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Theorem lmieu 24126
Description: Uniqueness of the line mirror point. Theorem 10.2 of [Schwabhauser] p. 88. (Contributed by Thierry Arnoux, 1-Dec-2019.)
Hypotheses
Ref Expression
ismid.p  |-  P  =  ( Base `  G
)
ismid.d  |-  .-  =  ( dist `  G )
ismid.i  |-  I  =  (Itv `  G )
ismid.g  |-  ( ph  ->  G  e. TarskiG )
ismid.1  |-  ( ph  ->  GDimTarskiG 2 )
lmieu.l  |-  L  =  (LineG `  G )
lmieu.1  |-  ( ph  ->  D  e.  ran  L
)
lmieu.a  |-  ( ph  ->  A  e.  P )
Assertion
Ref Expression
lmieu  |-  ( ph  ->  E! b  e.  P  ( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )
Distinct variable groups:    G, b    P, b    ph, b    A, b    D, b    L, b
Allowed substitution hints:    I( b)    .- ( b)

Proof of Theorem lmieu
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lmieu.a . . . 4  |-  ( ph  ->  A  e.  P )
21adantr 465 . . 3  |-  ( (
ph  /\  A  e.  D )  ->  A  e.  P )
3 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  -.  A  =  b
)
4 eqidd 2444 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  -> 
( A (midG `  G ) b )  =  ( A (midG `  G ) b ) )
5 ismid.p . . . . . . . . . . . . . . . 16  |-  P  =  ( Base `  G
)
6 ismid.d . . . . . . . . . . . . . . . 16  |-  .-  =  ( dist `  G )
7 ismid.i . . . . . . . . . . . . . . . 16  |-  I  =  (Itv `  G )
8 ismid.g . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G  e. TarskiG )
98ad4antr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  G  e. TarskiG )
10 ismid.1 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  GDimTarskiG 2 )
1110ad4antr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  GDimTarskiG 2 )
122ad3antrrr 729 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  A  e.  P )
13 simpllr 760 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  -> 
b  e.  P )
14 eqid 2443 . . . . . . . . . . . . . . . 16  |-  (pInvG `  G )  =  (pInvG `  G )
155, 6, 7, 9, 11, 12, 13midcl 24119 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  -> 
( A (midG `  G ) b )  e.  P )
165, 6, 7, 9, 11, 12, 13, 14, 15ismidb 24120 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  -> 
( b  =  ( ( (pInvG `  G
) `  ( A
(midG `  G )
b ) ) `  A )  <->  ( A
(midG `  G )
b )  =  ( A (midG `  G
) b ) ) )
174, 16mpbird 232 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  -> 
b  =  ( ( (pInvG `  G ) `  ( A (midG `  G ) b ) ) `  A ) )
1817adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  -> 
b  =  ( ( (pInvG `  G ) `  ( A (midG `  G ) b ) ) `  A ) )
19 lmieu.l . . . . . . . . . . . . . . . 16  |-  L  =  (LineG `  G )
209adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  ->  G  e. TarskiG )
21 lmieu.1 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  D  e.  ran  L
)
2221ad4antr 731 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  D  e.  ran  L )
2322adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  ->  D  e.  ran  L )
2412adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  ->  A  e.  P )
2513adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  -> 
b  e.  P )
263neqned 2646 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  A  =/=  b )
2726adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  ->  A  =/=  b )
285, 7, 19, 20, 24, 25, 27tgelrnln 23986 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  -> 
( A L b )  e.  ran  L
)
29 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  ->  D  =/=  ( A L b ) )
30 simp-4r 768 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  A  e.  D )
3130adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  ->  A  e.  D )
325, 7, 19, 20, 24, 25, 27tglinerflx1 23989 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  ->  A  e.  ( A L b ) )
3331, 32elind 3673 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  ->  A  e.  ( D  i^i  ( A L b ) ) )
34 simpllr 760 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  -> 
( A (midG `  G ) b )  e.  D )
355, 6, 7, 9, 11, 12, 13midbtwn 24121 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  -> 
( A (midG `  G ) b )  e.  ( A I b ) )
365, 7, 19, 9, 12, 13, 15, 26, 35btwnlng1 23975 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  -> 
( A (midG `  G ) b )  e.  ( A L b ) )
3736adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  -> 
( A (midG `  G ) b )  e.  ( A L b ) )
3834, 37elind 3673 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  -> 
( A (midG `  G ) b )  e.  ( D  i^i  ( A L b ) ) )
395, 7, 19, 20, 23, 28, 29, 33, 38tglineineq 23999 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  ->  A  =  ( A
(midG `  G )
b ) )
4039fveq2d 5860 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  -> 
( (pInvG `  G
) `  A )  =  ( (pInvG `  G ) `  ( A (midG `  G )
b ) ) )
4140fveq1d 5858 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  -> 
( ( (pInvG `  G ) `  A
) `  A )  =  ( ( (pInvG `  G ) `  ( A (midG `  G )
b ) ) `  A ) )
42 eqid 2443 . . . . . . . . . . . . . 14  |-  ( (pInvG `  G ) `  A
)  =  ( (pInvG `  G ) `  A
