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Theorem colperpexlem3 24631
Description: Lemma for colperpex 24632. Case 1 of theorem 8.21 of [Schwabhauser] p. 63. (Contributed by Thierry Arnoux, 20-Nov-2019.)
Hypotheses
Ref Expression
colperpex.p  |-  P  =  ( Base `  G
)
colperpex.d  |-  .-  =  ( dist `  G )
colperpex.i  |-  I  =  (Itv `  G )
colperpex.l  |-  L  =  (LineG `  G )
colperpex.g  |-  ( ph  ->  G  e. TarskiG )
colperpex.1  |-  ( ph  ->  A  e.  P )
colperpex.2  |-  ( ph  ->  B  e.  P )
colperpex.3  |-  ( ph  ->  C  e.  P )
colperpex.4  |-  ( ph  ->  A  =/=  B )
colperpexlem3.1  |-  ( ph  ->  -.  C  e.  ( A L B ) )
Assertion
Ref Expression
colperpexlem3  |-  ( ph  ->  E. p  e.  P  ( ( A L p ) (⟂G `  G
) ( A L B )  /\  E. t  e.  P  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( C I p ) ) ) )
Distinct variable groups:    .- , p, t    A, p, t    B, p, t    C, p, t    G, p, t    I, p, t    L, p, t    P, p, t    ph, p, t

Proof of Theorem colperpexlem3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 colperpex.p . . . 4  |-  P  =  ( Base `  G
)
2 colperpex.d . . . 4  |-  .-  =  ( dist `  G )
3 colperpex.i . . . 4  |-  I  =  (Itv `  G )
4 colperpex.l . . . 4  |-  L  =  (LineG `  G )
5 eqid 2429 . . . 4  |-  (pInvG `  G )  =  (pInvG `  G )
6 colperpex.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
76ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  G  e. TarskiG )
8 eqid 2429 . . . 4  |-  ( (pInvG `  G ) `  p
)  =  ( (pInvG `  G ) `  p
)
9 colperpex.1 . . . . . . . 8  |-  ( ph  ->  A  e.  P )
10 colperpex.2 . . . . . . . 8  |-  ( ph  ->  B  e.  P )
11 colperpex.4 . . . . . . . 8  |-  ( ph  ->  A  =/=  B )
121, 3, 4, 6, 9, 10, 11tgelrnln 24534 . . . . . . 7  |-  ( ph  ->  ( A L B )  e.  ran  L
)
1312ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( A L B )  e.  ran  L )
14 simplr 760 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  x  e.  ( A L B ) )
151, 4, 3, 7, 13, 14tglnpt 24454 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  x  e.  P
)
16 eqid 2429 . . . . 5  |-  ( (pInvG `  G ) `  x
)  =  ( (pInvG `  G ) `  x
)
17 colperpex.3 . . . . . 6  |-  ( ph  ->  C  e.  P )
1817ad2antrr 730 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  C  e.  P
)
191, 2, 3, 4, 5, 7, 15, 16, 18mircl 24566 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( ( (pInvG `  G ) `  x
) `  C )  e.  P )
209ad2antrr 730 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  A  e.  P
)
21 eqid 2429 . . . . 5  |-  ( (pInvG `  G ) `  A
)  =  ( (pInvG `  G ) `  A
)
221, 2, 3, 4, 5, 7, 20, 21, 18mircl 24566 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( ( (pInvG `  G ) `  A
) `  C )  e.  P )
231, 2, 3, 4, 5, 7, 20, 21, 18mircgr 24562 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( A  .-  ( ( (pInvG `  G ) `  A
) `  C )
)  =  ( A 
.-  C ) )
2410ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  B  e.  P
)
25 colperpexlem3.1 . . . . . . . . . . 11  |-  ( ph  ->  -.  C  e.  ( A L B ) )
2625ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  -.  C  e.  ( A L B ) )
27 nelne2 2761 . . . . . . . . . 10  |-  ( ( x  e.  ( A L B )  /\  -.  C  e.  ( A L B ) )  ->  x  =/=  C
)
2814, 26, 27syl2anc 665 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  x  =/=  C
)
291, 3, 4, 7, 15, 18, 28tgelrnln 24534 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( x L C )  e.  ran  L )
301, 3, 4, 7, 15, 18, 28tglinecom 24539 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( x L C )  =  ( C L x ) )
31 simpr 462 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( C L x ) (⟂G `  G
) ( A L B ) )
