Theorem tglineneq 24768

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 24749	. . . . 5 |- ( ph -> A =/= B )
10	1, 2, 3, 4, 5, 6, 9	tglinerflx1 24757	. . . 4 |- ( ph -> A e. ( A L B ) )
11	10	adantr 472	. . 3 |- ( ( ph /\ C = D ) -> A e. ( A L B ) )
12		simplr 770	. . . 4 |- ( ( ( ph /\ C = D ) /\ A e. ( C L D ) ) -> C = D )
13	4	adantr 472	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> G e. TarskiG )
14	7	adantr 472	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> C e. P )
15		tglineinteq.d	. . . . . . . 8 |- ( ph -> D e. P )
16	15	adantr 472	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> D e. P )
17		simpr 468	. . . . . . 7 |- ( ( ph /\ A e. ( C L D ) ) -> A e. ( C L D ) )
18	1, 3, 2, 13, 14, 16, 17	tglngne 24674	. . . . . 6 |- ( ( ph /\ A e. ( C L D ) ) -> C =/= D )
19	18	adantlr 729	. . . . 5 |- ( ( ( ph /\ C = D ) /\ A e. ( C L D ) ) -> C =/= D )
20	19	neneqd 2648	. . . 4 |- ( ( ( ph /\ C = D ) /\ A e. ( C L D ) ) -> -. C = D )
21	12, 20	pm2.65da 586	. . 3 |- ( ( ph /\ C = D ) -> -. A e. ( C L D ) )
22		nelne1 2739	. . 3 |- ( ( A e. ( A L B ) /\ -. A e. ( C L D ) ) -> ( A L B ) =/= ( C L D ) )
23	11, 21, 22	syl2anc 673	. 2 |- ( ( ph /\ C = D ) -> ( A L B ) =/= ( C L D ) )
24	4	ad2antrr 740	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> G e. TarskiG )
25	6	ad2antrr 740	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> B e. P )
26	7	ad2antrr 740	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. P )
27	5	ad2antrr 740	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> A e. P )
28		pm2.46 405	. . . . . . . . 9 |- ( -. ( A e. ( B L C ) \/ B = C ) -> -. B = C )
29	8, 28	syl 17	. . . . . . . 8 |- ( ph -> -. B = C )
30	29	neqned 2650	. . . . . . 7 |- ( ph -> B =/= C )
31	30	ad2antrr 740	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> B =/= C )
32	15	ad2antrr 740	. . . . . . . 8 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> D e. P )
33		simplr 770	. . . . . . . 8 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C =/= D )
34	1, 2, 3, 24, 26, 32, 33	tglinerflx1 24757	. . . . . . 7 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. ( C L D ) )
35		simpr 468	. . . . . . 7 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> ( A L B ) = ( C L D ) )
36	34, 35	eleqtrrd 2552	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> C e. ( A L B ) )
37	1, 3, 2, 24, 27, 25, 36	tglngne 24674	. . . . . 6 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> A =/= B )
38	1, 2, 3, 24, 25, 26, 27, 31, 36, 37	lnrot1 24747	. . . . 5 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> A e. ( B L C ) )
39	38	orcd 399	. . . 4 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> ( A e. ( B L C ) \/ B = C ) )
40	8	ad2antrr 740	. . . 4 |- ( ( ( ph /\ C =/= D ) /\ ( A L B ) = ( C L D ) ) -> -. ( A e. ( B L C ) \/ B = C ) )
41	39, 40	pm2.65da 586	. . 3 |- ( ( ph /\ C =/= D ) -> -. ( A L B ) = ( C L D ) )
42	41	neqned 2650	. 2 |- ( ( ph /\ C =/= D ) -> ( A L B ) =/= ( C L D ) )
43	23, 42	pm2.61dane 2730	1 |- ( ph -> ( A L B ) =/= ( C L D ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 375   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   ` cfv 5589  (class class class)co 6308   Basecbs 15199  TarskiGcstrkg 24557  Itvcitv 24563  LineGclng 24564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-trkgc 24575  df-trkgb 24576  df-trkgcb 24577  df-trkg 24580

This theorem is referenced by:  tglineinteq  24769  perpneq  24838