Theorem oppne3 25635

Description: Points lying on opposite sides of a line cannot be equal. (Contributed by Thierry Arnoux, 3-Aug-2020.)

Hypotheses

Ref	Expression
hpg.p	|- P = ( Base ` G )
hpg.d	|- .- = ( dist ` G )
hpg.i	|- I = (Itv ` G )
hpg.o	|- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
opphl.l	|- L = (LineG ` G )
opphl.d	|- ( ph -> D e. ran L )
opphl.g	|- ( ph -> G e. TarskiG )
oppcom.a	|- ( ph -> A e. P )
oppcom.b	|- ( ph -> B e. P )
oppcom.o	|- ( ph -> A O B )

Assertion

Ref	Expression
oppne3	|- ( ph -> A =/= B )

Distinct variable groups:   D, a, b   I, a, b   P, a, b   t, A   t, B   t, D   t, G   t, L   t, I   t, O   t, P   ph, t   t, .-   t, a, b

Allowed substitution hints:   ph( a, b)   A( a, b)   B( a, b)   G( a, b)   L( a, b)   .- ( a, b)   O( a, b)

Proof of Theorem oppne3

Step	Hyp	Ref	Expression
1		hpg.p	. . . . . 6 |- P = ( Base ` G )
2		eqid 2622	. . . . . 6 |- ( dist ` G ) = ( dist ` G )
3		hpg.i	. . . . . 6 |- I = (Itv ` G )
4		opphl.g	. . . . . . 7 |- ( ph -> G e. TarskiG )
5	4	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ph /\ A = B ) /\ t e. D ) /\ t e. ( A I B ) ) -> G e. TarskiG )
6		oppcom.a	. . . . . . 7 |- ( ph -> A e. P )
7	6	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ph /\ A = B ) /\ t e. D ) /\ t e. ( A I B ) ) -> A e. P )
8		opphl.l	. . . . . . 7 |- L = (LineG ` G )
9		opphl.d	. . . . . . . 8 |- ( ph -> D e. ran L )
10	9	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( ph /\ A = B ) /\ t e. D ) /\ t e. ( A I B ) ) -> D e. ran L )
11		simplr 792	. . . . . . 7 |- ( ( ( ( ph /\ A = B ) /\ t e. D ) /\ t e. ( A I B ) ) -> t e. D )
12	1, 8, 3, 5, 10, 11	tglnpt 25444	. . . . . 6 |- ( ( ( ( ph /\ A = B ) /\ t e. D ) /\ t e. ( A I B ) ) -> t e. P )
13		simpr 477	. . . . . . 7 |- ( ( ( ( ph /\ A = B ) /\ t e. D ) /\ t e. ( A I B ) ) -> t e. ( A I B ) )
14		simpllr 799	. . . . . . . 8 |- ( ( ( ( ph /\ A = B ) /\ t e. D ) /\ t e. ( A I B ) ) -> A = B )
15	14	oveq2d 6666	. . . . . . 7 |- ( ( ( ( ph /\ A = B ) /\ t e. D ) /\ t e. ( A I B ) ) -> ( A I A ) = ( A I B ) )
16	13, 15	eleqtrrd 2704	. . . . . 6 |- ( ( ( ( ph /\ A = B ) /\ t e. D ) /\ t e. ( A I B ) ) -> t e. ( A I A ) )
17	1, 2, 3, 5, 7, 12, 16	axtgbtwnid 25365	. . . . 5 |- ( ( ( ( ph /\ A = B ) /\ t e. D ) /\ t e. ( A I B ) ) -> A = t )
18	17, 11	eqeltrd 2701	. . . 4 |- ( ( ( ( ph /\ A = B ) /\ t e. D ) /\ t e. ( A I B ) ) -> A e. D )
19		oppcom.o	. . . . . . 7 |- ( ph -> A O B )
20		hpg.d	. . . . . . . 8 |- .- = ( dist ` G )
21		hpg.o	. . . . . . . 8 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
22		oppcom.b	. . . . . . . 8 |- ( ph -> B e. P )
23	1, 20, 3, 21, 6, 22	islnopp 25631	. . . . . . 7 |- ( ph -> ( A O B <-> ( ( -. A e. D /\ -. B e. D ) /\ E. t e. D t e. ( A I B ) ) ) )
24	19, 23	mpbid 222	. . . . . 6 |- ( ph -> ( ( -. A e. D /\ -. B e. D ) /\ E. t e. D t e. ( A I B ) ) )
25	24	simprd 479	. . . . 5 |- ( ph -> E. t e. D t e. ( A I B ) )
26	25	adantr 481	. . . 4 |- ( ( ph /\ A = B ) -> E. t e. D t e. ( A I B ) )
27	18, 26	r19.29a 3078	. . 3 |- ( ( ph /\ A = B ) -> A e. D )
28	1, 20, 3, 21, 8, 9, 4, 6, 22, 19	oppne1 25633	. . . 4 |- ( ph -> -. A e. D )
29	28	adantr 481	. . 3 |- ( ( ph /\ A = B ) -> -. A e. D )
30	27, 29	pm2.65da 600	. 2 |- ( ph -> -. A = B )
31	30	neqned 2801	1 |- ( ph -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   class class class wbr 4653   {copab 4712   ran crn 5115   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-trkgb 25348  df-trkg 25352

This theorem is referenced by:  colopp  25661  colhp  25662  trgcopyeulem  25697