Theorem islnoppd 25632

Description: Deduce that A and B lie on opposite sides of line L. (Contributed by Thierry Arnoux, 16-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 ) ) }
islnoppd.a	|- ( ph -> A e. P )
islnoppd.b	|- ( ph -> B e. P )
islnoppd.c	|- ( ph -> C e. D )
islnoppd.1	|- ( ph -> -. A e. D )
islnoppd.2	|- ( ph -> -. B e. D )
islnoppd.3	|- ( ph -> C e. ( A I B ) )

Assertion

Ref	Expression
islnoppd	|- ( ph -> A O B )

Distinct variable groups:   D, a, b   I, a, b   P, a, b   t, A   t, B   t, C   t, D, a, b   t, I   ph, t

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

Proof of Theorem islnoppd

Step	Hyp	Ref	Expression
1		islnoppd.1	. . 3 |- ( ph -> -. A e. D )
2		islnoppd.2	. . 3 |- ( ph -> -. B e. D )
3		islnoppd.c	. . . 4 |- ( ph -> C e. D )
4		simpr 477	. . . . 5 |- ( ( ph /\ t = C ) -> t = C )
5	4	eleq1d 2686	. . . 4 |- ( ( ph /\ t = C ) -> ( t e. ( A I B ) <-> C e. ( A I B ) ) )
6		islnoppd.3	. . . 4 |- ( ph -> C e. ( A I B ) )
7	3, 5, 6	rspcedvd 3317	. . 3 |- ( ph -> E. t e. D t e. ( A I B ) )
8	1, 2, 7	jca31 557	. 2 |- ( ph -> ( ( -. A e. D /\ -. B e. D ) /\ E. t e. D t e. ( A I B ) ) )
9		hpg.p	. . 3 |- P = ( Base ` G )
10		hpg.d	. . 3 |- .- = ( dist ` G )
11		hpg.i	. . 3 |- I = (Itv ` G )
12		hpg.o	. . 3 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
13		islnoppd.a	. . 3 |- ( ph -> A e. P )
14		islnoppd.b	. . 3 |- ( ph -> B e. P )
15	9, 10, 11, 12, 13, 14	islnopp 25631	. 2 |- ( ph -> ( A O B <-> ( ( -. A e. D /\ -. B e. D ) /\ E. t e. D t e. ( A I B ) ) ) )
16	8, 15	mpbird 247	1 |- ( ph -> A O B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   class class class wbr 4653   {copab 4712   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  Itvcitv 25335

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-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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  opphllem2  25640  outpasch  25647