3230, 31eqbrtrd 4446 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( x L C ) (⟂G `  G
) ( A L B ) )
331, 2, 3, 4, 7, 29, 13, 32perpcom 24615 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( A L B ) (⟂G `  G
) ( x L C ) )
341, 2, 3, 4, 7, 20, 24, 14, 18, 33perprag 24625 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  <" A x C ">  e.  (∟G `  G ) )
351, 2, 3, 4, 5, 7, 20, 15, 18israg 24599 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( <" A x C ">  e.  (∟G `  G )  <->  ( A  .-  C )  =  ( A  .-  ( ( (pInvG `  G ) `  x ) `  C
) ) ) )
3634, 35mpbid 213 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( A  .-  C )  =  ( A  .-  ( ( (pInvG `  G ) `  x ) `  C
) ) )
3723, 36eqtr2d 2471 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( A  .-  ( ( (pInvG `  G ) `  x
) `  C )
)  =  ( A 
.-  ( ( (pInvG `  G ) `  A
) `  C )
) )
381, 2, 3, 4, 5, 7, 8, 19, 22, 20, 37midexlem 24594 . . 3  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  E. p  e.  P  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )
397ad2antrr 730 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  G  e. TarskiG )
4022ad2antrr 730 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  -> 
( ( (pInvG `  G ) `  A
) `  C )  e.  P )
4120ad2antrr 730 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  A  e.  P )
4218ad2antrr 730 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  C  e.  P )
4319ad2antrr 730 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  -> 
( ( (pInvG `  G ) `  x
) `  C )  e.  P )
4415ad2antrr 730 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  x  e.  P )
45 simplr 760 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  p  e.  P )
461, 2, 3, 4, 5, 39, 41, 21, 42mirbtwn 24563 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  A  e.  ( (
( (pInvG `  G
) `  A ) `  C ) I C ) )
471, 2, 3, 4, 5, 39, 44, 16, 42mirbtwn 24563 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  x  e.  ( (
( (pInvG `  G
) `  x ) `  C ) I C ) )
481, 2, 3, 4, 5, 39, 45, 8, 43mirbtwn 24563 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  p  e.  ( (
( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) I ( ( (pInvG `  G ) `  x ) `  C
) ) )
49 simpr 462 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  -> 
( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )
5049eqcomd 2437 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  -> 
( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) )  =  ( ( (pInvG `  G
) `  A ) `  C ) )
5150oveq1d 6320 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  -> 
( ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) I ( ( (pInvG `  G
) `  x ) `  C ) )  =  ( ( ( (pInvG `  G ) `  A
) `  C )
I ( ( (pInvG `  G ) `  x
) `  C )
) )
5248, 51eleqtrd 2519 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  p  e.  ( (
( (pInvG `  G
) `  A ) `  C ) I ( ( (pInvG `  G
) `  x ) `  C ) ) )
531, 2, 3, 39, 40, 41, 42, 43, 44, 45, 46, 47, 52tgtrisegint 24406 . . . . . . 7  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  E. t  e.  P  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )
5439ad3antrrr 734 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  G  e. TarskiG )
5541ad3antrrr 734 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  A  e.  P )
56 simpllr 767 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  t  e.  P )
57 simplrr 769 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  t  e.  ( A I x ) )
58 simpr 462 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  x  =  A )
5958oveq2d 6321 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  ( A I x )  =  ( A I A ) )
