( x = ( <" A B C "> ` 1 ) \/ x ( K ` ( <" A B C "> ` 1 ) ) X ) ) ) ) <-> ( ( X e. P /\ <" A B C "> e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( A =/= B /\ C =/= B /\ X =/= B ) /\ E. x e. P ( x e. ( A I C ) /\ ( x = B \/ x ( K ` B ) X ) ) ) ) ) )
47	22, 46	bitrd 268	. 2 |- ( ph -> ( X { <. p , t >. | ( ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) } <" A B C "> <-> ( ( X e. P /\ <" A B C "> e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( A =/= B /\ C =/= B /\ X =/= B ) /\ E. x e. P ( x e. ( A I C ) /\ ( x = B \/ x ( K ` B ) X ) ) ) ) ) )
48		isinag.g	. . . 4 |- ( ph -> G e. V )
49		elex 3212	. . . 4 |- ( G e. V -> G e. _V )
50		fveq2 6191	. . . . . . . . . 10 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
51		isinag.p	. . . . . . . . . 10 |- P = ( Base ` G )
52	50, 51	syl6eqr 2674	. . . . . . . . 9 |- ( g = G -> ( Base ` g ) = P )
53	52	eleq2d 2687	. . . . . . . 8 |- ( g = G -> ( p e. ( Base ` g ) <-> p e. P ) )
54	52	oveq1d 6665	. . . . . . . . 9 |- ( g = G -> ( ( Base ` g ) ^m ( 0..^ 3 ) ) = ( P ^m ( 0..^ 3 ) ) )
55	54	eleq2d 2687	. . . . . . . 8 |- ( g = G -> ( t e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) <-> t e. ( P ^m ( 0..^ 3 ) ) ) )
56	53, 55	anbi12d 747	. . . . . . 7 |- ( g = G -> ( ( p e. ( Base ` g ) /\ t e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) ) <-> ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) ) )
57		fveq2 6191	. . . . . . . . . . . . 13 |- ( g = G -> (Itv ` g ) = (Itv ` G ) )
58		isinag.i	. . . . . . . . . . . . 13 |- I = (Itv ` G )
59	57, 58	syl6eqr 2674	. . . . . . . . . . . 12 |- ( g = G -> (Itv ` g ) = I )
60	59	oveqd 6667	. . . . . . . . . . 11 |- ( g = G -> ( ( t ` 0 ) (Itv ` g ) ( t ` 2 ) ) = ( ( t ` 0 ) I ( t ` 2 ) ) )
61	60	eleq2d 2687	. . . . . . . . . 10 |- ( g = G -> ( x e. ( ( t ` 0 ) (Itv ` g ) ( t ` 2 ) ) <-> x e. ( ( t ` 0 ) I ( t ` 2 ) ) ) )
62		fveq2 6191	. . . . . . . . . . . . . 14 |- ( g = G -> (hlG ` g ) = (hlG ` G ) )
63		isinag.k	. . . . . . . . . . . . . 14 |- K = (hlG ` G )
64	62, 63	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( g = G -> (hlG ` g ) = K )
65	64	fveq1d 6193	. . . . . . . . . . . 12 |- ( g = G -> ( (hlG ` g ) ` ( t ` 1 ) ) = ( K ` ( t ` 1 ) ) )
66	65	breqd 4664	. . . . . . . . . . 11 |- ( g = G -> ( x ( (hlG ` g ) ` ( t ` 1 ) ) p <-> x ( K ` ( t ` 1 ) ) p ) )
67	66	orbi2d 738	. . . . . . . . . 10 |- ( g = G -> ( ( x = ( t ` 1 ) \/ x ( (hlG ` g ) ` ( t ` 1 ) ) p ) <-> ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) )
68	61, 67	anbi12d 747	. . . . . . . . 9 |- ( g = G -> ( ( x e. ( ( t ` 0 ) (Itv ` g ) ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( (hlG ` g ) ` ( t ` 1 ) ) p ) ) <-> ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) )
69	52, 68	rexeqbidv 3153	. . . . . . . 8 |- ( g = G -> ( E. x e. ( Base ` g ) ( x e. ( ( t ` 0 ) (Itv ` g ) ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( (hlG ` g ) ` ( t ` 1 ) ) p ) ) <-> E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) )
70	69	anbi2d 740	. . . . . . 7 |- ( g = G -> ( ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. ( Base ` g ) ( x e. ( ( t ` 0 ) (Itv ` g ) ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( (hlG ` g ) ` ( t ` 1 ) ) p ) ) ) <-> ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) )
