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Theorem mideulem2 24776
Description: Lemma for opphllem 24777, which is itself used for mideu 24780. (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 6298 . . 3  |-  ( y  =  B  ->  ( R L y )  =  ( R L B ) )
21breq1d 4412 . 2  |-  ( y  =  B  ->  (
( R L y ) (⟂G `  G
) ( A L B )  <->  ( R L B ) (⟂G `  G
) ( A L B ) ) )
3 oveq2 6298 . . 3  |-  ( y  =  M  ->  ( R L y )  =  ( R L M ) )
43breq1d 4412 . 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 24675 . . 3  |-  ( ph  ->  ( A L B )  e.  ran  L
)
14 opphllem.1 . . 3  |-  ( ph  ->  R  e.  P )
1512adantr 467 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  A  =/=  B )
1615neneqd 2629 . . . . 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 24756 . . . . . . . . . 10  |-  ( ph  ->  ( A L O )  e.  ran  L
)
215, 7, 8, 9, 10, 17, 20tglnne 24673 . . . . . . . . 9  |-  ( ph  ->  A  =/=  O )
225, 6, 7, 9, 10, 17, 11, 14, 18, 21tgcgrneq 24527 . . . . . . . 8  |-  ( ph  ->  B  =/=  R )
2322adantr 467 . . . . . . 7  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  B  =/=  R )
2423necomd 2679 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  R  =/=  B )
2524neneqd 2629 . . . . 5  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  -.  R  =  B )
2616, 25jca 535 . . . 4  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  ( -.  A  =  B  /\  -.  R  =  B
) )
27 mideu.s . . . . . 6  |-  S  =  (pInvG `  G )
289adantr 467 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  G  e. TarskiG )
2910adantr 467 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  A  e.  P )
3011adantr 467 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  B  e.  P )
3114adantr 467 . . . . . 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 24756 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q L B )  e.  ran  L
)
355, 7, 8, 9, 32, 11, 34tglnne 24673 . . . . . . . . . . 11  |-  ( ph  ->  Q  =/=  B )
365, 7, 8, 9, 32, 11, 35tglinerflx2 24679 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( Q L B ) )
375, 6, 7, 8, 9, 13, 34, 33perpcom 24758 . . . . . . . . . . 11  |-  ( ph  ->  ( Q L B ) (⟂G `  G
) ( A L B ) )
385, 7, 8, 9, 10, 11, 12tglinecom 24680 . . . . . . . . . . 11  |-  ( ph  ->  ( A L B )  =  ( B L A ) )
3937, 38breqtrd 4427 . . . . . . . . . 10  |-  ( ph  ->  ( Q L B ) (⟂G `  G
) ( B L A ) )
405, 6, 7, 8, 9, 32, 11, 36, 10, 39perprag 24768 . . . . . . . . 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 24602 . . . . . . . . 9  |-  ( ph  ->  ( Q  e.  ( B L R )  \/  B  =  R ) )
435, 6, 7, 8, 27, 9, 32, 11, 10, 14, 40, 35, 42ragcol 24744 . . . . . . . 8  |-  ( ph  ->  <" R B A ">  e.  (∟G `  G ) )
445, 6, 7, 8, 27, 9, 14, 11, 10, 43ragcom 24743 . . . . . . 7  |-  ( ph  ->  <" A B R ">  e.  (∟G `  G ) )
4544adantr 467 . . . . . 6  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  <" A B R ">  e.  (∟G `  G ) )
46 simpr 463 . . . . . . 7  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  R  e.  ( A L B ) )
4746orcd 394 . . . . . 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 24751 . . . . 5  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  ( A  =  B  \/  R  =  B ) )
49 oran 499 . . . . 5  |-  ( ( A  =  B  \/  R  =  B )  <->  -.  ( -.  A  =  B  /\  -.  R  =  B ) )
5048, 49sylib 200 . . . 4  |-  ( (
ph  /\  R  e.  ( A L B ) )  ->  -.  ( -.  A  =  B  /\  -.  R  =  B ) )
5126, 50pm2.65da 580 . . 3  |-  ( ph  ->  -.  R  e.  ( A L B ) )
525, 6, 7, 8, 9, 13, 14, 51foot 24764 . 2  |-  ( ph  ->  E! y  e.  ( A L B ) ( R L y ) (⟂G `  G
) ( A L B ) )
535, 7, 8, 9, 10, 11, 12tglinerflx2 24679 . 2  |-  ( ph  ->  B  e.  ( A L B ) )
54 mideulem2.1 . . 3  |-  ( ph  ->  X  e.  P )
5512neneqd 2629 . . . . 5  |-  ( ph  ->  -.  A  =  B )
56 oveq2 6298 . . . . . . 7  |-  ( y  =  A  ->  ( R L y )  =  ( R L A ) )