6057, 59eleqtrd 2519 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  t  e.  ( A I A ) )
611, 2, 3, 54, 55, 56, 60axtgbtwnid 24377 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  A  =  t )
6261eqcomd 2437 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  t  =  A )
6362oveq1d 6320 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
t L p )  =  ( A L p ) )
6450ad3antrrr 734 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
)  =  ( ( (pInvG `  G ) `  A ) `  C
) )
6558fveq2d 5885 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
(pInvG `  G ) `  x )  =  ( (pInvG `  G ) `  A ) )
6665fveq1d 5883 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
( (pInvG `  G
) `  x ) `  C )  =  ( ( (pInvG `  G
) `  A ) `  C ) )
6764, 66eqtr4d 2473 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
)  =  ( ( (pInvG `  G ) `  x ) `  C
) )
6845ad3antrrr 734 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  p  e.  P )
6943ad3antrrr 734 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
( (pInvG `  G
) `  x ) `  C )  e.  P
)
701, 2, 3, 4, 5, 54, 68, 8, 69mirinv 24571 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) )  =  ( ( (pInvG `  G
) `  x ) `  C )  <->  p  =  ( ( (pInvG `  G ) `  x
) `  C )
) )
7167, 70mpbid 213 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  p  =  ( ( (pInvG `  G ) `  x
) `  C )
)
7244ad3antrrr 734 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  x  e.  P )
7358oveq1d 6320 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
x I x )  =  ( A I x ) )
7457, 73eleqtrrd 2520 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  t  e.  ( x I x ) )
751, 2, 3, 54, 72, 56, 74axtgbtwnid 24377 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  x  =  t )
7675eqcomd 2437 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  t  =  x )
7771, 76oveq12d 6323 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
p L t )  =  ( ( ( (pInvG `  G ) `  x ) `  C
) L x ) )
7834ad2antrr 730 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  <" A x C ">  e.  (∟G `  G ) )
791, 2, 3, 4, 5, 39, 45, 8, 43, 50mircom 24568 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  -> 
( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  A ) `  C
) )  =  ( ( (pInvG `  G
) `  x ) `  C ) )
8028ad2antrr 730 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  x  =/=  C )
811, 2, 3, 4, 39, 5, 21, 16, 8, 41, 44, 42, 45, 78, 79, 80colperpexlem2 24630 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  A  =/=  p )
8281ad3antrrr 734 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  A  =/=  p )
8362, 82eqnetrd 2724 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  t  =/=  p )
841, 3, 4, 54, 56, 68, 83tglinecom 24539 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
t L p )  =  ( p L t ) )
8542ad3antrrr 734 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  C  e.  P )
8680ad3antrrr 734 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  x  =/=  C )
8754adantr 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  /\  t  e.  P )  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  /\  (
( (pInvG `  G
) `  x ) `  C )  =  x )  ->  G  e. TarskiG )
8872adantr 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  /\  t  e.  P )  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  /\  (
( (pInvG `  G
) `  x ) `  C )  =  x )  ->  x  e.  P )
8985adantr 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  /\  t  e.  P )  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  /\  (
( (pInvG `  G
) `  x ) `  C )  =  x )  ->  C  e.  P )
