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Theorem mideulem2 24855
Description: Lemma for opphllem 24856, which is itself used for mideu 24859. (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 6316 . . 3  |-  ( y  =  B  ->  ( R L y )  =  ( R L B ) )
21breq1d 4405 . 2  |-  ( y  =  B  ->  (
( R L y ) (⟂G `  G
) ( A L B )  <->  ( R L B ) (⟂G `  G
) ( A L B ) ) )
3 oveq2 6316 . . 3  |-  ( y  =  M  ->  ( R L y )  =  ( R L M ) )
43breq1d 4405 . 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 24754 . . 3  |-  ( ph  ->  ( A L B )  e.  ran  L
)
14 opphllem.1 . . 3  |-  ( ph  ->  R  e.  P )
1512adantr 472 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  A  =/=  B )
1615neneqd 2648 . . . . 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 24835 . . . . . . . . . 10  |-  ( ph  ->  ( A L O )  e.  ran  L
)
215, 7, 8, 9, 10, 17, 20tglnne 24752 . . . . . . . . 9  |-  ( ph  ->  A  =/=  O )
225, 6, 7, 9, 10, 17, 11, 14, 18, 21tgcgrneq 24606 . . . . . . . 8  |-  ( ph  ->  B  =/=  R )
2322adantr 472 . . . . . . 7  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  B  =/=  R )
2423necomd 2698 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  R  =/=  B )
2524neneqd 2648 . . . . 5  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  -.  R  =  B )
2616, 25jca 541 . . . 4  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  ( -.  A  =  B  /\  -.  R  =  B
) )
27 mideu.s . . . . . 6  |-  S  =  (pInvG `  G )
289adantr 472 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  G  e. TarskiG )
2910adantr 472 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  A  e.  P )
3011adantr 472 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  B  e.  P )
3114adantr 472 . . . . . 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 24835 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q L B )  e.  ran  L
)
355, 7, 8, 9, 32, 11, 34tglnne 24752 . . . . . . . . . . 11  |-  ( ph  ->  Q  =/=  B )
365, 7, 8, 9, 32, 11, 35tglinerflx2 24758 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( Q L B ) )
375, 6, 7, 8, 9, 13, 34, 33perpcom 24837 . . . . . . . . . . 11  |-  ( ph  ->  ( Q L B ) (⟂G `  G
) ( A L B ) )
385, 7, 8, 9, 10, 11, 12tglinecom 24759 . . . . . . . . . . 11  |-  ( ph  ->  ( A L B )  =  ( B L A ) )
3937, 38breqtrd 4420 . . . . . . . . . 10  |-  ( ph  ->  ( Q L B ) (⟂G `  G
) ( B L A ) )
405, 6, 7, 8, 9, 32, 11, 36, 10, 39perprag 24847 . . . . . . . . 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 24681 . . . . . . . . 9  |-  ( ph  ->  ( Q  e.  ( B L R )  \/  B  =  R ) )
435, 6, 7, 8, 27, 9, 32, 11, 10, 14, 40, 35, 42ragcol 24823 . . . . . . . 8  |-  ( ph  ->  <" R B A ">  e.  (∟G `  G ) )
445, 6, 7, 8, 27, 9, 14, 11, 10, 43ragcom 24822 . . . . . . 7  |-  ( ph  ->  <" A B R ">  e.  (∟G `  G ) )
4544adantr 472 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  <" A B R ">  e.  (∟G `  G ) )
46 simpr 468 . . . . . . 7  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  R  e.  ( A L B ) )
4746orcd 399 . . . . . 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 24830 . . . . 5  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  ( A  =  B  \/  R  =  B ) )
49 oran 504 . . . . 5  |-  ( ( A  =  B  \/  R  =  B )  <->  -.  ( -.  A  =  B  /\  -.  R  =  B ) )
5048, 49sylib 201 . . . 4  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  -.  ( -.  A  =  B  /\  -.  R  =  B ) )
5126, 50pm2.65da 586 . . 3  |-  ( ph  ->  -.  R  e.  ( A L B ) )
525, 6, 7, 8, 9, 13, 14, 51foot 24843 . 2  |-  ( ph  ->  E! y  e.  ( A L B ) ( R L y ) (⟂G `  G
) ( A L B ) )
535, 7, 8, 9, 10, 11, 12tglinerflx2 24758 . 2  |-  ( ph  ->  B  e.  ( A L B ) )
54 mideulem2.1 . . 3  |-  ( ph  ->  X  e.  P )
5512neneqd 2648 . . . . 5  |-  ( ph  ->  -.  A  =  B )