)
435, 6, 7, 19, 14, 20, 24, 42mircinv 24024 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  -> 
( ( (pInvG `  G ) `  A
) `  A )  =  A )
4418, 41, 433eqtr2rd 2491 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  ->  A  =  b )
453, 44mtand 659 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  -.  D  =/=  ( A L b ) )
468ad5antr 733 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D (⟂G `  G )
( A L b ) )  ->  G  e. TarskiG )
4721ad5antr 733 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D (⟂G `  G )
( A L b ) )  ->  D  e.  ran  L )
48 nne 2644 . . . . . . . . . . . . . . 15  |-  ( -.  D  =/=  ( A L b )  <->  D  =  ( A L b ) )
4945, 48sylib 196 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  D  =  ( A L b ) )
5049adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D (⟂G `  G )
( A L b ) )  ->  D  =  ( A L b ) )
5150, 47eqeltrrd 2532 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D (⟂G `  G )
( A L b ) )  ->  ( A L b )  e. 
ran  L )
52 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D (⟂G `  G )
( A L b ) )  ->  D
(⟂G `  G ) ( A L b ) )
535, 6, 7, 19, 46, 47, 51, 52perpneq 24067 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D (⟂G `  G )
( A L b ) )  ->  D  =/=  ( A L b ) )
5445, 53mtand 659 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  -.  D (⟂G `  G
) ( A L b ) )
5554ex 434 . . . . . . . . 9  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  ->  ( -.  A  =  b  ->  -.  D (⟂G `  G
) ( A L b ) ) )
5655con4d 105 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  ->  ( D
(⟂G `  G ) ( A L b )  ->  A  =  b ) )
57 idd 24 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  ->  ( A  =  b  ->  A  =  b ) )
5856, 57jaod 380 . . . . . . 7  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  ->  ( ( D (⟂G `  G )
( A L b )  \/  A  =  b )  ->  A  =  b ) )
5958impr 619 . . . . . 6  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  A  =  b )
6059eqcomd 2451 . . . . 5  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  b  =  A )
61 simpr 461 . . . . . . . . 9  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  b  =  A )
6261oveq2d 6297 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  ( A (midG `  G )
b )  =  ( A (midG `  G
) A ) )
635, 6, 7, 8, 10, 1, 1midid 24123 . . . . . . . . 9  |-  ( ph  ->  ( A (midG `  G ) A )  =  A )
6463ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  ( A (midG `  G ) A )  =  A )
6562, 64eqtrd 2484 . . . . . . 7  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  ( A (midG `  G )
b )  =  A )
66 simpllr 760 . . . . . . 7  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  A  e.  D )
6765, 66eqeltrd 2531 . . . . . 6  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  ( A (midG `  G )
b )  e.  D
)
6861eqcomd 2451 . . . . . . 7  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  A  =  b )
6968olcd 393 . . . . . 6  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) )
7067, 69jca 532 . . . . 5  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  (
( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )
7160, 70impbida 832 . . . 4  |-  ( ( ( ph  /\  A  e.  D )  /\  b  e.  P )  ->  (
( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) )  <->  b  =  A ) )
7271ralrimiva 2857 . . 3  |-  ( (
ph  /\  A  e.  D )  ->  A. b  e.  P  ( (
( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) )  <->  b  =  A ) )
73 reu6i 3276 . . 3  |-  ( ( A  e.  P  /\  A. b  e.  P  ( ( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) )  <->  b  =  A ) )  ->  E! b  e.  P  ( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )
742, 72, 73syl2anc 661 . 2  |-  ( (
ph  /\  A  e.  D )  ->  E! b  e.  P  (
( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )
758adantr 465 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  D )  ->  G  e. TarskiG )
7675ad2antrr 725 . . . . 5  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  G  e. TarskiG )
7721adantr 465 . . . . . . 7  |-  ( (
ph  /\  -.  A  e.  D )  ->  D  e.  ran  L )
7877ad2antrr 725 . . . . . 6  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  D  e.  ran  L )
79 simplr 755 . . . . . 6  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  x  e.  D
)
805, 19, 7, 76, 78, 79tglnpt 23912 . . . . 5  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  x  e.  P
)
81 eqid 2443 . . . . 5  |-  ( (pInvG `  G ) `  x
)  =  ( (pInvG `  G ) `  x
)
821adantr 465 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  D )  ->  A  e.  P )
8382ad2antrr 725 . . . . 5  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  A  e.  P
)
845, 6, 7, 19, 14, 76, 80, 81, 83mircl 24018 . . . 4  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  ( ( (pInvG `  G ) `  x
) `  A )  e.  P )
85 oveq2 6289 . . . . . . . . . 10  |-  ( x  =  ( A (midG `  G ) b )  ->  ( A L x )  =  ( A L ( A (midG `  G )
b ) ) )
8685breq1d 4447 . . . . . . . . 9  |-  ( x  =  ( A (midG `  G ) b )  ->  ( ( A L x ) (⟂G `  G ) D  <->  ( A L ( A (midG `  G ) b ) ) (⟂G `  G