901, 2, 3, 4, 5, 87, 88, 16mircinv 24572 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  /\  t  e.  P )  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  /\  (
( (pInvG `  G
) `  x ) `  C )  =  x )  ->  ( (
(pInvG `  G ) `  x ) `  x
)  =  x )
91 simpr 462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  /\  t  e.  P )  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  /\  (
( (pInvG `  G
) `  x ) `  C )  =  x )  ->  ( (
(pInvG `  G ) `  x ) `  C
)  =  x )
9290, 91eqtr4d 2473 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  /\  t  e.  P )  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  /\  (
( (pInvG `  G
) `  x ) `  C )  =  x )  ->  ( (
(pInvG `  G ) `  x ) `  x
)  =  ( ( (pInvG `  G ) `  x ) `  C
) )
931, 2, 3, 4, 5, 87, 88, 16, 88, 89, 92mireq 24570 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  /\  t  e.  P )  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  /\  (
( (pInvG `  G
) `  x ) `  C )  =  x )  ->  x  =  C )
9486adantr 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  /\  t  e.  P )  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  /\  (
( (pInvG `  G
) `  x ) `  C )  =  x )  ->  x  =/=  C )
9594neneqd 2632 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  /\  t  e.  P )  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  /\  (
( (pInvG `  G
) `  x ) `  C )  =  x )  ->  -.  x  =  C )
9693, 95pm2.65da 578 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  -.  ( ( (pInvG `  G ) `  x
) `  C )  =  x )
9796neqned 2634 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
( (pInvG `  G
) `  x ) `  C )  =/=  x
)
9847ad3antrrr 734 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  x  e.  ( ( ( (pInvG `  G ) `  x
) `  C )
I C ) )
991, 3, 4, 54, 72, 85, 69, 86, 98btwnlng2 24524 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
( (pInvG `  G
) `  x ) `  C )  e.  ( x L C ) )
1001, 3, 4, 54, 72, 85, 86, 69, 97, 99tglineelsb2 24536 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
x L C )  =  ( x L ( ( (pInvG `  G ) `  x
) `  C )
) )
10128necomd 2702 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  C  =/=  x
)
102101ad5antr 738 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  C  =/=  x )
1031, 3, 4, 54, 85, 72, 102tglinecom 24539 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  ( C L x )  =  ( x L C ) )
1041, 3, 4, 54, 69, 72, 97tglinecom 24539 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
( ( (pInvG `  G ) `  x
) `  C ) L x )  =  ( x L ( ( (pInvG `  G
) `  x ) `  C ) ) )
105100, 103, 1043eqtr4d 2480 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  ( C L x )  =  ( ( ( (pInvG `  G ) `  x
) `  C ) L x ) )
10677, 84, 1053eqtr4d 2480 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
t L p )  =  ( C L x ) )
10763, 106eqtr3d 2472 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  ( A L p )  =  ( C L x ) )
10831ad5antr 738 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  ( C L x ) (⟂G `  G ) ( A L B ) )
109107, 108eqbrtrd 4446 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  ( A L p ) (⟂G `  G ) ( A L B ) )
11039ad3antrrr 734 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  G  e. TarskiG )
11141ad3antrrr 734 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  A  e.  P )
11245ad3antrrr 734 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  p  e.  P )
11381ad3antrrr 734 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  A  =/=  p )
1141, 3, 4, 110, 111, 112, 113tgelrnln 24534 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  ( A L p )  e. 
ran  L )
11513ad5antr 738 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  ( A L B )  e. 