5756breq1d 4412 . . . . . 6  |-  ( y  =  A  ->  (
( R L y ) (⟂G `  G
) ( A L B )  <->  ( R L A ) (⟂G `  G
) ( A L B ) ) )
5852adantr 467 . . . . . 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 24678 . . . . . . 7  |-  ( ph  ->  A  e.  ( A L B ) )
6059adantr 467 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  A  e.  ( A L B ) )
6153adantr 467 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  B  e.  ( A L B ) )
629adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  G  e. TarskiG )
6314adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  R  e.  P )
6410adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  A  e.  P )
6551, 55jca 535 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  R  e.  ( A L B )  /\  -.  A  =  B ) )
66 pm4.56 498 . . . . . . . . . . . 12  |-  ( ( -.  R  e.  ( A L B )  /\  -.  A  =  B )  <->  -.  ( R  e.  ( A L B )  \/  A  =  B ) )
6765, 66sylib 200 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( R  e.  ( A L B )  \/  A  =  B ) )
685, 7, 8, 9, 14, 10, 11, 67ncolne1 24670 . . . . . . . . . 10  |-  ( ph  ->  R  =/=  A )
6968adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  R  =/=  A )
705, 7, 8, 62, 63, 64, 69tglinecom 24680 . . . . . . . 8  |-  ( (
ph  /\  X  =  A )  ->  ( R L A )  =  ( A L R ) )
7169necomd 2679 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  A  =/=  R )
7217adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  O  e.  P )
7321necomd 2679 . . . . . . . . . 10  |-  ( ph  ->  O  =/=  A )
7473adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  O  =/=  A )
7554adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =  A )  ->  X  e.  P )
76 simpr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  X  =  A )  ->  X  =  A )
7776, 71eqnetrd 2691 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =  A )  ->  X  =/=  R )
78 mideulem2.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X  e.  ( R I O ) )
795, 6, 7, 9, 14, 54, 17, 78tgbtwncom 24532 . . . . . . . . . . . . . . 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 24693 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X  e.  ( A L B ) )
8412necomd 2679 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  B  =/=  A )
8584neneqd 2629 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  -.  B  =  A )
8685adantr 467 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  -.  B  =  A )
8773neneqd 2629 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  -.  O  =  A )
8887adantr 467 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  -.  O  =  A )
8986, 88jca 535 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  ( -.  B  =  A  /\  -.  O  =  A
) )
909adantr 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  G  e. TarskiG )
9111adantr 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  B  e.  P )
9210adantr 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  A  e.  P )
9317adantr 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  O  e.  P )
945, 7, 8, 9, 11, 10, 84tglinerflx2 24679 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  A  e.  ( B L A ) )
9538, 19eqbrtrrd 4425 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( B L A ) (⟂G `  G
) ( A L O ) )
965, 6, 7, 8, 9, 11, 10, 94, 17, 95perprag 24768 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  <" B A O ">  e.  (∟G `  G ) )
9796adantr 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  <" B A O ">  e.  (∟G `  G ) )
98 simpr 463 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  O  e.  ( B L A ) )
9998orcd 394 . . . . . . . . . . . . . . . . . . . 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 24751 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  ( B  =  A  \/  O  =  A ) )
