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Theorem mideulem2 24086
Description: Lemma for opphllem 24087, which is itself used for mideu 24090. (Contributed by Thierry Arnoux, 19-Feb-2020.)
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 )
mideu.s  |-  S  =  (pInvG `  G )
mideu.1  |-  ( ph  ->  A  e.  P )
mideu.2  |-  ( ph  ->  B  e.  P )
mideulem.1  |-  ( ph  ->  A  =/=  B )
mideulem.2  |-  ( ph  ->  Q  e.  P )
mideulem.3  |-  ( ph  ->  O  e.  P )
mideulem.4  |-  ( ph  ->  T  e.  P )
mideulem.5  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( Q L B ) )
mideulem.6  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( A L O ) )
mideulem.7  |-  ( ph  ->  T  e.  ( A L B ) )
mideulem.8  |-  ( ph  ->  T  e.  ( Q I O ) )
opphllem.1  |-  ( ph  ->  R  e.  P )
opphllem.2  |-  ( ph  ->  R  e.  ( B I Q ) )
opphllem.3  |-  ( ph  ->  ( A  .-  O
)  =  ( B 
.-  R ) )
mideulem2.1  |-  ( ph  ->  X  e.  P )
mideulem2.2  |-  ( ph  ->  X  e.  ( T I B ) )
mideulem2.3  |-  ( ph  ->  X  e.  ( R I O ) )
mideulem2.4  |-  ( ph  ->  Z  e.  P )
mideulem2.5  |-  ( ph  ->  X  e.  ( ( ( S `  A
) `  O )
I Z ) )
mideulem2.6  |-  ( ph  ->  ( X  .-  Z
)  =  ( X 
.-  R ) )
mideulem2.7  |-  ( ph  ->  M  e.  P )
mideulem2.8  |-  ( ph  ->  R  =  ( ( S `  M ) `
 Z ) )
Assertion
Ref Expression
mideulem2  |-  ( ph  ->  B  =  M )

Proof of Theorem mideulem2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 oveq2 6289 . . 3  |-  ( y  =  B  ->  ( R L y )  =  ( R L B ) )
21breq1d 4447 . 2  |-  ( y  =  B  ->  (
( R L y ) (⟂G `  G
) ( A L B )  <->  ( R L B ) (⟂G `  G
) ( A L B ) ) )
3 oveq2 6289 . . 3  |-  ( y  =  M  ->  ( R L y )  =  ( R L M ) )
43breq1d 4447 . 2  |-  ( y  =  M  ->  (
( R L y ) (⟂G `  G
) ( A L B )  <->  ( R L M ) (⟂G `  G
) ( A L B ) ) )
5 colperpex.p . . 3  |-  P  =  ( Base `  G
)
6 colperpex.d . . 3  |-  .-  =  ( dist `  G )
7 colperpex.i . . 3  |-  I  =  (Itv `  G )
8 colperpex.l . . 3  |-  L  =  (LineG `  G )
9 colperpex.g . . 3  |-  ( ph  ->  G  e. TarskiG )
10 mideu.1 . . . 4  |-  ( ph  ->  A  e.  P )
11 mideu.2 . . . 4  |-  ( ph  ->  B  e.  P )
12 mideulem.1 . . . 4  |-  ( ph  ->  A  =/=  B )
135, 7, 8, 9, 10, 11, 12tgelrnln 23988 . . 3  |-  ( ph  ->  ( A L B )  e.  ran  L
)
14 opphllem.1 . . 3  |-  ( ph  ->  R  e.  P )
1512adantr 465 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  A  =/=  B )
1615neneqd 2645 . . . . 5  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  -.  A  =  B )
17 mideulem.3 . . . . . . . . 9  |-  ( ph  ->  O  e.  P )
18 opphllem.3 . . . . . . . . 9  |-  ( ph  ->  ( A  .-  O
)  =  ( B 
.-  R ) )
19 mideulem.6 . . . . . . . . . . 11  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( A L O ) )
208, 9, 19perpln2 24066 . . . . . . . . . 10  |-  ( ph  ->  ( A L O )  e.  ran  L
)
215, 7, 8, 9, 10, 17, 20tglnne 23986 . . . . . . . . 9  |-  ( ph  ->  A  =/=  O )
225, 6, 7, 9, 10, 17, 11, 14, 18, 21tgcgrneq 23852 . . . . . . . 8  |-  ( ph  ->  B  =/=  R )
2322adantr 465 . . . . . . 7  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  B  =/=  R )
2423necomd 2714 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  R  =/=  B )
2524neneqd 2645 . . . . 5  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  -.  R  =  B )
2616, 25jca 532 . . . 4  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  ( -.  A  =  B  /\  -.  R  =  B
) )
27 mideu.s . . . . . 6  |-  S  =  (pInvG `  G )
289adantr 465 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  G  e. TarskiG )
2910adantr 465 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  A  e.  P )
3011adantr 465 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  B  e.  P )
3114adantr 465 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  R  e.  P )
32 mideulem.2 . . . . . . . . 9  |-  ( ph  ->  Q  e.  P )
33 mideulem.5 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( Q L B ) )
348, 9, 33perpln2 24066 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q L B )  e.  ran  L
)
355, 7, 8, 9, 32, 11, 34tglnne 23986 . . . . . . . . . . 11  |-  ( ph  ->  Q  =/=  B )