56 oveq2 6316 . . . . . . 7  |-  ( y  =  A  ->  ( R L y )  =  ( R L A ) )
5756breq1d 4405 . . . . . 6  |-  ( y  =  A  ->  (
( R L y ) (⟂G `  G
) ( A L B )  <->  ( R L A ) (⟂G `  G
) ( A L B ) ) )
5852adantr 472 . . . . . 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 24757 . . . . . . 7  |-  ( ph  ->  A  e.  ( A L B ) )
6059adantr 472 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  A  e.  ( A L B ) )
6153adantr 472 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  B  e.  ( A L B ) )
629adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  G  e. TarskiG )
6314adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  R  e.  P )
6410adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  A  e.  P )
6551, 55jca 541 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  R  e.  ( A L B )  /\  -.  A  =  B ) )
66 pm4.56 503 . . . . . . . . . . . 12  |-  ( ( -.  R  e.  ( A L B )  /\  -.  A  =  B )  <->  -.  ( R  e.  ( A L B )  \/  A  =  B ) )
6765, 66sylib 201 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( R  e.  ( A L B )  \/  A  =  B ) )
685, 7, 8, 9, 14, 10, 11, 67ncolne1 24749 . . . . . . . . . 10  |-  ( ph  ->  R  =/=  A )
6968adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  R  =/=  A )
705, 7, 8, 62, 63, 64, 69tglinecom 24759 . . . . . . . 8  |-  ( (
ph  /\  X  =  A )  ->  ( R L A )  =  ( A L R ) )
7169necomd 2698 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  A  =/=  R )
7217adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  O  e.  P )
7321necomd 2698 . . . . . . . . . 10  |-  ( ph  ->  O  =/=  A )
7473adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  O  =/=  A )
7554adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =  A )  ->  X  e.  P )
76 simpr 468 . . . . . . . . . . . 12  |-  ( (
ph  /\  X  =  A )  ->  X  =  A )
7776, 71eqnetrd 2710 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =  A )  ->  X  =/=  R )
78 mideulem2.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X  e.  ( R I O ) )
795, 6, 7, 9, 14, 54, 17, 78tgbtwncom 24611 . . . . . . . . . . . . . . 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 24772 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X  e.  ( A L B ) )
8412necomd 2698 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  B  =/=  A )
8584neneqd 2648 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  -.  B  =  A )
8685adantr 472 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  -.  B  =  A )
8773neneqd 2648 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  -.  O  =  A )
8887adantr 472 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  -.  O  =  A )
8986, 88jca 541 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  ( -.  B  =  A  /\  -.  O  =  A
) )
909adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  G  e. TarskiG )
9111adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  B  e.  P )
9210adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  A  e.  P )
9317adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  O  e.  P )
945, 7, 8, 9, 11, 10, 84tglinerflx2 24758 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  A  e.  ( B L A ) )
9538, 19eqbrtrrd 4418 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( B L A ) (⟂G `  G
) ( A L O ) )
965, 6, 7, 8, 9, 11, 10, 94, 17, 95perprag 24847 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  <" B A O ">  e.  (∟G `  G ) )
9796adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  <" B A O ">  e.  (∟G `  G ) )
98 simpr 468 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  O  e.  ( B L A ) )
9998orcd 399 . . . . . . . . . . . . . . . . . . . 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 24830 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  ( B  =  A  \/  O  =  A ) )
