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Theorem opphl 25646
Description: If two points A and C lie on the opposite side of a line D then any point of the half line ( R A) also lies opposite to C. Theorem 9.5 of [Schwabhauser] p. 69. (Contributed by Thierry Arnoux, 3-Mar-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 )
opphl.k |- K = (hlG ` G )
opphl.a |- ( ph -> A e. P )
opphl.b |- ( ph -> B e. P )
opphl.c |- ( ph -> C e. P )
opphl.1 |- ( ph -> A O C )
opphl.2 |- ( ph -> R e. D )
opphl.3 |- ( ph -> A ( K ` R ) B )
Assertion
Ref Expression
opphl |- ( ph -> B O C )
Distinct variable groups:   D, a, b   I, a, b   P, a, b   t, A   t, B   t, D   t, R   t, C   t, G   t, L   t, I   t, K   t, O   t, P   ph, t   t, .-   t, a, b
Allowed substitution hints:   ph( a, b)   A( a, b)   B( a, b)   C( a, b)   R( a, b)   G( a, b)   K( a, b)   L( a, b)   .- ( a, b)   O( a, b)
Proof of Theorem opphl
Dummy variables m x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hpg.p . . . . . 6 |- P = ( Base ` G )
2 hpg.d . . . . . 6 |- .- = ( dist ` G )
3 hpg.i . . . . . 6 |- I = (Itv ` G )
4 hpg.o . . . . . 6 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
5 opphl.l . . . . . 6 |- L = (LineG ` G )
6 opphl.d . . . . . . 7 |- ( ph -> D e. ran L )
76ad8antr 776 . . . . . 6 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> D e. ran L )
8 opphl.g . . . . . . 7 |- ( ph -> G e. TarskiG )
98ad8antr 776 . . . . . 6 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> G e. TarskiG )
10 opphl.k . . . . . 6 |- K = (hlG ` G )
11 eqid 2622 . . . . . 6 |- ( (pInvG ` G ) ` m ) = ( (pInvG ` G ) ` m )
12 opphl.b . . . . . . 7 |- ( ph -> B e. P )
1312ad8antr 776 . . . . . 6 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> B e. P )
14 eqid 2622 . . . . . . 7 |- (pInvG ` G ) = (pInvG ` G )
15 simp-4r 807 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> m e. P )
16 opphl.a . . . . . . . 8 |- ( ph -> A e. P )
1716ad8antr 776 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> A e. P )
181, 2, 3, 5, 14, 9, 15, 11, 17mircl 25556 . . . . . 6 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( ( (pInvG ` G ) ` m ) ` A ) e. P )
19 simplr 792 . . . . . 6 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> y e. D )
20 simp-6r 811 . . . . . 6 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> z e. D )
21 opphl.2 . . . . . . . 8 |- ( ph -> R e. D )
2221ad8antr 776 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> R e. D )
23 opphl.c . . . . . . . . 9 |- ( ph -> C e. P )
2423ad8antr 776 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> C e. P )
25 simp-8r 815 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> x e. D )
26 opphl.1 . . . . . . . . 9 |- ( ph -> A O C )
2726ad8antr 776 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> A O C )
28 simp-7r 813 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( A L x ) (⟂G ` G ) D )
295, 9, 28perpln1 25605 . . . . . . . . 9 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( A L x ) e. ran L )
301, 2, 3, 5, 9, 29, 7, 28perpcom 25608 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> D (⟂G ` G ) ( A L x ) )
31 simp-5r 809 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( C L z ) (⟂G ` G ) D )
325, 9, 31perpln1 25605 . . . . . . . . 9 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( C L z ) e. ran L )
331, 2, 3, 5, 9, 32, 7, 31perpcom 25608 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> D (⟂G ` G ) ( C L z ) )
341, 5, 3, 9, 7, 25tglnpt 25444 . . . . . . . . 9 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> x e. P )
351, 3, 5, 9, 17, 34, 29tglnne 25523 . . . . . . . . 9 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> A =/= x )
361, 3, 10, 17, 17, 34, 9, 35hlid 25504 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> A ( K ` x ) A )
37 simpllr 799 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> z = ( ( (pInvG ` G ) ` m ) ` x ) )
