Theorem tglineneq 25539

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 25520	. . . . 5 |- ( ph -> A =/= B )
10	1, 2, 3, 4, 5, 6, 9	tglinerflx1 25528	. . . 4 |- ( ph -> A e. ( A L B ) )
11	10	adantr 481	. . 3 |- ( ( ph /\ C = D ) -> A e. ( A L B ) )
12		simplr 792	. . . 4 |- ( ( ( ph /\ C = D ) /\ A e. ( C L D ) ) -> C = D )
13	4	adantr 481	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> G e. TarskiG )
14	7	adantr 481	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> C e. P )
15		tglineinteq.d	. . . . . . . 8 |- ( ph -> D e. P )
16	15	adantr 481	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> D e. P )
17		simpr 477	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> A e. ( C L D ) )
18	1, 3, 2, 13, 14, 16, 17	tglngne 25445	. . . . . 6 |- ( ( ph /\ A e. ( C L D ) ) -> C =/= D )
19	18	adantlr 751	. . . . 5 |- ( ( ( ph /\ C = D ) /\ A e. ( C L D ) ) -> C =/= D )
20	19	neneqd 2799	. . . 4 |- ( ( ( ph /\ C = D ) /\ A e. ( C L D ) ) -> -. C = D )
21	12, 20	pm2.65da 600	. . 3 |- ( ( ph /\ C = D ) -> -. A e. ( C L D ) )
22		nelne1 2890	. . 3 |- ( ( A e. ( A L B ) /\ -. A e. ( C L D ) ) -> ( A L B ) =/= ( C L D ) )
23	11, 21, 22	syl2anc 693	. 2 |- ( ( ph /\ C = D ) -> ( A L B ) =/= ( C L D ) )
24	4	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> G e. TarskiG )
25	6	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> B e. P )
26	7	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. P )
27	5	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> A e. P )
28		pm2.46 413	. . . . . . . . 9 |- ( -. ( A e. ( B L C ) \/ B = C ) -> -. B = C )
29	8, 28	syl 17	. . . . . . . 8 |- ( ph -> -. B = C )
30	29	neqned 2801	. . . . . . 7 |- ( ph -> B =/= C )
31	30	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> B =/= C )
32	15	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> D e. P )
33		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C =/= D )
34	1, 2, 3, 24, 26, 32, 33	tglinerflx1 25528	. . . . . . 7 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. ( C L D ) )
35		simpr 477	. . . . . . 7 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> ( A L B ) = ( C L D ) )
36	34, 35	eleqtrrd 2704	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. ( A L B ) )
37	1, 3, 2, 24, 27, 25, 36	tglngne 25445	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> A =/= B )
38	1, 2, 3, 24, 25, 26, 27, 31, 36, 37	lnrot1 25518	. . . . 5 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> A e. ( B L C ) )
39	38	orcd 407	. . . 4 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> ( A e. ( B L C ) \/ B = C ) )
40	8	ad2antrr 762	. . . 4 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> -. ( A e. ( B L C ) \/ B = C ) )
41	39, 40	pm2.65da 600	. . 3 |- ( ( ph /\ C =/= D ) -> -. ( A L B ) = ( C L D ) )
42	41	neqned 2801	. 2 |- ( ( ph /\ C =/= D ) -> ( A L B ) =/= ( C L D ) )
43	23, 42	pm2.61dane 2881	1 |- ( ph -> ( A L B ) =/= ( C L D ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336

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-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-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-ral 2917  df-rex 2918  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352

This theorem is referenced by:  tglineinteq  25540  perpneq  25609