365, 7, 8, 9, 32, 11, 35tglinerflx2 23992 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( Q L B ) )
375, 6, 7, 8, 9, 13, 34, 33perpcom 24068 . . . . . . . . . . 11  |-  ( ph  ->  ( Q L B ) (⟂G `  G
) ( A L B ) )
385, 7, 8, 9, 10, 11, 12tglinecom 23993 . . . . . . . . . . 11  |-  ( ph  ->  ( A L B )  =  ( B L A ) )
3937, 38breqtrd 4461 . . . . . . . . . 10  |-  ( ph  ->  ( Q L B ) (⟂G `  G
) ( B L A ) )
405, 6, 7, 8, 9, 32, 11, 36, 10, 39perprag 24078 . . . . . . . . 9  |-  ( ph  ->  <" Q B A ">  e.  (∟G `  G ) )
41 opphllem.2 . . . . . . . . . 10  |-  ( ph  ->  R  e.  ( B I Q ) )
425, 8, 7, 9, 11, 14, 32, 41btwncolg3 23922 . . . . . . . . 9  |-  ( ph  ->  ( Q  e.  ( B L R )  \/  B  =  R ) )
435, 6, 7, 8, 27, 9, 32, 11, 10, 14, 40, 35, 42ragcol 24054 . . . . . . . 8  |-  ( ph  ->  <" R B A ">  e.  (∟G `  G ) )
445, 6, 7, 8, 27, 9, 14, 11, 10, 43ragcom 24053 . . . . . . 7  |-  ( ph  ->  <" A B R ">  e.  (∟G `  G ) )
4544adantr 465 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  <" A B R ">  e.  (∟G `  G ) )
46 simpr 461 . . . . . . 7  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  R  e.  ( A L B ) )
4746orcd 392 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  ( R  e.  ( A L B )  \/  A  =  B ) )
485, 6, 7, 8, 27, 28, 29, 30, 31, 45, 47ragflat3 24061 . . . . 5  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  ( A  =  B  \/  R  =  B ) )
49 oran 496 . . . . 5  |-  ( ( A  =  B  \/  R  =  B )  <->  -.  ( -.  A  =  B  /\  -.  R  =  B ) )
5048, 49sylib 196 . . . 4  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  -.  ( -.  A  =  B  /\  -.  R  =  B ) )
5126, 50pm2.65da 576 . . 3  |-  ( ph  ->  -.  R  e.  ( A L B ) )
525, 6, 7, 8, 9, 13, 14, 51foot 24074 . 2  |-  ( ph  ->  E! y  e.  ( A L B ) ( R L y ) (⟂G `  G
) ( A L B ) )
535, 7, 8, 9, 10, 11, 12tglinerflx2 23992 . 2  |-  ( ph  ->  B  e.  ( A L B ) )
54 mideulem2.1 . . 3  |-  ( ph  ->  X  e.  P )
5512neneqd 2645 . . . . 5  |-  ( ph  ->  -.  A  =  B )
56 oveq2 6289 . . . . . . 7  |-  ( y  =  A  ->  ( R L y )  =  ( R L A ) )
5756breq1d 4447 . . . . . 6  |-  ( y  =  A  ->  (
( R L y ) (⟂G `  G
) ( A L B )  <->  ( R L A ) (⟂G `  G
) ( A L B ) ) )
5852adantr 465 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  E! y  e.  ( A L B ) ( R L y ) (⟂G `  G ) ( A L B ) )
595, 7, 8, 9, 10, 11, 12tglinerflx1 23991 . . . . . . 7  |-  ( ph  ->  A  e.  ( A L B ) )
6059adantr 465 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  A  e.  ( A L B ) )
6153adantr 465 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  B  e.  ( A L B ) )
629adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  G  e. TarskiG )
6314adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  R  e.  P )
6410adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  A  e.  P )
6551, 55jca 532 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  R  e.  ( A L B )  /\  -.  A  =  B ) )
66 pm4.56 495 . . . . . . . . . . . 12  |-  ( ( -.  R  e.  ( A L B )  /\  -.  A  =  B )  <->  -.  ( R  e.  ( A L B )  \/  A  =  B ) )
6765, 66sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( R  e.  ( A L B )  \/  A  =  B ) )
685, 7, 8, 9, 14, 10, 11, 67ncolne1 23983 . . . . . . . . . 10  |-  ( ph  ->  R  =/=  A )
6968adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  R  =/=  A )
705, 7, 8, 62, 63, 64, 69tglinecom 23993 . . . . . . . 8  |-  ( (
ph  /\  X  =  A )  ->  ( R L A )  =  ( A L R ) )
7169necomd 2714 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  A  =/=  R )
7217adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  O  e.  P )
7321necomd 2714 . . . . . . . . . 10  |-  ( ph  ->  O  =/=  A )
7473adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  O  =/=  A )
7554adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =  A )  ->  X  e.  P )
76 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  X  =  A )  ->  X  =  A )
7776, 71eqnetrd 2736 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =  A )  ->  X  =/=  R )
78 mideulem2.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X  e.  ( R I O ) )
795, 6, 7, 9, 14, 54, 17, 78tgbtwncom 23857 . . . . . . . . . . . . . . 15  |-  ( ph  ->  X  e.  ( O I R ) )