101 oran 504 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  =  A  \/  O  =  A )  <->  -.  ( -.  B  =  A  /\  -.  O  =  A ) )
102100, 101sylib 201 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  -.  ( -.  B  =  A  /\  -.  O  =  A ) )
10389, 102pm2.65da 586 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  -.  O  e.  ( B L A ) )
104103, 38neleqtrrd 2571 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  -.  O  e.  ( A L B ) )
105 nelne2 2740 . . . . . . . . . . . . . . . 16  |-  ( ( X  e.  ( A L B )  /\  -.  O  e.  ( A L B ) )  ->  X  =/=  O
)
10683, 104, 105syl2anc 673 . . . . . . . . . . . . . . 15  |-  ( ph  ->  X  =/=  O )
1075, 6, 7, 9, 17, 54, 14, 79, 106tgbtwnne 24613 . . . . . . . . . . . . . 14  |-  ( ph  ->  O  =/=  R )
108107adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  X  =  A )  ->  O  =/=  R )
109108necomd 2698 . . . . . . . . . . . 12  |-  ( (
ph  /\  X  =  A )  ->  R  =/=  O )
11078adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  X  =  A )  ->  X  e.  ( R I O ) )
1115, 7, 8, 62, 63, 72, 75, 109, 110btwnlng1 24743 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =  A )  ->  X  e.  ( R L O ) )
1125, 7, 8, 62, 75, 63, 72, 77, 111, 109lnrot2 24748 . . . . . . . . . 10  |-  ( (
ph  /\  X  =  A )  ->  O  e.  ( X L R ) )
11376oveq1d 6323 . . . . . . . . . 10  |-  ( (
ph  /\  X  =  A )  ->  ( X L R )  =  ( A L R ) )
114112, 113eleqtrd 2551 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  O  e.  ( A L R ) )
1155, 7, 8, 62, 64, 63, 71, 72, 74, 114tglineelsb2 24756 . . . . . . . 8  |-  ( (
ph  /\  X  =  A )  ->  ( A L R )  =  ( A L O ) )
11670, 115eqtrd 2505 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( R L A )  =  ( A L O ) )
1175, 6, 7, 8, 9, 13, 20, 19perpcom 24837 . . . . . . . 8  |-  ( ph  ->  ( A L O ) (⟂G `  G
) ( A L B ) )
118117adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( A L O ) (⟂G `  G ) ( A L B ) )
119116, 118eqbrtrd 4416 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  ( R L A ) (⟂G `  G ) ( A L B ) )
12013adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( A L B )  e. 
ran  L )
12122necomd 2698 . . . . . . . . 9  |-  ( ph  ->  R  =/=  B )
1225, 7, 8, 9, 14, 11, 121tgelrnln 24754 . . . . . . . 8  |-  ( ph  ->  ( R L B )  e.  ran  L
)
123122adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( R L B )  e. 
ran  L )
1245, 7, 8, 9, 14, 11, 121tglinerflx2 24758 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( R L B ) )
12553, 124elind 3609 . . . . . . . . 9  |-  ( ph  ->  B  e.  ( ( A L B )  i^i  ( R L B ) ) )
1265, 7, 8, 9, 14, 11, 121tglinerflx1 24757 . . . . . . . . 9  |-  ( ph  ->  R  e.  ( R L B ) )
1275, 6, 7, 8, 9, 13, 122, 125, 59, 126, 12, 121, 44ragperp 24841 . . . . . . . 8  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( R L B ) )
128127adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( A L B ) (⟂G `  G ) ( R L B ) )
1295, 6, 7, 8, 62, 120, 123, 128perpcom 24837 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  ( R L B ) (⟂G `  G ) ( A L B ) )
13057, 2, 58, 60, 61, 119, 129reu2eqd 3223 . . . . 5  |-  ( (
ph  /\  X  =  A )  ->  A  =  B )
13155, 130mtand 671 . . . 4  |-  ( ph  ->  -.  X  =  A )
132131neqned 2650 . . 3  |-  ( ph  ->  X  =/=  A )
133 mideulem2.7 . . 3  |-  ( ph  ->  M  e.  P )
134132necomd 2698 . . . 4  |-  ( ph  ->  A  =/=  X )
135 eqid 2471 . . . . 5  |-  ( S `
 A )  =  ( S `  A
)
136 eqid 2471 . . . . 5  |-  ( S `
 M )  =  ( S `  M
)
1375, 6, 7, 8, 27, 9, 10, 135, 17mircl 24785 . . . . 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 399 . . . . . . . . 9  |-  ( ph  ->  ( X  e.  ( A L B )  \/  A  =  B ) )
1415, 8, 7, 9, 10, 11, 54, 140colcom 24682 . . . . . . . 8  |-  ( ph  ->  ( X  e.  ( B L A )  \/  B  =  A ) )
