Theorem ishlg 25497

Description: Rays : Definition 6.1 of [Schwabhauser] p. 43. With this definition, A ( K ` C ) B means that A and B are on the same ray with initial point C. This follows the same notation as Schwabhauser where rays are first defined as a relation. It is possible to recover the ray itself using e.g. ( ( K ` C ) " { A } ) (Contributed by Thierry Arnoux, 21-Dec-2019.)

Hypotheses

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

Assertion

Ref	Expression
ishlg	|- ( ph -> ( A ( K ` C ) B <-> ( A =/= C /\ B =/= C /\ ( A e. ( C I B ) \/ B e. ( C I A ) ) ) ) )

Proof of Theorem ishlg

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

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . 6 |- ( ( a = A /\ b = B ) -> a = A )
2	1	neeq1d 2853	. . . . 5 |- ( ( a = A /\ b = B ) -> ( a =/= C <-> A =/= C ) )
3		simpr 477	. . . . . 6 |- ( ( a = A /\ b = B ) -> b = B )
4	3	neeq1d 2853	. . . . 5 |- ( ( a = A /\ b = B ) -> ( b =/= C <-> B =/= C ) )
5	3	oveq2d 6666	. . . . . . 7 |- ( ( a = A /\ b = B ) -> ( C I b ) = ( C I B ) )
6	1, 5	eleq12d 2695	. . . . . 6 |- ( ( a = A /\ b = B ) -> ( a e. ( C I b ) <-> A e. ( C I B ) ) )
7	1	oveq2d 6666	. . . . . . 7 |- ( ( a = A /\ b = B ) -> ( C I a ) = ( C I A ) )
8	3, 7	eleq12d 2695	. . . . . 6 |- ( ( a = A /\ b = B ) -> ( b e. ( C I a ) <-> B e. ( C I A ) ) )
9	6, 8	orbi12d 746	. . . . 5 |- ( ( a = A /\ b = B ) -> ( ( a e. ( C I b ) \/ b e. ( C I a ) ) <-> ( A e. ( C I B ) \/ B e. ( C I A ) ) ) )
10	2, 4, 9	3anbi123d 1399	. . . 4 |- ( ( a = A /\ b = B ) -> ( ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) <-> ( A =/= C /\ B =/= C /\ ( A e. ( C I B ) \/ B e. ( C I A ) ) ) ) )
11		eqid 2622	. . . 4 |- { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) } = { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) }
12	10, 11	brab2a 5194	. . 3 |- ( A { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) } B <-> ( ( A e. P /\ B e. P ) /\ ( A =/= C /\ B =/= C /\ ( A e. ( C I B ) \/ B e. ( C I A ) ) ) ) )
13	12	a1i 11	. 2 |- ( ph -> ( A { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) } B <-> ( ( A e. P /\ B e. P ) /\ ( A =/= C /\ B =/= C /\ ( A e. ( C I B ) \/ B e. ( C I A ) ) ) ) ) )
14		ishlg.k	. . . . 5 |- K = (hlG ` G )
15		ishlg.g	. . . . . 6 |- ( ph -> G e. V )
16		elex 3212	. . . . . 6 |- ( G e. V -> G e. _V )
17		fveq2 6191	. . . . . . . . 9 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
18		ishlg.p	. . . . . . . . 9 |- P = ( Base ` G )
19	17, 18	syl6eqr 2674	. . . . . . . 8 |- ( g = G -> ( Base ` g ) = P )
20	19	eleq2d 2687	. . . . . . . . . . 11 |- ( g = G -> ( a e. ( Base ` g ) <-> a e. P ) )
21	19	eleq2d 2687	. . . . . . . . . . 11 |- ( g = G -> ( b e. ( Base ` g ) <-> b e. P ) )
22	20, 21	anbi12d 747	. . . . . . . . . 10 |- ( g = G -> ( ( a e. ( Base ` g ) /\ b e. ( Base ` g ) ) <-> ( a e. P /\ b e. P ) ) )
23		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( g = G -> (Itv ` g ) = (Itv ` G ) )
24		ishlg.i	. . . . . . . . . . . . . . 15 |- I = (Itv ` G )