80 mideulem.4 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  T  e.  P )
81 mideulem.7 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  T  e.  ( A L B ) )
82 mideulem2.2 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  e.  ( T I B ) )
835, 7, 8, 9, 80, 10, 11, 54, 81, 82coltr3 24007 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X  e.  ( A L B ) )
8412necomd 2714 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  B  =/=  A )
8584neneqd 2645 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  -.  B  =  A )
8685adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  -.  B  =  A )
8773neneqd 2645 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  -.  O  =  A )
8887adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  -.  O  =  A )
8986, 88jca 532 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  ( -.  B  =  A  /\  -.  O  =  A
) )
909adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  G  e. TarskiG )
9111adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  B  e.  P )
9210adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  A  e.  P )
9317adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  O  e.  P )
945, 7, 8, 9, 11, 10, 84tglinerflx2 23992 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  A  e.  ( B L A ) )
9538, 19eqbrtrrd 4459 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( B L A ) (⟂G `  G
) ( A L O ) )
965, 6, 7, 8, 9, 11, 10, 94, 17, 95perprag 24078 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  <" B A O ">  e.  (∟G `  G ) )
9796adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  <" B A O ">  e.  (∟G `  G ) )
98 simpr 461 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  O  e.  ( B L A ) )
9998orcd 392 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  ( O  e.  ( B L A )  \/  B  =  A ) )
1005, 6, 7, 8, 27, 90, 91, 92, 93, 97, 99ragflat3 24061 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  ( B  =  A  \/  O  =  A ) )
101 oran 496 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  =  A  \/  O  =  A )  <->  -.  ( -.  B  =  A  /\  -.  O  =  A ) )
102100, 101sylib 196 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  -.  ( -.  B  =  A  /\  -.  O  =  A ) )
10389, 102pm2.65da 576 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  -.  O  e.  ( B L A ) )
104103, 38neleqtrrd 2556 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  -.  O  e.  ( A L B ) )
105 nelne2 2773 . . . . . . . . . . . . . . . 16  |-  ( ( X  e.  ( A L B )  /\  -.  O  e.  ( A L B ) )  ->  X  =/=  O
)
10683, 104, 105syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  X  =/=  O )
1075, 6, 7, 9, 17, 54, 14, 79, 106tgbtwnne 23859 . . . . . . . . . . . . . 14  |-  ( ph  ->  O  =/=  R )
108107adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  X  =  A )  ->  O  =/=  R )
109108necomd 2714 . . . . . . . . . . . 12  |-  ( (
ph  /\  X  =  A )  ->  R  =/=  O )
11078adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  X  =  A )  ->  X  e.  ( R I O ) )
1115, 7, 8, 62, 63, 72, 75, 109, 110btwnlng1 23977 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =  A )  ->  X  e.  ( R L O ) )
1125, 7, 8, 62, 75, 63, 72, 77, 111, 109lnrot2 23982 . . . . . . . . . 10  |-  ( (
ph  /\  X  =  A )  ->  O  e.  ( X L R ) )
11376oveq1d 6296 . . . . . . . . . 10  |-  ( (
ph  /\  X  =  A )  ->  ( X L R )  =  ( A L R ) )
114112, 113eleqtrd 2533 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  O  e.  ( A L R ) )
1155, 7, 8, 62, 64, 63, 71, 72, 74, 114tglineelsb2 23990 . . . . . . . 8  |-  ( (
ph  /\  X  =  A )  ->  ( A L R )  =  ( A L O ) )
11670, 115eqtrd 2484 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( R L A )  =  ( A L O ) )
1175, 6, 7, 8, 9, 13, 20, 19perpcom 24068 . . . . . . . 8  |-  ( ph  ->  ( A L O ) (⟂G `  G
) ( A L B ) )
118117adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( A L O ) (⟂G `  G ) ( A L B ) )
119116, 118eqbrtrd 4457 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  ( R L A ) (⟂G `  G ) ( A L B ) )
12013adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( A L B )  e. 
