Theorem tglineneq 23765

Description: Given three non-colinear points, build two different lines. (Contributed by Thierry Arnoux, 6-Aug-2019.)

Hypotheses

Ref	Expression
tglineintmo.p	|- P = ( Base ` G )
tglineintmo.i	|- I = (Itv ` G )
tglineintmo.l	|- L = (LineG ` G )
tglineintmo.g	|- ( ph -> G e. TarskiG )
tglineinteq.a	|- ( ph -> A e. P )
tglineinteq.b	|- ( ph -> B e. P )
tglineinteq.c	|- ( ph -> C e. P )
tglineinteq.d	|- ( ph -> D e. P )
tglineinteq.e	|- ( ph -> -. ( A e. ( B L C ) \/ B = C ) )

Assertion

Ref	Expression
tglineneq	|- ( ph -> ( A L B ) =/= ( C L D ) )

Proof of Theorem tglineneq

Step	Hyp	Ref	Expression
1		tglineintmo.p	. . . . 5 |- P = ( Base ` G )
2		tglineintmo.i	. . . . 5 |- I = (Itv ` G )
3		tglineintmo.l	. . . . 5 |- L = (LineG ` G )
4		tglineintmo.g	. . . . 5 |- ( ph -> G e. TarskiG )
5		tglineinteq.a	. . . . 5 |- ( ph -> A e. P )
6		tglineinteq.b	. . . . 5 |- ( ph -> B e. P )
7		tglineinteq.c	. . . . . 6 |- ( ph -> C e. P )
8		tglineinteq.e	. . . . . 6 |- ( ph -> -. ( A e. ( B L C ) \/ B = C ) )
9	1, 2, 3, 4, 5, 6, 7, 8	ncolne1 23747	. . . . 5 |- ( ph -> A =/= B )
10		eqid 2467	. . . . . 6 |- ( dist ` G ) = ( dist ` G )
11	1, 10, 2, 4, 5, 6	tgbtwntriv1 23638	. . . . 5 |- ( ph -> A e. ( A I B ) )
12	1, 2, 3, 4, 5, 6, 5, 9, 11	btwnlng1 23741	. . . 4 |- ( ph -> A e. ( A L B ) )
13	12	adantr 465	. . 3 |- ( ( ph /\ C = D ) -> A e. ( A L B ) )
14		simplr 754	. . . 4 |- ( ( ( ph /\ C = D ) /\ A e. ( C L D ) ) -> C = D )
15	4	adantr 465	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> G e. TarskiG )
16	7	adantr 465	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> C e. P )
17		tglineinteq.d	. . . . . . . 8 |- ( ph -> D e. P )
18	17	adantr 465	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> D e. P )
19		simpr 461	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> A e. ( C L D ) )
20	1, 3, 2, 15, 16, 18, 19	tglngne 23693	. . . . . 6 |- ( ( ph /\ A e. ( C L D ) ) -> C =/= D )
21	20	adantlr 714	. . . . 5 |- ( ( ( ph /\ C = D ) /\ A e. ( C L D ) ) -> C =/= D )
22	21	neneqd 2669	. . . 4 |- ( ( ( ph /\ C = D ) /\ A e. ( C L D ) ) -> -. C = D )
23	14, 22	pm2.65da 576	. . 3 |- ( ( ph /\ C = D ) -> -. A e. ( C L D ) )
24		nelne1 2796	. . 3 |- ( ( A e. ( A L B ) /\ -. A e. ( C L D ) ) -> ( A L B ) =/= ( C L D ) )
25	13, 23, 24	syl2anc 661	. 2 |- ( ( ph /\ C = D ) -> ( A L B ) =/= ( C L D ) )
26	4	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> G e. TarskiG )
27	6	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> B e. P )
28	7	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. P )
29	5	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> A e. P )
30		pm2.46 398	. . . . . . . . 9 |- ( -. ( A e. ( B L C ) \/ B = C ) -> -. B = C )
31	8, 30	syl 16	. . . . . . . 8 |- ( ph -> -. B = C )
32	31	neqned 2670	. . . . . . 7 |- ( ph -> B =/= C )
33	32	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> B =/= C )
34	17	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> D e. P )
35		simplr 754	. . . . . . . 8 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C =/= D )
36	1, 10, 2, 4, 7, 17	tgbtwntriv1 23638	. . . . . . . . 9 |- ( ph -> C e. ( C I D ) )
37	36	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. ( C I D ) )
38	1, 2, 3, 26, 28, 34, 28, 35, 37	btwnlng1 23741	. . . . . . 7 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. ( C L D ) )
39		simpr 461	. . . . . . 7 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> ( A L B ) = ( C L D ) )
40	38, 39	eleqtrrd 2558	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. ( A L B ) )
41	1, 3, 2, 26, 29, 27, 40	tglngne 23693	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> A =/= B )
42	1, 2, 3, 26, 27, 28, 29, 33, 40, 41	lnrot1 23745	. . . . 5 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> A e. ( B L C ) )
43	42	orcd 392	. . . 4 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> ( A e. ( B L C ) \/ B = C ) )
44	8	ad2antrr 725	. . . 4 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> -. ( A e. ( B L C ) \/ B = C ) )
45	43, 44	pm2.65da 576	. . 3 |- ( ( ph /\ C =/= D ) -> -. ( A L B ) = ( C L D ) )
46	45	neqned 2670	. 2 |- ( ( ph /\ C =/= D ) -> ( A L B ) =/= ( C L D ) )
47	25, 46	pm2.61dane 2785	1 |- ( ph -> ( A L B ) =/= ( C L D ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   ` cfv 5588  (class class class)co 6284   Basecbs 14490   distcds 14564  TarskiGcstrkg 23581  Itvcitv 23588  LineGclng 23589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-trkgc 23600  df-trkgb 23601  df-trkgcb 23602  df-trkg 23606

This theorem is referenced by:  tglineinteq  23766  perpneq  23827