101 oran 499 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  =  A  \/  O  =  A )  <->  -.  ( -.  B  =  A  /\  -.  O  =  A ) )
102100, 101sylib 200 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  O  e.  ( B L A ) )  ->  -.  ( -.  B  =  A  /\  -.  O  =  A ) )
10389, 102pm2.65da 580 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  -.  O  e.  ( B L A ) )
104103, 38neleqtrrd 2551 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  -.  O  e.  ( A L B ) )
105 nelne2 2721 . . . . . . . . . . . . . . . 16  |-  ( ( X  e.  ( A L B )  /\  -.  O  e.  ( A L B ) )  ->  X  =/=  O
)
10683, 104, 105syl2anc 667 . . . . . . . . . . . . . . 15  |-  ( ph  ->  X  =/=  O )
1075, 6, 7, 9, 17, 54, 14, 79, 106tgbtwnne 24534 . . . . . . . . . . . . . 14  |-  ( ph  ->  O  =/=  R )
108107adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  X  =  A )  ->  O  =/=  R )
109108necomd 2679 . . . . . . . . . . . 12  |-  ( (
ph  /\  X  =  A )  ->  R  =/=  O )
11078adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  X  =  A )  ->  X  e.  ( R I O ) )
1115, 7, 8, 62, 63, 72, 75, 109, 110btwnlng1 24664 . . . . . . . . . . 11  |-  ( (
ph  /\  X  =  A )  ->  X  e.  ( R L O ) )
1125, 7, 8, 62, 75, 63, 72, 77, 111, 109lnrot2 24669 . . . . . . . . . 10  |-  ( (
ph  /\  X  =  A )  ->  O  e.  ( X L R ) )
11376oveq1d 6305 . . . . . . . . . 10  |-  ( (
ph  /\  X  =  A )  ->  ( X L R )  =  ( A L R ) )
114112, 113eleqtrd 2531 . . . . . . . . 9  |-  ( (
ph  /\  X  =  A )  ->  O  e.  ( A L R ) )
1155, 7, 8, 62, 64, 63, 71, 72, 74, 114tglineelsb2 24677 . . . . . . . 8  |-  ( (
ph  /\  X  =  A )  ->  ( A L R )  =  ( A L O ) )
11670, 115eqtrd 2485 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( R L A )  =  ( A L O ) )
1175, 6, 7, 8, 9, 13, 20, 19perpcom 24758 . . . . . . . 8  |-  ( ph  ->  ( A L O ) (⟂G `  G
) ( A L B ) )
118117adantr 467 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( A L O ) (⟂G `  G ) ( A L B ) )
119116, 118eqbrtrd 4423 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  ( R L A ) (⟂G `  G ) ( A L B ) )
12013adantr 467 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( A L B )  e. 
ran  L )
12122necomd 2679 . . . . . . . . 9  |-  ( ph  ->  R  =/=  B )
1225, 7, 8, 9, 14, 11, 121tgelrnln 24675 . . . . . . . 8  |-  ( ph  ->  ( R L B )  e.  ran  L
)
123122adantr 467 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( R L B )  e. 
ran  L )
1245, 7, 8, 9, 14, 11, 121tglinerflx2 24679 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( R L B ) )
12553, 124elind 3618 . . . . . . . . 9  |-  ( ph  ->  B  e.  ( ( A L B )  i^i  ( R L B ) ) )
1265, 7, 8, 9, 14, 11, 121tglinerflx1 24678 . . . . . . . . 9  |-  ( ph  ->  R  e.  ( R L B ) )
1275, 6, 7, 8, 9, 13, 122, 125, 59, 126, 12, 121, 44ragperp 24762 . . . . . . . 8  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( R L B ) )
128127adantr 467 . . . . . . 7  |-  ( (
ph  /\  X  =  A )  ->  ( A L B ) (⟂G `  G ) ( R L B ) )
1295, 6, 7, 8, 62, 120, 123, 128perpcom 24758 . . . . . 6  |-  ( (
ph  /\  X  =  A )  ->  ( R L B ) (⟂G `  G ) ( A L B ) )
13057, 2, 58, 60, 61, 119, 129reu2eqd 3235 . . . . 5  |-  ( (
ph  /\  X  =  A )  ->  A  =  B )
13155, 130mtand 665 . . . 4  |-  ( ph  ->  -.  X  =  A )
132131neqned 2631 . . 3  |-  ( ph  ->  X  =/=  A )
133 mideulem2.7 . . 3  |-  ( ph  ->  M  e.  P )
134132necomd 2679 . . . 4  |-  ( ph  ->  A  =/=  X )
135 eqid 2451 . . . . 5  |-  ( S `
 A )  =  ( S `  A
)
136 eqid 2451 . . . . 5  |-  ( S `
 M )  =  ( S `  M
)
1375, 6, 7, 8, 27, 9, 10, 135, 17mircl 24706 . . . . 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 394 . . . . . . . . 9  |-  ( ph  ->  ( X  e.  ( A L B )  \/  A  =  B ) )
1415, 8, 7, 9, 10, 11, 54, 140colcom 24603 . . . . . . . 8  |-  ( ph  ->  ( X  e.  ( B L A )  \/  B  =  A ) )