25	23, 24	syl6eqr 2674	. . . . . . . . . . . . . 14 |- ( g = G -> (Itv ` g ) = I )
26	25	oveqd 6667	. . . . . . . . . . . . 13 |- ( g = G -> ( c (Itv ` g ) b ) = ( c I b ) )
27	26	eleq2d 2687	. . . . . . . . . . . 12 |- ( g = G -> ( a e. ( c (Itv ` g ) b ) <-> a e. ( c I b ) ) )
28	25	oveqd 6667	. . . . . . . . . . . . 13 |- ( g = G -> ( c (Itv ` g ) a ) = ( c I a ) )
29	28	eleq2d 2687	. . . . . . . . . . . 12 |- ( g = G -> ( b e. ( c (Itv ` g ) a ) <-> b e. ( c I a ) ) )
30	27, 29	orbi12d 746	. . . . . . . . . . 11 |- ( g = G -> ( ( a e. ( c (Itv ` g ) b ) \/ b e. ( c (Itv ` g ) a ) ) <-> ( a e. ( c I b ) \/ b e. ( c I a ) ) ) )
31	30	3anbi3d 1405	. . . . . . . . . 10 |- ( g = G -> ( ( a =/= c /\ b =/= c /\ ( a e. ( c (Itv ` g ) b ) \/ b e. ( c (Itv ` g ) a ) ) ) <-> ( a =/= c /\ b =/= c /\ ( a e. ( c I b ) \/ b e. ( c I a ) ) ) ) )
32	22, 31	anbi12d 747	. . . . . . . . 9 |- ( g = G -> ( ( ( a e. ( Base ` g ) /\ b e. ( Base ` g ) ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c (Itv ` g ) b ) \/ b e. ( c (Itv ` g ) a ) ) ) ) <-> ( ( a e. P /\ b e. P ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c I b ) \/ b e. ( c I a ) ) ) ) ) )
33	32	opabbidv 4716	. . . . . . . 8 |- ( g = G -> { <. a , b >. | ( ( a e. ( Base ` g ) /\ b e. ( Base ` g ) ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c (Itv ` g ) b ) \/ b e. ( c (Itv ` g ) a ) ) ) ) } = { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c I b ) \/ b e. ( c I a ) ) ) ) } )
34	19, 33	mpteq12dv 4733	. . . . . . 7 |- ( g = G -> ( c e. ( Base ` g ) |-> { <. a , b >. | ( ( a e. ( Base ` g ) /\ b e. ( Base ` g ) ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c (Itv ` g ) b ) \/ b e. ( c (Itv ` g ) a ) ) ) ) } ) = ( c e. P |-> { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c I b ) \/ b e. ( c I a ) ) ) ) } ) )
35		df-hlg 25496	. . . . . . 7 |- hlG = ( g e. _V |-> ( c e. ( Base ` g ) |-> { <. a , b >. | ( ( a e. ( Base ` g ) /\ b e. ( Base ` g ) ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c (Itv ` g ) b ) \/ b e. ( c (Itv ` g ) a ) ) ) ) } ) )
36		fvex 6201	. . . . . . . . 9 |- ( Base ` G ) e. _V
37	18, 36	eqeltri 2697	. . . . . . . 8 |- P e. _V
38	37	mptex 6486	. . . . . . 7 |- ( c e. P |-> { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c I b ) \/ b e. ( c I a ) ) ) ) } ) e. _V
39	34, 35, 38	fvmpt 6282	. . . . . 6 |- ( G e. _V -> (hlG ` G ) = ( c e. P |-> { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c I b ) \/ b e. ( c I a ) ) ) ) } ) )
40	15, 16, 39	3syl 18	. . . . 5 |- ( ph -> (hlG ` G ) = ( c e. P |-> { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c I b ) \/ b e. ( c I a ) ) ) ) } ) )
41	14, 40	syl5eq 2668	. . . 4 |- ( ph -> K = ( c e. P |-> { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c I b ) \/ b e. ( c I a ) ) ) ) } ) )
42		neeq2 2857	. . . . . . . 8 |- ( c = C -> ( a =/= c <-> a =/= C ) )
43		neeq2 2857	. . . . . . . 8 |- ( c = C -> ( b =/= c <-> b =/= C ) )