ran  L )
1161, 3, 4, 110, 111, 112, 113tglinerflx1 24537 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  A  e.  ( A L p ) )
11711ad2antrr 730 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  A  =/=  B
)
1181, 3, 4, 7, 20, 24, 117tglinerflx1 24537 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  A  e.  ( A L B ) )
119118ad5antr 738 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  A  e.  ( A L B ) )
120116, 119elind 3656 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  A  e.  ( ( A L p )  i^i  ( A L B ) ) )
1211, 3, 4, 110, 111, 112, 113tglinerflx2 24538 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  p  e.  ( A L p ) )
12214ad5antr 738 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  x  e.  ( A L B ) )
123113necomd 2702 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  p  =/=  A )
124 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  x  =/=  A )
12544ad3antrrr 734 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  x  e.  P )
1261, 2, 3, 4, 39, 5, 21, 16, 8, 41, 44, 42, 45, 78, 79colperpexlem1 24629 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  <" x A p ">  e.  (∟G `  G ) )
127126ad3antrrr 734 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  <" x A p ">  e.  (∟G `  G )
)
1281, 2, 3, 4, 5, 110, 125, 111, 112, 127ragcom 24600 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  <" p A x ">  e.  (∟G `  G )
)
1291, 2, 3, 4, 110, 114, 115, 120, 121, 122, 123, 124, 128ragperp 24619 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  ( A L p ) (⟂G `  G ) ( A L B ) )
130109, 129pm2.61dane 2749 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  ( (
(pInvG `  G ) `  A ) `  C
)  =  ( ( (pInvG `  G ) `  p ) `  (
( (pInvG `  G
) `  x ) `  C ) ) )  /\  t  e.  P
)  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  -> 
( A L p ) (⟂G `  G
) ( A L B ) )
131118ad5antr 738 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  A  e.  ( A L B ) )
13262, 131eqeltrd 2517 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  t  e.  ( A L B ) )
133132orcd 393 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =  A )  ->  (
t  e.  ( A L B )  \/  A  =  B ) )
13424ad5antr 738 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  B  e.  P )
135117ad5antr 738 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  A  =/=  B )
136 simpllr 767 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  t  e.  P )
137124necomd 2702 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  A  =/=  x )
138 simplrr 769 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  t  e.  ( A I x ) )
1391, 3, 4, 110, 111, 125, 136, 137, 138btwnlng1 24523 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  t  e.  ( A L x ) )
1401, 3, 4, 110, 111, 134, 135, 125, 124, 122, 136, 139tglineeltr 24535 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  t  e.  ( A L B ) )
141140orcd 393 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  /\  (
( (pInvG `  G
) `  A ) `  C )  =  ( ( (pInvG `  G
) `  p ) `  ( ( (pInvG `  G ) `  x
) `  C )
) )  /\  t  e.  P )  /\  (
t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  /\  x  =/= 
A )  ->  (
t  e.  ( A L B )  \/  A  =  B ) )
142133, 141pm2.61dane 2749 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  ( (
(pInvG `  G ) `  A ) `  C
)  =  ( ( (pInvG `  G ) `  p ) `  (
( (pInvG `  G
) `  x ) `  C ) ) )  /\  t  e.  P
)  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  -> 
( t  e.  ( A L B )  \/  A  =  B ) )
14339ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  ( (
(pInvG `  G ) `  A ) `  C
)  =  ( ( (pInvG `  G ) `  p ) `  (
( (pInvG `  G
) `  x ) `  C ) ) )  /\  t  e.  P
)  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  ->  G  e. TarskiG )
14445ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  ( (
(pInvG `  G ) `  A ) `  C
)  =  ( ( (pInvG `  G ) `  p ) `  (
( (pInvG `  G
) `  x ) `  C ) ) )  /\  t  e.  P
)  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  ->  p  e.  P )
145 simplr 760 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  ( (
(pInvG `  G ) `  A ) `  C
)  =  ( ( (pInvG `  G ) `  p ) `  (
( (pInvG `  G
) `  x ) `  C ) ) )  /\  t  e.  P
)  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  -> 
t  e.  P )