ran  L )
12122necomd 2714 . . . . . . . . 9  |-  ( ph  ->  R  =/=  B )
1225, 7, 8, 9, 14, 11, 121tgelrnln 23988 . . . . . . . 8  |-  ( ph  ->  ( R L B )  e.  ran  L
)
123122adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( R L B )  e. 
ran  L )
1245, 7, 8, 9, 14, 11, 121tglinerflx2 23992 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( R L B ) )
12553, 124elind 3673 . . . . . . . . 9  |-  ( ph  ->  B  e.  ( ( A L B )  i^i  ( R L B ) ) )
1265, 7, 8, 9, 14, 11, 121tglinerflx1 23991 . . . . . . . . 9  |-  ( ph  ->  R  e.  ( R L B ) )
1275, 6, 7, 8, 9, 13, 122, 125, 59, 126, 12, 121, 44ragperp 24072 . . . . . . . 8  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( R L B ) )
128127adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( A L B ) (⟂G `  G ) ( R L B ) )
1295, 6, 7, 8, 62, 120, 123, 128perpcom 24068 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  ( R L B ) (⟂G `  G ) ( A L B ) )
13057, 2, 58, 60, 61, 119, 129reu2eqd 3282 . . . . 5  |-  ( (
ph  /\  X  =  A )  ->  A  =  B )
13155, 130mtand 659 . . . 4  |-  ( ph  ->  -.  X  =  A )
132131neqned 2646 . . 3  |-  ( ph  ->  X  =/=  A )
133 mideulem2.7 . . 3  |-  ( ph  ->  M  e.  P )
134132necomd 2714 . . . 4  |-  ( ph  ->  A  =/=  X )
135 eqid 2443 . . . . 5  |-  ( S `
 A )  =  ( S `  A
)
136 eqid 2443 . . . . 5  |-  ( S `
 M )  =  ( S `  M
)
1375, 6, 7, 8, 27, 9, 10, 135, 17mircl 24020 . . . . 5  |-  ( ph  ->  ( ( S `  A ) `  O
)  e.  P )
138 mideulem2.4 . . . . 5  |-  ( ph  ->  Z  e.  P )
139 mideulem2.5 . . . . 5  |-  ( ph  ->  X  e.  ( ( ( S `  A
) `  O )
I Z ) )
14083orcd 392 . . . . . . . . 9  |-  ( ph  ->  ( X  e.  ( A L B )  \/  A  =  B ) )
1415, 8, 7, 9, 10, 11, 54, 140colcom 23923 . . . . . . . 8  |-  ( ph  ->  ( X  e.  ( B L A )  \/  B  =  A ) )
1425, 8, 7, 9, 11, 10, 54, 141colrot1 23924 . . . . . . 7  |-  ( ph  ->  ( B  e.  ( A L X )  \/  A  =  X ) )
1435, 6, 7, 8, 27, 9, 11, 10, 17, 54, 96, 84, 142ragcol 24054 . . . . . 6  |-  ( ph  ->  <" X A O ">  e.  (∟G `  G ) )
1445, 6, 7, 8, 27, 9, 54, 10, 17israg 24052 . . . . . 6  |-  ( ph  ->  ( <" X A O ">  e.  (∟G `  G )  <->  ( X  .-  O )  =  ( X  .-  ( ( S `  A ) `
 O ) ) ) )
145143, 144mpbid 210 . . . . 5  |-  ( ph  ->  ( X  .-  O
)  =  ( X 
.-  ( ( S `
 A ) `  O ) ) )
146 mideulem2.6 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  =  ( X 
.-  R ) )
147146eqcomd 2451 . . . . 5  |-  ( ph  ->  ( X  .-  R
)  =  ( X 
.-  Z ) )
148 eqidd 2444 . . . . 5  |-  ( ph  ->  ( ( S `  A ) `  O
)  =  ( ( S `  A ) `
 O ) )
149 mideulem2.8 . . . . . . . 8  |-  ( ph  ->  R  =  ( ( S `  M ) `
 Z ) )
150149eqcomd 2451 . . . . . . 7  |-  ( ph  ->  ( ( S `  M ) `  Z
)  =  R )
1515, 6, 7, 8, 27, 9, 133, 136, 138, 150mircom 24022 . . . . . 6  |-  ( ph  ->  ( ( S `  M ) `  R
)  =  Z )
152151eqcomd 2451 . . . . 5  |-  ( ph  ->  Z  =  ( ( S `  M ) `
 R ) )
1535, 6, 7, 8, 27, 9, 135, 136, 17, 137, 54, 14, 138, 10, 133, 79, 139, 145, 147, 148, 152krippen 24046 . . . 4  |-  ( ph  ->  X  e.  ( A I M ) )
1545, 7, 8, 9, 10, 54, 133, 134, 153btwnlng3 23979 . . 3  |-  ( ph  ->  M  e.  ( A L X ) )
1555, 7, 8, 9, 10, 11, 12, 54, 132, 83, 133, 154tglineeltr 23989 . 2  |-  ( ph  ->  M  e.  ( A L B ) )