1425, 8, 7, 9, 11, 10, 54, 141colrot1 24683 . . . . . . 7  |-  ( ph  ->  ( B  e.  ( A L X )  \/  A  =  X ) )
1435, 6, 7, 8, 27, 9, 11, 10, 17, 54, 96, 84, 142ragcol 24823 . . . . . 6  |-  ( ph  ->  <" X A O ">  e.  (∟G `  G ) )
1445, 6, 7, 8, 27, 9, 54, 10, 17israg 24821 . . . . . 6  |-  ( ph  ->  ( <" X A O ">  e.  (∟G `  G )  <->  ( X  .-  O )  =  ( X  .-  ( ( S `  A ) `
 O ) ) ) )
145143, 144mpbid 215 . . . . 5  |-  ( ph  ->  ( X  .-  O
)  =  ( X 
.-  ( ( S `
 A ) `  O ) ) )
146 mideulem2.6 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  =  ( X 
.-  R ) )
147146eqcomd 2477 . . . . 5  |-  ( ph  ->  ( X  .-  R
)  =  ( X 
.-  Z ) )
148 eqidd 2472 . . . . 5  |-  ( ph  ->  ( ( S `  A ) `  O
)  =  ( ( S `  A ) `
 O ) )
149 mideulem2.8 . . . . . . . 8  |-  ( ph  ->  R  =  ( ( S `  M ) `
 Z ) )
150149eqcomd 2477 . . . . . . 7  |-  ( ph  ->  ( ( S `  M ) `  Z
)  =  R )
1515, 6, 7, 8, 27, 9, 133, 136, 138, 150mircom 24787 . . . . . 6  |-  ( ph  ->  ( ( S `  M ) `  R
)  =  Z )
152151eqcomd 2477 . . . . 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 24815 . . . 4  |-  ( ph  ->  X  e.  ( A I M ) )
1545, 7, 8, 9, 10, 54, 133, 134, 153btwnlng3 24745 . . 3  |-  ( ph  ->  M  e.  ( A L X ) )
1555, 7, 8, 9, 10, 11, 12, 54, 132, 83, 133, 154tglineeltr 24755 . 2  |-  ( ph  ->  M  e.  ( A L B ) )
1565, 6, 7, 8, 9, 13, 122, 127perpcom 24837 . 2  |-  ( ph  ->  ( R L B ) (⟂G `  G
) ( A L B ) )
157 nelne2 2740 . . . . . 6  |-  ( ( M  e.  ( A L B )  /\  -.  R  e.  ( A L B ) )  ->  M  =/=  R
)
158155, 51, 157syl2anc 673 . . . . 5  |-  ( ph  ->  M  =/=  R )
159158necomd 2698 . . . 4  |-  ( ph  ->  R  =/=  M )
1605, 7, 8, 9, 14, 133, 159tgelrnln 24754 . . 3  |-  ( ph  ->  ( R L M )  e.  ran  L
)
1615, 7, 8, 9, 14, 133, 159tglinerflx2 24758 . . . . 5  |-  ( ph  ->  M  e.  ( R L M ) )
162155, 161elind 3609 . . . 4  |-  ( ph  ->  M  e.  ( ( A L B )  i^i  ( R L M ) ) )
1635, 7, 8, 9, 14, 133, 159tglinerflx1 24757 . . . 4  |-  ( ph  ->  R  e.  ( R L M ) )
164 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  M  =  X )  ->  M  =  X )
1659adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  G  e. TarskiG )
166133adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  M  e.  P )
16710adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  A  e.  P )
16817adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  O  e.  P )
169137adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  M  =  X )  ->  (
( S `  A
) `  O )  e.  P )
170145adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  ( X  .-  O )  =  ( X  .-  (
( S `  A
) `  O )
) )
171164oveq1d 6323 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  O )  =  ( X  .-  O
) )
172164oveq1d 6323 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  ( ( S `
 A ) `  O ) )  =  ( X  .-  (
( S `  A
) `  O )
) )
173170, 171, 1723eqtr4rd 2516 . . . . . . . . . . 11  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  ( ( S `
 A ) `  O ) )  =  ( M  .-  O
) )
174138adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  Z  e.  P )
17514adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  R  e.  P )
176149adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  M  =  X )  ->  R  =  ( ( S `
 M ) `  Z ) )
177176oveq2d 6324 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  R )  =  ( M  .-  (
( S `  M
) `  Z )
) )
1785, 6, 7, 8, 27, 165, 166, 136, 174mircgr 24781 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  ( ( S `
 M ) `  Z ) )  =  ( M  .-  Z
) )
179177, 178eqtrd 2505 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  R )  =  ( M  .-  Z
) )
1805, 6, 7, 165, 166, 175, 166, 174, 179tgcgrcomlr 24603 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  ( R  .-  M )  =  ( Z  .-  M
) )
18183adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  M  =  X )  ->  X  e.  ( A L B ) )