3837eqcomd 2628 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( ( (pInvG ` G ) ` m ) ` x ) = z )
391, 2, 3, 4, 5, 7, 9, 10, 11, 17, 24, 25, 20, 15, 27, 30, 33, 17, 38opphllem6 25644 . . . . . . . . 9 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( A ( K ` x ) A <-> ( ( (pInvG ` G ) ` m ) ` A ) ( K ` z ) C ) )
4036, 39mpbid 222 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( ( (pInvG ` G ) ` m ) ` A ) ( K ` z ) C )
411, 2, 3, 4, 5, 7, 9, 10, 11, 17, 24, 25, 20, 15, 27, 30, 33, 17, 18, 36, 40opphllem5 25643 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> A O ( ( (pInvG ` G ) ` m ) ` A ) )
4238, 20eqeltrd 2701 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( ( (pInvG ` G ) ` m ) ` x ) e. D )
431, 2, 3, 5, 14, 9, 11, 7, 15, 25, 42mirln2 25572 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> m e. D )
441, 2, 3, 5, 14, 9, 15, 11, 17mirmir 25557 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( ( (pInvG ` G ) ` m ) ` ( ( (pInvG ` G ) ` m ) ` A ) ) = A )
4544eqcomd 2628 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> A = ( ( (pInvG ` G ) ` m ) ` ( ( (pInvG ` G ) ` m ) ` A ) ) )
461, 5, 3, 8, 6, 21tglnpt 25444 . . . . . . . . 9 |- ( ph -> R e. P )
47 opphl.3 . . . . . . . . 9 |- ( ph -> A ( K ` R ) B )
481, 3, 10, 16, 12, 46, 8, 47hlne1 25500 . . . . . . . 8 |- ( ph -> A =/= R )
4948ad8antr 776 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> A =/= R )
501, 3, 10, 16, 12, 46, 8, 5, 47hlln 25502 . . . . . . . . 9 |- ( ph -> A e. ( B L R ) )
511, 5, 3, 8, 12, 46, 50tglngne 25445 . . . . . . . 8 |- ( ph -> B =/= R )
5251ad8antr 776 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> B =/= R )
531, 3, 10, 16, 12, 46, 8ishlg 25497 . . . . . . . . . 10 |- ( ph -> ( A ( K ` R ) B <-> ( A =/= R /\ B =/= R /\ ( A e. ( R I B ) \/ B e. ( R I A ) ) ) ) )
5447, 53mpbid 222 . . . . . . . . 9 |- ( ph -> ( A =/= R /\ B =/= R /\ ( A e. ( R I B ) \/ B e. ( R I A ) ) ) )
5554simp3d 1075 . . . . . . . 8 |- ( ph -> ( A e. ( R I B ) \/ B e. ( R I A ) ) )
5655ad8antr 776 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( A e. ( R I B ) \/ B e. ( R I A ) ) )
571, 2, 3, 4, 5, 7, 9, 11, 17, 13, 18, 22, 41, 43, 45, 49, 52, 56opphllem2 25640 . . . . . 6 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> B O ( ( (pInvG ` G ) ` m ) ` A ) )
58 simpr 477 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( B L y ) (⟂G ` G ) D )
595, 9, 58perpln1 25605 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( B L y ) e. ran L )
601, 2, 3, 5, 9, 59, 7, 58perpcom 25608 . . . . . 6 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> D (⟂G ` G ) ( B L y ) )
611, 5, 3, 9, 7, 20tglnpt 25444 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> z e. P )
621, 3, 10, 18, 24, 61, 9, 40hlne1 25500 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( ( (pInvG ` G ) ` m ) ` A ) =/= z )
631, 3, 10, 18, 24, 61, 9, 5, 40hlln 25502 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( ( (pInvG ` G ) ` m ) ` A ) e. ( C L z ) )
641, 5, 3, 9, 24, 61, 63tglngne 25445 . . . . . . . . 9 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> C =/= z )
651, 3, 5, 9, 24, 61, 64tglinerflx2 25529 . . . . . . . 8 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> z e. ( C L z ) )
661, 3, 5, 9, 18, 61, 62, 62, 32, 63, 65tglinethru 25531 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> ( C L z ) = ( ( ( (pInvG ` G ) ` m ) ` A ) L z ) )
6733, 66breqtrd 4679 . . . . . 6 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> D (⟂G ` G ) ( ( ( (pInvG ` G ) ` m ) ` A ) L z ) )
681, 5, 3, 9, 7, 19tglnpt 25444 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> y e. P )
691, 3, 5, 9, 13, 68, 59tglnne 25523 . . . . . . 7 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> B =/= y )
701, 3, 10, 13, 17, 68, 9, 69hlid 25504 . . . . . 6 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> B ( K ` y ) B )