1565, 6, 7, 8, 9, 13, 122, 127perpcom 24068 . 2  |-  ( ph  ->  ( R L B ) (⟂G `  G
) ( A L B ) )
157 nelne2 2773 . . . . . 6  |-  ( ( M  e.  ( A L B )  /\  -.  R  e.  ( A L B ) )  ->  M  =/=  R
)
158155, 51, 157syl2anc 661 . . . . 5  |-  ( ph  ->  M  =/=  R )
159158necomd 2714 . . . 4  |-  ( ph  ->  R  =/=  M )
1605, 7, 8, 9, 14, 133, 159tgelrnln 23988 . . 3  |-  ( ph  ->  ( R L M )  e.  ran  L
)
1615, 7, 8, 9, 14, 133, 159tglinerflx2 23992 . . . . 5  |-  ( ph  ->  M  e.  ( R L M ) )
162155, 161elind 3673 . . . 4  |-  ( ph  ->  M  e.  ( ( A L B )  i^i  ( R L M ) ) )
1635, 7, 8, 9, 14, 133, 159tglinerflx1 23991 . . . 4  |-  ( ph  ->  R  e.  ( R L M ) )
164 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  M  =  X )  ->  M  =  X )
1659adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  G  e. TarskiG )
166133adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  M  e.  P )
16710adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  A  e.  P )
16817adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  O  e.  P )
169137adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  M  =  X )  ->  (
( S `  A
) `  O )  e.  P )
170145adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  ( X  .-  O )  =  ( X  .-  (
( S `  A
) `  O )
) )
171164oveq1d 6296 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  O )  =  ( X  .-  O
) )
172164oveq1d 6296 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  ( ( S `
 A ) `  O ) )  =  ( X  .-  (
( S `  A
) `  O )
) )
173170, 171, 1723eqtr4rd 2495 . . . . . . . . . . 11  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  ( ( S `
 A ) `  O ) )  =  ( M  .-  O
) )
174138adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  Z  e.  P )
17514adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  R  e.  P )
176149adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  M  =  X )  ->  R  =  ( ( S `
 M ) `  Z ) )
177176oveq2d 6297 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  R )  =  ( M  .-  (
( S `  M
) `  Z )
) )
1785, 6, 7, 8, 27, 165, 166, 136, 174mircgr 24016 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  ( ( S `
 M ) `  Z ) )  =  ( M  .-  Z
) )
179177, 178eqtrd 2484 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  R )  =  ( M  .-  Z
) )
1805, 6, 7, 165, 166, 175, 166, 174, 179tgcgrcomlr 23849 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  ( R  .-  M )  =  ( Z  .-  M
) )
18183adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  M  =  X )  ->  X  e.  ( A L B ) )
182164, 181eqeltrd 2531 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( A L B ) )
18351adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  =  X )  ->  -.  R  e.  ( A L B ) )
184182, 183, 157syl2anc 661 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  =  X )  ->  M  =/=  R )
185184necomd 2714 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  R  =/=  M )
1865, 6, 7, 165, 175, 166, 174, 166, 180, 185tgcgrneq 23852 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  Z  =/=  M )
1875, 6, 7, 8, 27, 9, 133, 136, 138mirbtwn 24017 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ( ( ( S `  M
) `  Z )
I Z ) )