) D ) )
87 simprl 756 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( A
(midG `  G )
b )  e.  D
)
88 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  e.  D )  ->  -.  A  e.  D )
895, 6, 7, 19, 75, 77, 82, 88foot 24072 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  A  e.  D )  ->  E! x  e.  D  ( A L x ) (⟂G `  G ) D )
90 reurmo 3061 . . . . . . . . . . 11  |-  ( E! x  e.  D  ( A L x ) (⟂G `  G ) D  ->  E* x  e.  D  ( A L x ) (⟂G `  G
) D )
9189, 90syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  e.  D )  ->  E* x  e.  D  ( A L x ) (⟂G `  G ) D )
9291ad4antr 731 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  E* x  e.  D  ( A L x ) (⟂G `  G ) D )
9379ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  x  e.  D )
94 simpllr 760 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( A L x ) (⟂G `  G ) D )
9576ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  G  e. TarskiG )
9683ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  A  e.  P )
97 simplr 755 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  b  e.  P )
9810ad5antr 733 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  GDimTarskiG 2 )
995, 6, 7, 95, 98, 96, 97midcl 24119 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( A
(midG `  G )
b )  e.  P
)
1005, 6, 7, 95, 98, 96, 97midbtwn 24121 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( A
(midG `  G )
b )  e.  ( A I b ) )
10188ad4antr 731 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  -.  A  e.  D )
102 nelne2 2773 . . . . . . . . . . . . 13  |-  ( ( ( A (midG `  G ) b )  e.  D  /\  -.  A  e.  D )  ->  ( A (midG `  G ) b )  =/=  A )
10387, 101, 102syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( A
(midG `  G )
b )  =/=  A
)
1045, 6, 7, 95, 96, 99, 97, 100, 103tgbtwnne 23857 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  A  =/=  b )
1055, 7, 19, 95, 96, 97, 99, 104, 100btwnlng1 23975 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( A
(midG `  G )
b )  e.  ( A L b ) )
1065, 7, 19, 95, 96, 97, 104, 99, 103, 105tglineelsb2 23988 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( A L b )  =  ( A L ( A (midG `  G
) b ) ) )
10778ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  D  e.  ran  L )
1085, 7, 19, 95, 96, 97, 104tgelrnln 23986 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( A L b )  e. 
ran  L )
109104neneqd 2645 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  -.  A  =  b )
110 simprr 757 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( D
(⟂G `  G ) ( A L b )  \/  A  =  b ) )
111110orcomd 388 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( A  =  b  \/  D
(⟂G `  G ) ( A L b ) ) )
112111ord 377 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( -.  A  =  b  ->  D (⟂G `  G )
( A L b ) ) )
113109, 112mpd 15 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  D (⟂G `  G ) ( A L b ) )
1145, 6, 7, 19, 95, 107, 108, 113perpcom 24066 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( A L b ) (⟂G `  G ) D )
115106, 114eqbrtrrd 4459 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( A L ( A (midG `  G ) b ) ) (⟂G `  G
) D )
11686, 87, 92, 93, 94, 115rmoi2 3419 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  x  =  ( A (midG `  G
) b ) )
117116eqcomd 2451 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( A
(midG `  G )
b )  =  x )
11880ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  x  e.  P )
1195, 6, 7, 95, 98, 96, 97, 14, 118ismidb 24120 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  ( b  =  ( ( (pInvG `  G ) `  x
) `  A )  <->  ( A (midG `  G
) b )  =  x ) )
120117, 119mpbird 232 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )  ->  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)
121 simpr 461 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)
12276ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  G  e. TarskiG )
12310ad5antr 733 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  GDimTarskiG 2 )
12483ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  A  e.  P )
125 simplr 755 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  b  e.  P )
12680ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  x  e.  P )
1275, 6, 7, 122, 123, 124, 125, 14, 126ismidb 24120 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  ( b  =  ( ( (pInvG `  G ) `  x
) `  A )  <->  ( A (midG `  G
) b )  =  x ) )
128121, 127mpbid 210 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  ( A
(midG `  G )
b )  =  x )
12979ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  x  e.  D )
130128, 129eqeltrd 2531 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  ( A
(midG `  G )
b )  e.  D
)
131122adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  G  e. TarskiG )
132 simp-4r 768 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  ( A L x ) (⟂G `  G ) D )
13319, 131, 132perpln1 24063 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  ( A L x )  e. 