44		oveq1 6657	. . . . . . . . . 10 |- ( c = C -> ( c I b ) = ( C I b ) )
45	44	eleq2d 2687	. . . . . . . . 9 |- ( c = C -> ( a e. ( c I b ) <-> a e. ( C I b ) ) )
46		oveq1 6657	. . . . . . . . . 10 |- ( c = C -> ( c I a ) = ( C I a ) )
47	46	eleq2d 2687	. . . . . . . . 9 |- ( c = C -> ( b e. ( c I a ) <-> b e. ( C I a ) ) )
48	45, 47	orbi12d 746	. . . . . . . 8 |- ( c = C -> ( ( a e. ( c I b ) \/ b e. ( c I a ) ) <-> ( a e. ( C I b ) \/ b e. ( C I a ) ) ) )
49	42, 43, 48	3anbi123d 1399	. . . . . . 7 |- ( c = C -> ( ( a =/= c /\ b =/= c /\ ( a e. ( c I b ) \/ b e. ( c I a ) ) ) <-> ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) )
50	49	anbi2d 740	. . . . . 6 |- ( c = C -> ( ( ( a e. P /\ b e. P ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c I b ) \/ b e. ( c I a ) ) ) ) <-> ( ( a e. P /\ b e. P ) /\ ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) ) )
51	50	opabbidv 4716	. . . . 5 |- ( c = C -> { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c I b ) \/ b e. ( c I a ) ) ) ) } = { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) } )
52	51	adantl 482	. . . 4 |- ( ( ph /\ c = C ) -> { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= c /\ b =/= c /\ ( a e. ( c I b ) \/ b e. ( c I a ) ) ) ) } = { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) } )
53		ishlg.c	. . . 4 |- ( ph -> C e. P )
54	37, 37	xpex 6962	. . . . . 6 |- ( P X. P ) e. _V
55		opabssxp 5193	. . . . . 6 |- { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) } C_ ( P X. P )
56	54, 55	ssexi 4803	. . . . 5 |- { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) } e. _V
57	56	a1i 11	. . . 4 |- ( ph -> { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) } e. _V )
58	41, 52, 53, 57	fvmptd 6288	. . 3 |- ( ph -> ( K ` C ) = { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) } )
59	58	breqd 4664	. 2 |- ( ph -> ( A ( K ` C ) B <-> A { <. a , b >. | ( ( a e. P /\ b e. P ) /\ ( a =/= C /\ b =/= C /\ ( a e. ( C I b ) \/ b e. ( C I a ) ) ) ) } B ) )
60		ishlg.a	. . . 4 |- ( ph -> A e. P )
61		ishlg.b	. . . 4 |- ( ph -> B e. P )
62	60, 61	jca 554	. . 3 |- ( ph -> ( A e. P /\ B e. P ) )
63	62	biantrurd 529	. 2 |- ( ph -> ( ( A =/= C /\ B =/= C /\ ( A e. ( C I B ) \/ B e. ( C I A ) ) ) <-> ( ( A e. P /\ B e. P ) /\ ( A =/= C /\ B =/= C /\ ( A e. ( C I B ) \/ B e. ( C I A ) ) ) ) ) )
64	13, 59, 63	3bitr4d 300	1 |- ( ph -> ( A ( K ` C ) B <-> ( A =/= C /\ B =/= C /\ ( A e. ( C I B ) \/ B e. ( C I A ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Itvcitv 25335  hlGchlg 25495

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-hlg 25496

This theorem is referenced by:  hlcomb  25498  hlne1  25500  hlne2  25501  hlln  25502  hlid  25504  hltr  25505  hlbtwn  25506  btwnhl1  25507  btwnhl2  25508  btwnhl  25509  lnhl  25510  hlcgrex  25511  mirhl  25574  mirbtwnhl  25575  mirhl2  25576  hlperpnel  25617  opphllem4  25642  opphl  25646  hlpasch  25648  lnopp2hpgb  25655  cgracgr  25710  cgraswap  25712  cgrahl  25718  cgracol  25719