14642ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  ( (
(pInvG `  G ) `  A ) `  C
)  =  ( ( (pInvG `  G ) `  p ) `  (
( (pInvG `  G
) `  x ) `  C ) ) )  /\  t  e.  P
)  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  ->  C  e.  P )
147 simprl 762 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  ( (
(pInvG `  G ) `  A ) `  C
)  =  ( ( (pInvG `  G ) `  p ) `  (
( (pInvG `  G
) `  x ) `  C ) ) )  /\  t  e.  P
)  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  -> 
t  e.  ( p I C ) )
1481, 2, 3, 143, 144, 145, 146, 147tgbtwncom 24395 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  ( (
(pInvG `  G ) `  A ) `  C
)  =  ( ( (pInvG `  G ) `  p ) `  (
( (pInvG `  G
) `  x ) `  C ) ) )  /\  t  e.  P
)  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  -> 
t  e.  ( C I p ) )
149130, 142, 148jca32 537 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  ( (
(pInvG `  G ) `  A ) `  C
)  =  ( ( (pInvG `  G ) `  p ) `  (
( (pInvG `  G
) `  x ) `  C ) ) )  /\  t  e.  P
)  /\  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) ) )  -> 
( ( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( C I p ) ) ) )
150149ex 435 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  ( (
(pInvG `  G ) `  A ) `  C
)  =  ( ( (pInvG `  G ) `  p ) `  (
( (pInvG `  G
) `  x ) `  C ) ) )  /\  t  e.  P
)  ->  ( (
t  e.  ( p I C )  /\  t  e.  ( A I x ) )  ->  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( C I p ) ) ) ) )
151150reximdva 2907 . . . . . . 7  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  -> 
( E. t  e.  P  ( t  e.  ( p I C )  /\  t  e.  ( A I x ) )  ->  E. t  e.  P  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( C I p ) ) ) ) )
15253, 151mpd 15 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  ->  E. t  e.  P  ( ( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( C I p ) ) ) )
153 r19.42v 2990 . . . . . 6  |-  ( E. t  e.  P  ( ( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( C I p ) ) )  <->  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  E. t  e.  P  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( C I p ) ) ) )
154152, 153sylib 199 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  ( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) ) )  -> 
( ( A L p ) (⟂G `  G
) ( A L B )  /\  E. t  e.  P  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( C I p ) ) ) )
155154ex 435 . . . 4  |-  ( ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G )
( A L B ) )  /\  p  e.  P )  ->  (
( ( (pInvG `  G ) `  A
) `  C )  =  ( ( (pInvG `  G ) `  p
) `  ( (
(pInvG `  G ) `  x ) `  C
) )  ->  (
( A L p ) (⟂G `  G
) ( A L B )  /\  E. t  e.  P  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( C I p ) ) ) ) )
156155reximdva 2907 . . 3  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  ( E. p  e.  P  ( (
(pInvG `  G ) `  A ) `  C
)  =  ( ( (pInvG `  G ) `  p ) `  (
( (pInvG `  G
) `  x ) `  C ) )  ->  E. p  e.  P  ( ( A L p ) (⟂G `  G
) ( A L B )  /\  E. t  e.  P  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( C I p ) ) ) ) )
15738, 156mpd 15 . 2  |-  ( ( ( ph  /\  x  e.  ( A L B ) )  /\  ( C L x ) (⟂G `  G ) ( A L B ) )  ->  E. p  e.  P  ( ( A L p ) (⟂G `  G
) ( A L B )  /\  E. t  e.  P  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( C I p ) ) ) )
1581, 2, 3, 4, 6, 12, 17, 25footex 24620 . 2  |-  ( ph  ->  E. x  e.  ( A L B ) ( C L x ) (⟂G `  G
) ( A L B ) )
159157, 158r19.29a 2977 1  |-  ( ph  ->  E. p  e.  P  ( ( A L p ) (⟂G `  G
) ( A L B )  /\  E. t  e.  P  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( C I p ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783   class class class wbr 4426   ran crn 4855   ` cfv 5601  (class class class)co 6305   <"cs3 12923   Basecbs 15084   distcds 15161  TarskiGcstrkg 24341  Itvcitv 24347  LineGclng 24348  pInvGcmir 24557  ∟Gcrag 24595  ⟂Gcperpg 24597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-concat 12653  df-s1 12654  df-s2 12929  df-s3 12930  df-trkgc 24359  df-trkgb 24360  df-trkgcb 24361  df-trkg 24364  df-cgrg 24419  df-leg 24488  df-mir 24558  df-rag 24596  df-perpg 24598
This theorem is referenced by:  colperpex  24632
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