71	56, 70	anbi12d 747	. . . . . 6 |- ( g = G -> ( ( ( p e. ( Base ` g ) /\ t e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. ( Base ` g ) ( x e. ( ( t ` 0 ) (Itv ` g ) ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( (hlG ` g ) ` ( t ` 1 ) ) p ) ) ) ) <-> ( ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) ) )
72	71	opabbidv 4716	. . . . 5 |- ( g = G -> { <. p , t >. | ( ( p e. ( Base ` g ) /\ t e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. ( Base ` g ) ( x e. ( ( t ` 0 ) (Itv ` g ) ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( (hlG ` g ) ` ( t ` 1 ) ) p ) ) ) ) } = { <. p , t >. | ( ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) } )
73		df-inag 25728	. . . . 5 |- inA = ( g e. _V |-> { <. p , t >. | ( ( p e. ( Base ` g ) /\ t e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. ( Base ` g ) ( x e. ( ( t ` 0 ) (Itv ` g ) ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( (hlG ` g ) ` ( t ` 1 ) ) p ) ) ) ) } )
74		fvex 6201	. . . . . . . 8 |- ( Base ` G ) e. _V
75	51, 74	eqeltri 2697	. . . . . . 7 |- P e. _V
76		ovex 6678	. . . . . . 7 |- ( P ^m ( 0..^ 3 ) ) e. _V
77	75, 76	xpex 6962	. . . . . 6 |- ( P X. ( P ^m ( 0..^ 3 ) ) ) e. _V
78		opabssxp 5193	. . . . . 6 |- { <. p , t >. | ( ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) } C_ ( P X. ( P ^m ( 0..^ 3 ) ) )
79	77, 78	ssexi 4803	. . . . 5 |- { <. p , t >. | ( ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) } e. _V
80	72, 73, 79	fvmpt 6282	. . . 4 |- ( G e. _V -> (inA ` G ) = { <. p , t >. | ( ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) } )
81	48, 49, 80	3syl 18	. . 3 |- ( ph -> (inA ` G ) = { <. p , t >. | ( ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) } )
82	81	breqd 4664	. 2 |- ( ph -> ( X (inA ` G ) <" A B C "> <-> X { <. p , t >. | ( ( p e. P /\ t e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( ( t ` 0 ) =/= ( t ` 1 ) /\ ( t ` 2 ) =/= ( t ` 1 ) /\ p =/= ( t ` 1 ) ) /\ E. x e. P ( x e. ( ( t ` 0 ) I ( t ` 2 ) ) /\ ( x = ( t ` 1 ) \/ x ( K ` ( t ` 1 ) ) p ) ) ) ) } <" A B C "> ) )
83		isinag.x	. . . 4 |- ( ph -> X e. P )
84	24, 27, 31	s3cld 13617	. . . . . 6 |- ( ph -> <" A B C "> e. Word P )
85		s3len 13639	. . . . . . 7 |- ( # ` <" A B C "> ) = 3
86	85	a1i 11	. . . . . 6 |- ( ph -> ( # ` <" A B C "> ) = 3 )
87	84, 86	jca 554	. . . . 5 |- ( ph -> ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) )
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	75, 88, 89	mp2an 708	. . . . 5 |- ( ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) <-> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) )
91	87, 90	sylib 208	. . . 4 |- ( ph -> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) )
92	83, 91	jca 554	. . 3 |- ( ph -> ( X e. P /\ <" A B C "> e. ( P ^m ( 0..^ 3 ) ) ) )
93	92	biantrurd 529	. 2 |- ( ph -> ( ( ( A =/= B /\ C =/= B /\ X =/= B ) /\ E. x e. P ( x e. ( A I C ) /\ ( x = B \/ x ( K ` B ) X ) ) ) <-> ( ( X e. P /\ <" A B C "> e. ( P ^m ( 0..^ 3 ) ) ) /\ ( ( A =/= B /\ C =/= B /\ X =/= B ) /\ E. x e. P ( x e. ( A I C ) /\ ( x = B \/ x ( K ` B ) X ) ) ) ) ) )
94	47, 82, 93	3bitr4d 300	1 |- ( ph -> ( X (inA ` G ) <" A B C "> <-> ( ( A =/= B /\ C =/= B /\ X =/= B ) /\ E. x e. P ( x e. ( A I C ) /\ ( x = B \/ x ( K ` B ) X ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   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  Itvcitv 25335  hlGchlg 25495  inAcinag 25726

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-inag 25728

This theorem is referenced by:  inagswap  25730  inaghl  25731