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Theorem oppcom 25636
Description: Commutativity rule for "opposite" Theorem 9.2 of [Schwabhauser] p. 67. (Contributed by Thierry Arnoux, 19-Dec-2019.)
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
oppcom |- ( ph -> B O A )
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 oppcom
StepHypRef Expression
1 oppcom.o . . . . . 6 |- ( ph -> A O B )
2 hpg.p . . . . . . 7 |- P = ( Base ` G )
3 hpg.d . . . . . . 7 |- .- = ( dist ` G )
4 hpg.i . . . . . . 7 |- I = (Itv ` G )
5 hpg.o . . . . . . 7 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
6 oppcom.a . . . . . . 7 |- ( ph -> A e. P )
7 oppcom.b . . . . . . 7 |- ( ph -> B e. P )
82, 3, 4, 5, 6, 7islnopp 25631 . . . . . 6 |- ( ph -> ( A O B <-> ( ( -. A e. D /\ -. B e. D ) /\ E. t e. D t e. ( A I B ) ) ) )
91, 8mpbid 222 . . . . 5 |- ( ph -> ( ( -. A e. D /\ -. B e. D ) /\ E. t e. D t e. ( A I B ) ) )
109simpld 475 . . . 4 |- ( ph -> ( -. A e. D /\ -. B e. D ) )
1110simprd 479 . . 3 |- ( ph -> -. B e. D )
1210simpld 475 . . 3 |- ( ph -> -. A e. D )
139simprd 479 . . . 4 |- ( ph -> E. t e. D t e. ( A I B ) )
14 opphl.g . . . . . . . 8 |- ( ph -> G e. TarskiG )
1514ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ t e. D ) /\ t e. ( A I B ) ) -> G e. TarskiG )
166ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ t e. D ) /\ t e. ( A I B ) ) -> A e. P )
17 opphl.l . . . . . . . . 9 |- L = (LineG ` G )
1814adantr 481 . . . . . . . . 9 |- ( ( ph /\ t e. D ) -> G e. TarskiG )
19 opphl.d . . . . . . . . . 10 |- ( ph -> D e. ran L )
2019adantr 481 . . . . . . . . 9 |- ( ( ph /\ t e. D ) -> D e. ran L )
21 simpr 477 . . . . . . . . 9 |- ( ( ph /\ t e. D ) -> t e. D )
222, 17, 4, 18, 20, 21tglnpt 25444 . . . . . . . 8 |- ( ( ph /\ t e. D ) -> t e. P )
2322adantr 481 . . . . . . 7 |- ( ( ( ph /\ t e. D ) /\ t e. ( A I B ) ) -> t e. P )
247ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ t e. D ) /\ t e. ( A I B ) ) -> B e. P )
25 simpr 477 . . . . . . 7 |- ( ( ( ph /\ t e. D ) /\ t e. ( A I B ) ) -> t e. ( A I B ) )
262, 3, 4, 15, 16, 23, 24, 25tgbtwncom 25383 . . . . . 6 |- ( ( ( ph /\ t e. D ) /\ t e. ( A I B ) ) -> t e. ( B I A ) )
2714ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ t e. D ) /\ t e. ( B I A ) ) -> G e. TarskiG )
287ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ t e. D ) /\ t e. ( B I A ) ) -> B e. P )
2922adantr 481 . . . . . . 7 |- ( ( ( ph /\ t e. D ) /\ t e. ( B I A ) ) -> t e. P )
306ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ t e. D ) /\ t e. ( B I A ) ) -> A e. P )
31 simpr 477 . . . . . . 7 |- ( ( ( ph /\ t e. D ) /\ t e. ( B I A ) ) -> t e. ( B I A ) )
322, 3, 4, 27, 28, 29, 30, 31tgbtwncom 25383 . . . . . 6 |- ( ( ( ph /\ t e. D ) /\ t e. ( B I A ) ) -> t e. ( A I B ) )
3326, 32impbida 877 . . . . 5 |- ( ( ph /\ t e. D ) -> ( t e. ( A I B ) <-> t e. ( B I A ) ) )
3433rexbidva 3049 . . . 4 |- ( ph -> ( E. t e. D t e. ( A I B ) <-> E. t e. D t e. ( B I A ) ) )
3513, 34mpbid 222 . . 3 |- ( ph -> E. t e. D t e. ( B I A ) )
3611, 12, 35jca31 557 . 2 |- ( ph -> ( ( -. B e. D /\ -. A e. D ) /\ E. t e. D t e. ( B I A ) ) )
372, 3, 4, 5, 7, 6islnopp 25631 . 2 |- ( ph -> ( B O A <-> ( ( -. B e. D /\ -. A e. D ) /\ E. t e. D t e. ( B I A ) ) ) )
3836, 37mpbird 247 1 |- ( ph -> B O A )
Colors of variables: wff setvar class
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   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-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-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352
This theorem is referenced by:  opphllem2  25640  opphllem4  25642  opphllem5  25643  opphllem6  25644  lnperpex  25695
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