1425, 8, 7, 9, 11, 10, 54, 141colrot1 24604 . . . . . . 7  |-  ( ph  ->  ( B  e.  ( A L X )  \/  A  =  X ) )
1435, 6, 7, 8, 27, 9, 11, 10, 17, 54, 96, 84, 142ragcol 24744 . . . . . 6  |-  ( ph  ->  <" X A O ">  e.  (∟G `  G ) )
1445, 6, 7, 8, 27, 9, 54, 10, 17israg 24742 . . . . . 6  |-  ( ph  ->  ( <" X A O ">  e.  (∟G `  G )  <->  ( X  .-  O )  =  ( X  .-  ( ( S `  A ) `
 O ) ) ) )
145143, 144mpbid 214 . . . . 5  |-  ( ph  ->  ( X  .-  O
)  =  ( X 
.-  ( ( S `
 A ) `  O ) ) )
146 mideulem2.6 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  =  ( X 
.-  R ) )
147146eqcomd 2457 . . . . 5  |-  ( ph  ->  ( X  .-  R
)  =  ( X 
.-  Z ) )
148 eqidd 2452 . . . . 5  |-  ( ph  ->  ( ( S `  A ) `  O
)  =  ( ( S `  A ) `
 O ) )
149 mideulem2.8 . . . . . . . 8  |-  ( ph  ->  R  =  ( ( S `  M ) `
 Z ) )
150149eqcomd 2457 . . . . . . 7  |-  ( ph  ->  ( ( S `  M ) `  Z
)  =  R )
1515, 6, 7, 8, 27, 9, 133, 136, 138, 150mircom 24708 . . . . . 6  |-  ( ph  ->  ( ( S `  M ) `  R
)  =  Z )
152151eqcomd 2457 . . . . 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 24736 . . . 4  |-  ( ph  ->  X  e.  ( A I M ) )
1545, 7, 8, 9, 10, 54, 133, 134, 153btwnlng3 24666 . . 3  |-  ( ph  ->  M  e.  ( A L X ) )
1555, 7, 8, 9, 10, 11, 12, 54, 132, 83, 133, 154tglineeltr 24676 . 2  |-  ( ph  ->  M  e.  ( A L B ) )
1565, 6, 7, 8, 9, 13, 122, 127perpcom 24758 . 2  |-  ( ph  ->  ( R L B ) (⟂G `  G
) ( A L B ) )
157 nelne2 2721 . . . . . 6  |-  ( ( M  e.  ( A L B )  /\  -.  R  e.  ( A L B ) )  ->  M  =/=  R
)
158155, 51, 157syl2anc 667 . . . . 5  |-  ( ph  ->  M  =/=  R )
159158necomd 2679 . . . 4  |-  ( ph  ->  R  =/=  M )
1605, 7, 8, 9, 14, 133, 159tgelrnln 24675 . . 3  |-  ( ph  ->  ( R L M )  e.  ran  L
)
1615, 7, 8, 9, 14, 133, 159tglinerflx2 24679 . . . . 5  |-  ( ph  ->  M  e.  ( R L M ) )
162155, 161elind 3618 . . . 4  |-  ( ph  ->  M  e.  ( ( A L B )  i^i  ( R L M ) ) )
1635, 7, 8, 9, 14, 133, 159tglinerflx1 24678 . . . 4  |-  ( ph  ->  R  e.  ( R L M ) )
164 simpr 463 . . . . . . . 8  |-  ( (
ph  /\  M  =  X )  ->  M  =  X )
1659adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  G  e. TarskiG )
166133adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  M  e.  P )
16710adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  A  e.  P )
16817adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  O  e.  P )
169137adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  M  =  X )  ->  (
( S `  A
) `  O )  e.  P )
170145adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  ( X  .-  O )  =  ( X  .-  (
( S `  A
) `  O )
) )
171164oveq1d 6305 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  O )  =  ( X  .-  O
) )
172164oveq1d 6305 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  ( ( S `
 A ) `  O ) )  =  ( X  .-  (
( S `  A
) `  O )
) )
173170, 171, 1723eqtr4rd 2496 . . . . . . . . . . 11  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  ( ( S `
 A ) `  O ) )  =  ( M  .-  O
) )
174138adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  Z  e.  P )
17514adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  R  e.  P )
176149adantr 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  M  =  X )  ->  R  =  ( ( S `
 M ) `  Z ) )
177176oveq2d 6306 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  R )  =  ( M  .-  (
( S `  M
) `  Z )
) )
1785, 6, 7, 8, 27, 165, 166, 136, 174mircgr 24702 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  ( ( S `
 M ) `  Z ) )  =  ( M  .-  Z
) )
179177, 178eqtrd 2485 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  =  X )  ->  ( M  .-  R )  =  ( M  .-  Z
) )
1805, 6, 7, 165, 166, 175, 166, 174, 179tgcgrcomlr 24524 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  ( R  .-  M )  =  ( Z  .-  M
) )
18183adantr 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  M  =  X )  ->  X  e.  ( A L B ) )