188149oveq1d 6296 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( R I Z )  =  ( ( ( S `  M
) `  Z )
I Z ) )
189187, 188eleqtrrd 2534 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  ( R I Z ) )
190189adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( R I Z ) )
1915, 6, 7, 165, 175, 166, 174, 190tgbtwncom 23857 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( Z I R ) )
192139adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  =  X )  ->  X  e.  ( ( ( S `
 A ) `  O ) I Z ) )
193164, 192eqeltrd 2531 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( ( ( S `
 A ) `  O ) I Z ) )
1945, 6, 7, 165, 169, 166, 174, 193tgbtwncom 23857 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( Z I ( ( S `  A
) `  O )
) )
19578adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  X  e.  ( R I O ) )
196164, 195eqeltrd 2531 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( R I O ) )
1975, 7, 165, 174, 166, 175, 169, 168, 186, 185, 191, 194, 196tgbtwnconn22 23944 . . . . . . . . . . 11  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( ( ( S `
 A ) `  O ) I O ) )
1985, 6, 7, 8, 27, 165, 166, 136, 168, 169, 173, 197ismir 24018 . . . . . . . . . 10  |-  ( (
ph  /\  M  =  X )  ->  (
( S `  A
) `  O )  =  ( ( S `
 M ) `  O ) )
199198eqcomd 2451 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  (
( S `  M
) `  O )  =  ( ( S `
 A ) `  O ) )
2005, 6, 7, 8, 27, 165, 166, 167, 168, 199miduniq1 24041 . . . . . . . 8  |-  ( (
ph  /\  M  =  X )  ->  M  =  A )
201164, 200eqtr3d 2486 . . . . . . 7  |-  ( (
ph  /\  M  =  X )  ->  X  =  A )
202131, 201mtand 659 . . . . . 6  |-  ( ph  ->  -.  M  =  X )
203202neqned 2646 . . . . 5  |-  ( ph  ->  M  =/=  X )
204203necomd 2714 . . . 4  |-  ( ph  ->  X  =/=  M )
205151oveq2d 6297 . . . . . 6  |-  ( ph  ->  ( X  .-  (
( S `  M
) `  R )
)  =  ( X 
.-  Z ) )
206205, 146eqtr2d 2485 . . . . 5  |-  ( ph  ->  ( X  .-  R
)  =  ( X 
.-  ( ( S `
 M ) `  R ) ) )
2075, 6, 7, 8, 27, 9, 54, 133, 14israg 24052 . . . . 5  |-  ( ph  ->  ( <" X M R ">  e.  (∟G `  G )  <->  ( X  .-  R )  =  ( X  .-  ( ( S `  M ) `
 R ) ) ) )
208206, 207mpbird 232 . . . 4  |-  ( ph  ->  <" X M R ">  e.  (∟G `  G ) )
2095, 6, 7, 8, 9, 13, 160, 162, 83, 163, 204, 159, 208ragperp 24072 . . 3  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( R L M ) )
2105, 6, 7, 8, 9, 13, 160, 209perpcom 24068 . 2  |-  ( ph  ->  ( R L M ) (⟂G `  G
) ( A L B ) )
2112, 4, 52, 53, 155, 156, 210reu2eqd 3282 1  |-  ( ph  ->  B  =  M )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   E!wreu 2795   class class class wbr 4437   ran crn 4990   ` cfv 5578  (class class class)co 6281   <"cs3 12789   Basecbs 14614   distcds 14688  TarskiGcstrkg 23803  Itvcitv 23810  LineGclng 23811  pInvGcmir 24011  ∟Gcrag 24048  ⟂Gcperpg 24050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-hash 12388  df-word 12524  df-concat 12526  df-s1 12527  df-s2 12795  df-s3 12796  df-trkgc 23822  df-trkgb 23823  df-trkgcb 23824  df-trkg 23828  df-cgrg 23881  df-leg 23948  df-mir 24012  df-rag 24049  df-perpg 24051
This theorem is referenced by:  opphllem  24087
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