ran  L )
13478ad3antrrr 729 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  D  e.  ran  L )
1355, 6, 7, 19, 131, 133, 134, 132perpcom 24066 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  D
(⟂G `  G ) ( A L x ) )
136124adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  A  e.  P )
137126adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  x  e.  P )
1385, 7, 19, 131, 136, 137, 133tglnne 23984 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  A  =/=  x )
139 simpllr 760 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  b  e.  P )
140 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  A  =/=  b )
141140necomd 2714 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  b  =/=  A )
1425, 6, 7, 19, 14, 131, 137, 81, 136mirbtwn 24015 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  x  e.  ( ( ( (pInvG `  G ) `  x
) `  A )
I A ) )
143 simplr 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)
144143oveq1d 6296 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  (
b I A )  =  ( ( ( (pInvG `  G ) `  x ) `  A
) I A ) )
145142, 144eleqtrrd 2534 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  x  e.  ( b I A ) )
1465, 7, 19, 131, 139, 136, 137, 141, 145btwnlng1 23975 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  x  e.  ( b L A ) )
1475, 7, 19, 131, 136, 137, 139, 138, 146, 141lnrot1 23979 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  b  e.  ( A L x ) )
1485, 7, 19, 131, 136, 137, 138, 139, 141, 147tglineelsb2 23988 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  ( A L x )  =  ( A L b ) )
149135, 148breqtrd 4461 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  -.  A  e.  D )  /\  x  e.  D
)  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  /\  A  =/=  b )  ->  D
(⟂G `  G ) ( A L b ) )
150149ex 434 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  ( A  =/=  b  ->  D (⟂G `  G ) ( A L b ) ) )
151150necon1bd 2661 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  ( -.  D (⟂G `  G )
( A L b )  ->  A  =  b ) )
152151orrd 378 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  ( D
(⟂G `  G ) ( A L b )  \/  A  =  b ) )
153130, 152jca 532 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  -.  A  e.  D )  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  /\  b  =  ( ( (pInvG `  G ) `  x
) `  A )
)  ->  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )
154120, 153impbida 832 . . . . 5  |-  ( ( ( ( ( ph  /\ 
-.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  /\  b  e.  P
)  ->  ( (
( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) )  <->  b  =  ( ( (pInvG `  G ) `  x
) `  A )
) )
155154ralrimiva 2857 . . . 4  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  A. b  e.  P  ( ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) )  <-> 
b  =  ( ( (pInvG `  G ) `  x ) `  A
) ) )
156 reu6i 3276 . . . 4  |-  ( ( ( ( (pInvG `  G ) `  x
) `  A )  e.  P  /\  A. b  e.  P  ( (
( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) )  <->  b  =  ( ( (pInvG `  G ) `  x
) `  A )
) )  ->  E! b  e.  P  (
( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )
15784, 155, 156syl2anc 661 . . 3  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  E! b  e.  P  ( ( A (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( A L b )  \/  A  =  b ) ) )
1585, 6, 7, 19, 75, 77, 82, 88footex 24071 . . 3  |-  ( (
ph  /\  -.  A  e.  D )  ->  E. x  e.  D  ( A L x ) (⟂G `  G ) D )
159157, 158r19.29a 2985 . 2  |-  ( (
ph  /\  -.  A  e.  D )  ->  E! b  e.  P  (
( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )
16074, 159pm2.61dan 791 1  |-  ( ph  ->  E! b  e.  P  ( ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E!wreu 2795   E*wrmo 2796   class class class wbr 4437   ran crn 4990   ` cfv 5578  (class class class)co 6281   2c2 10592   Basecbs 14613   distcds 14687  TarskiGcstrkg 23801  DimTarskiGcstrkgld 23805  Itvcitv 23808  LineGclng 23809  pInvGcmir 24009  ⟂Gcperpg 24048  midGcmid 24114
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 11092  df-fz 11683  df-fzo 11806  df-hash 12387  df-word 12523  df-concat 12525  df-s1 12526  df-s2 12794  df-s3 12795  df-trkgc 23820  df-trkgb 23821  df-trkgcb 23822  df-trkgld 23824  df-trkg 23826  df-cgrg 23879  df-leg 23946  df-mir 24010  df-rag 24047  df-perpg 24049  df-mid 24116
This theorem is referenced by:  lmif  24127  islmib  24129
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