182164, 181eqeltrd 2529 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( A L B ) )
18351adantr 467 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  =  X )  ->  -.  R  e.  ( A L B ) )
184182, 183, 157syl2anc 667 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  =  X )  ->  M  =/=  R )
185184necomd 2679 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  R  =/=  M )
1865, 6, 7, 165, 175, 166, 174, 166, 180, 185tgcgrneq 24527 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  Z  =/=  M )
1875, 6, 7, 8, 27, 9, 133, 136, 138mirbtwn 24703 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ( ( ( S `  M
) `  Z )
I Z ) )
188149oveq1d 6305 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( R I Z )  =  ( ( ( S `  M
) `  Z )
I Z ) )
189187, 188eleqtrrd 2532 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  ( R I Z ) )
190189adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( R I Z ) )
1915, 6, 7, 165, 175, 166, 174, 190tgbtwncom 24532 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( Z I R ) )
192139adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  =  X )  ->  X  e.  ( ( ( S `
 A ) `  O ) I Z ) )
193164, 192eqeltrd 2529 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( ( ( S `
 A ) `  O ) I Z ) )
1945, 6, 7, 165, 169, 166, 174, 193tgbtwncom 24532 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( Z I ( ( S `  A
) `  O )
) )
19578adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  =  X )  ->  X  e.  ( R I O ) )
196164, 195eqeltrd 2529 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( R I O ) )
1975, 7, 165, 174, 166, 175, 169, 168, 186, 185, 191, 194, 196tgbtwnconn22 24624 . . . . . . . . . . 11  |-  ( (
ph  /\  M  =  X )  ->  M  e.  ( ( ( S `
 A ) `  O ) I O ) )
1985, 6, 7, 8, 27, 165, 166, 136, 168, 169, 173, 197ismir 24704 . . . . . . . . . 10  |-  ( (
ph  /\  M  =  X )  ->  (
( S `  A
) `  O )  =  ( ( S `
 M ) `  O ) )
199198eqcomd 2457 . . . . . . . . 9  |-  ( (
ph  /\  M  =  X )  ->  (
( S `  M
) `  O )  =  ( ( S `
 A ) `  O ) )
2005, 6, 7, 8, 27, 165, 166, 167, 168, 199miduniq1 24731 . . . . . . . 8  |-  ( (
ph  /\  M  =  X )  ->  M  =  A )
201164, 200eqtr3d 2487 . . . . . . 7  |-  ( (
ph  /\  M  =  X )  ->  X  =  A )
202131, 201mtand 665 . . . . . 6  |-  ( ph  ->  -.  M  =  X )
203202neqned 2631 . . . . 5  |-  ( ph  ->  M  =/=  X )
204203necomd 2679 . . . 4  |-  ( ph  ->  X  =/=  M )
205151oveq2d 6306 . . . . . 6  |-  ( ph  ->  ( X  .-  (
( S `  M
) `  R )
)  =  ( X 
.-  Z ) )
206205, 146eqtr2d 2486 . . . . 5  |-  ( ph  ->  ( X  .-  R
)  =  ( X 
.-  ( ( S `
 M ) `  R ) ) )
2075, 6, 7, 8, 27, 9, 54, 133, 14israg 24742 . . . . 5  |-  ( ph  ->  ( <" X M R ">  e.  (∟G `  G )  <->  ( X  .-  R )  =  ( X  .-  ( ( S `  M ) `
 R ) ) ) )
208206, 207mpbird 236 . . . 4  |-  ( ph  ->  <" X M R ">  e.  (∟G `  G ) )
2095, 6, 7, 8, 9, 13, 160, 162, 83, 163, 204, 159, 208ragperp 24762 . . 3  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( R L M ) )
2105, 6, 7, 8, 9, 13, 160, 209perpcom 24758 . 2  |-  ( ph  ->  ( R L M ) (⟂G `  G
) ( A L B ) )
2112, 4, 52, 53, 155, 156, 210reu2eqd 3235 1  |-  ( ph  ->  B  =  M )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   E!wreu 2739   class class class wbr 4402   ran crn 4835   ` cfv 5582  (class class class)co 6290   <"cs3 12938   Basecbs 15121   distcds 15199  TarskiGcstrkg 24478  Itvcitv 24484  LineGclng 24485  pInvGcmir 24697  ∟Gcrag 24738  ⟂Gcperpg 24740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-concat 12666  df-s1 12667  df-s2 12944  df-s3 12945  df-trkgc 24496  df-trkgb 24497  df-trkgcb 24498  df-trkg 24501  df-cgrg 24556  df-leg 24628  df-mir 24698  df-rag 24739  df-perpg 24741
This theorem is referenced by:  opphllem  24777
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