182164, 181eqeltrd 2549 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( A L B ) )
18351adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  =  X )  ->  -.  R  e.  ( A L B ) )
184182, 183, 157syl2anc 673 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  =  X )  ->  M  =/=  R )
185184necomd 2698 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  R  =/=  M )
1865, 6, 7, 165, 175, 166, 174, 166, 180, 185tgcgrneq 24606 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  Z  =/=  M )
1875, 6, 7, 8, 27, 9, 133, 136, 138mirbtwn 24782 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ( ( ( S `  M
) `  Z )
I Z ) )
188149oveq1d 6323 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( R I Z )  =  ( ( ( S `  M
) `  Z )
I Z ) )
189187, 188eleqtrrd 2552 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  ( R I Z ) )
190189adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( R I Z ) )
1915, 6, 7, 165, 175, 166, 174, 190tgbtwncom 24611 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( Z I R ) )
192139adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  =  X )  ->  X  e.  ( ( ( S `
 A ) `  O ) I Z ) )
193164, 192eqeltrd 2549 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( ( ( S `
 A ) `  O ) I Z ) )
1945, 6, 7, 165, 169, 166, 174, 193tgbtwncom 24611 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( Z I ( ( S `  A
) `  O )
) )
19578adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  X  e.  ( R I O ) )
196164, 195eqeltrd 2549 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( R I O ) )
1975, 7, 165, 174, 166, 175, 169, 168, 186, 185, 191, 194, 196tgbtwnconn22 24703 . . . . . . . . . . 11  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( ( ( S `
 A ) `  O ) I O ) )
1985, 6, 7, 8, 27, 165, 166, 136, 168, 169, 173, 197ismir 24783 . . . . . . . . . 10  |-  ( (
ph  /\  M  =  X )  ->  (
( S `  A
) `  O )  =  ( ( S `
 M ) `  O ) )
199198eqcomd 2477 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  (
( S `  M
) `  O )  =  ( ( S `
 A ) `  O ) )
2005, 6, 7, 8, 27, 165, 166, 167, 168, 199miduniq1 24810 . . . . . . . 8  |-  ( (
ph  /\  M  =  X )  ->  M  =  A )
201164, 200eqtr3d 2507 . . . . . . 7  |-  ( (
ph  /\  M  =  X )  ->  X  =  A )
202131, 201mtand 671 . . . . . 6  |-  ( ph  ->  -.  M  =  X )
203202neqned 2650 . . . . 5  |-  ( ph  ->  M  =/=  X )
204203necomd 2698 . . . 4  |-  ( ph  ->  X  =/=  M )
205151oveq2d 6324 . . . . . 6  |-  ( ph  ->  ( X  .-  (
( S `  M
) `  R )
)  =  ( X 
.-  Z ) )
206205, 146eqtr2d 2506 . . . . 5  |-  ( ph  ->  ( X  .-  R
)  =  ( X 
.-  ( ( S `
 M ) `  R ) ) )
2075, 6, 7, 8, 27, 9, 54, 133, 14israg 24821 . . . . 5  |-  ( ph  ->  ( <" X M R ">  e.  (∟G `  G )  <->  ( X  .-  R )  =  ( X  .-  ( ( S `  M ) `
 R ) ) ) )
208206, 207mpbird 240 . . . 4  |-  ( ph  ->  <" X M R ">  e.  (∟G `  G ) )
2095, 6, 7, 8, 9, 13, 160, 162, 83, 163, 204, 159, 208ragperp 24841 . . 3  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( R L M ) )
2105, 6, 7, 8, 9, 13, 160, 209perpcom 24837 . 2  |-  ( ph  ->  ( R L M ) (⟂G `  G
) ( A L B ) )
2112, 4, 52, 53, 155, 156, 210reu2eqd 3223 1  |-  ( ph  ->  B  =  M )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   E!wreu 2758   class class class wbr 4395   ran crn 4840   ` cfv 5589  (class class class)co 6308   <"cs3 12997   Basecbs 15199   distcds 15277  TarskiGcstrkg 24557  Itvcitv 24563  LineGclng 24564  pInvGcmir 24776  ∟Gcrag 24817  ⟂Gcperpg 24819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003  df-s3 13004  df-trkgc 24575  df-trkgb 24576  df-trkgcb 24577  df-trkg 24580  df-cgrg 24635  df-leg 24707  df-mir 24777  df-rag 24818  df-perpg 24820
This theorem is referenced by:  opphllem  24856
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