Theorem tglineneq 24548

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 24529	. . . . 5 |- ( ph -> A =/= B )
10	1, 2, 3, 4, 5, 6, 9	tglinerflx1 24537	. . . 4 |- ( ph -> A e. ( A L B ) )
11	10	adantr 466	. . 3 |- ( ( ph /\ C = D ) -> A e. ( A L B ) )
12		simplr 760	. . . 4 |- ( ( ( ph /\ C = D ) /\ A e. ( C L D ) ) -> C = D )
13	4	adantr 466	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> G e. TarskiG )
14	7	adantr 466	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> C e. P )
15		tglineinteq.d	. . . . . . . 8 |- ( ph -> D e. P )
16	15	adantr 466	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> D e. P )
17		simpr 462	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> A e. ( C L D ) )
18	1, 3, 2, 13, 14, 16, 17	tglngne 24455	. . . . . 6 |- ( ( ph /\ A e. ( C L D ) ) -> C =/= D )
19	18	adantlr 719	. . . . 5 |- ( ( ( ph /\ C = D ) /\ A e. ( C L D ) ) -> C =/= D )
20	19	neneqd 2632	. . . 4 |- ( ( ( ph /\ C = D ) /\ A e. ( C L D ) ) -> -. C = D )
21	12, 20	pm2.65da 578	. . 3 |- ( ( ph /\ C = D ) -> -. A e. ( C L D ) )
22		nelne1 2760	. . 3 |- ( ( A e. ( A L B ) /\ -. A e. ( C L D ) ) -> ( A L B ) =/= ( C L D ) )
23	11, 21, 22	syl2anc 665	. 2 |- ( ( ph /\ C = D ) -> ( A L B ) =/= ( C L D ) )
24	4	ad2antrr 730	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> G e. TarskiG )
25	6	ad2antrr 730	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> B e. P )
26	7	ad2antrr 730	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. P )
27	5	ad2antrr 730	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> A e. P )
28		pm2.46 399	. . . . . . . . 9 |- ( -. ( A e. ( B L C ) \/ B = C ) -> -. B = C )
29	8, 28	syl 17	. . . . . . . 8 |- ( ph -> -. B = C )
30	29	neqned 2634	. . . . . . 7 |- ( ph -> B =/= C )
31	30	ad2antrr 730	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> B =/= C )
32	15	ad2antrr 730	. . . . . . . 8 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> D e. P )
33		simplr 760	. . . . . . . 8 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C =/= D )
34	1, 2, 3, 24, 26, 32, 33	tglinerflx1 24537	. . . . . . 7 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. ( C L D ) )
35		simpr 462	. . . . . . 7 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> ( A L B ) = ( C L D ) )
36	34, 35	eleqtrrd 2520	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. ( A L B ) )
37	1, 3, 2, 24, 27, 25, 36	tglngne 24455	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> A =/= B )
38	1, 2, 3, 24, 25, 26, 27, 31, 36, 37	lnrot1 24527	. . . . 5 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> A e. ( B L C ) )
39	38	orcd 393	. . . 4 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> ( A e. ( B L C ) \/ B = C ) )
40	8	ad2antrr 730	. . . 4 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> -. ( A e. ( B L C ) \/ B = C ) )
41	39, 40	pm2.65da 578	. . 3 |- ( ( ph /\ C =/= D ) -> -. ( A L B ) = ( C L D ) )
42	41	neqned 2634	. 2 |- ( ( ph /\ C =/= D ) -> ( A L B ) =/= ( C L D ) )
43	23, 42	pm2.61dane 2749	1 |- ( ph -> ( A L B ) =/= ( C L D ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 369   /\ wa 370   = wceq 1437   e. wcel 1870   =/= wne 2625   ` cfv 5601  (class class class)co 6305   Basecbs 15084  TarskiGcstrkg 24341  Itvcitv 24347  LineGclng 24348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-trkgc 24359  df-trkgb 24360  df-trkgcb 24361  df-trkg 24364

This theorem is referenced by:  tglineinteq  24549  perpneq  24616