711, 3, 10, 18, 24, 61, 9, 40hlcomd 25499 . . . . . 6 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> C ( K ` z ) ( ( (pInvG ` G ) ` m ) ` A ) )
721, 2, 3, 4, 5, 7, 9, 10, 11, 13, 18, 19, 20, 15, 57, 60, 67, 13, 24, 70, 71opphllem5 25643 . . . . 5 |- ( ( ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) /\ y e. D ) /\ ( B L y ) (⟂G ` G ) D ) -> B O C )
731, 2, 3, 4, 5, 6, 8, 16, 23, 26oppne1 25633 . . . . . . . 8 |- ( ph -> -. A e. D )
7450adantr 481 . . . . . . . . 9 |- ( ( ph /\ B e. D ) -> A e. ( B L R ) )
758adantr 481 . . . . . . . . . 10 |- ( ( ph /\ B e. D ) -> G e. TarskiG )
7612adantr 481 . . . . . . . . . 10 |- ( ( ph /\ B e. D ) -> B e. P )
7746adantr 481 . . . . . . . . . 10 |- ( ( ph /\ B e. D ) -> R e. P )
7851adantr 481 . . . . . . . . . 10 |- ( ( ph /\ B e. D ) -> B =/= R )
796adantr 481 . . . . . . . . . 10 |- ( ( ph /\ B e. D ) -> D e. ran L )
80 simpr 477 . . . . . . . . . 10 |- ( ( ph /\ B e. D ) -> B e. D )
8121adantr 481 . . . . . . . . . 10 |- ( ( ph /\ B e. D ) -> R e. D )
821, 3, 5, 75, 76, 77, 78, 78, 79, 80, 81tglinethru 25531 . . . . . . . . 9 |- ( ( ph /\ B e. D ) -> D = ( B L R ) )
8374, 82eleqtrrd 2704 . . . . . . . 8 |- ( ( ph /\ B e. D ) -> A e. D )
8473, 83mtand 691 . . . . . . 7 |- ( ph -> -. B e. D )
851, 2, 3, 5, 8, 6, 12, 84footex 25613 . . . . . 6 |- ( ph -> E. y e. D ( B L y ) (⟂G ` G ) D )
8685ad6antr 772 . . . . 5 |- ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) -> E. y e. D ( B L y ) (⟂G ` G ) D )
8772, 86r19.29a 3078 . . . 4 |- ( ( ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) /\ m e. P ) /\ z = ( ( (pInvG ` G ) ` m ) ` x ) ) -> B O C )
888ad4antr 768 . . . . 5 |- ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) -> G e. TarskiG )
896ad4antr 768 . . . . . 6 |- ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) -> D e. ran L )
90 simp-4r 807 . . . . . 6 |- ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) -> x e. D )
911, 5, 3, 88, 89, 90tglnpt 25444 . . . . 5 |- ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) -> x e. P )
92 simplr 792 . . . . . 6 |- ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) -> z e. D )
931, 5, 3, 88, 89, 92tglnpt 25444 . . . . 5 |- ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) -> z e. P )
941, 2, 3, 4, 5, 6, 8, 16, 23, 26opptgdim2 25637 . . . . . 6 |- ( ph -> GDimTarskiG≥ 2 )
9594ad4antr 768 . . . . 5 |- ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) -> GDimTarskiG≥ 2 )
961, 2, 3, 5, 88, 14, 91, 93, 95midex 25629 . . . 4 |- ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) -> E. m e. P z = ( ( (pInvG ` G ) ` m ) ` x ) )
9787, 96r19.29a 3078 . . 3 |- ( ( ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) /\ z e. D ) /\ ( C L z ) (⟂G ` G ) D ) -> B O C )
981, 2, 3, 4, 5, 6, 8, 16, 23, 26oppne2 25634 . . . . 5 |- ( ph -> -. C e. D )
991, 2, 3, 5, 8, 6, 23, 98footex 25613 . . . 4 |- ( ph -> E. z e. D ( C L z ) (⟂G ` G ) D )
10099ad2antrr 762 . . 3 |- ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> E. z e. D ( C L z ) (⟂G ` G ) D )
10197, 100r19.29a 3078 . 2 |- ( ( ( ph /\ x e. D ) /\ ( A L x ) (⟂G ` G ) D ) -> B O C )
1021, 2, 3, 5, 8, 6, 16, 73footex 25613 . 2 |- ( ph -> E. x e. D ( A L x ) (⟂G ` G ) D )
103101, 102r19.29a 3078 1 |- ( ph -> B O C )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = 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   2c2 11070   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  DimTarskiG≥cstrkgld 25333  Itvcitv 25335  LineGclng 25336  hlGchlg 25495  pInvGcmir 25547  ⟂Gcperpg 25590
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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkgld 25351  df-trkg 25352  df-cgrg 25406  df-leg 25478  df-hlg 25496  df-mir 25548  df-rag 25589  df-perpg 25591
This theorem is referenced by:  outpasch  25647  lnopp2hpgb  25655
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