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Theorem lmieu 24826
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 467 . . 3  |-  ( (
ph  /\  A  e.  D )  ->  A  e.  P )
3 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  -.  A  =  b
)
4 eqidd 2452 . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  GDimTarskiG 2 )
122ad3antrrr 736 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  A  e.  P )
13 simpllr 769 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  -> 
b  e.  P )
14 eqid 2451 . . . . . . . . . . . . . . . 16  |-  (pInvG `  G )  =  (pInvG `  G )
155, 6, 7, 9, 11, 12, 13midcl 24819 . . . . . . . . . . . . . . . 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 24820 . . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  D  e.  ran  L )
2322adantr 467 . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  ->  A  e.  P )
2513adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  -> 
b  e.  P )
263neqned 2631 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  A  =/=  b )
2726adantr 467 . . . . . . . . . . . . . . . . 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 24675 . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  A  e.  D )
3130adantr 467 . . . . . . . . . . . . . . . . 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 24678 . . . . . . . . . . . . . . . . 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 3618 . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . 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 24821 . . . . . . . . . . . . . . . . . . 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 24664 . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . 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 3618 . . . . . . . . . . . . . . . 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 24688 . . . . . . . . . . . . . . 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 5869 . . . . . . . . . . . . . 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 5867 . . . . . . . . . . . . 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 2451 . . . . . . . . . . . . . 14  |-  ( (pInvG `  G ) `  A
)  =  ( (pInvG `  G ) `  A
)
435, 6, 7, 19, 14, 20, 24, 42mircinv 24713 . . . . . . . . . . . . 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 2492 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  A  e.  D )  /\  b  e.  P )  /\  ( A (midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  /\  D  =/=  ( A L b ) )  ->  A  =  b )
453, 44mtand 665 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  -.  D  =/=  ( A L b ) )
468ad5antr 740 . . . . . . . . . . . 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 740 . . . . . . . . . . . 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 2628 . . . . . . . . . . . . . . 15  |-  ( -.  D  =/=  ( A L b )  <->  D  =  ( A L b ) )
4945, 48sylib 200 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  D  =  ( A L b ) )
5049adantr 467 . . . . . . . . . . . . 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 2530 . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 24759 . . . . . . . . . . 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 665 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  /\  -.  A  =  b )  ->  -.  D (⟂G `  G
) ( A L b ) )
5554ex 436 . . . . . . . . 9  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  ->  ( -.  A  =  b  ->  -.  D (⟂G `  G
) ( A L b ) ) )
5655con4d 109 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  ->  ( D
(⟂G `  G ) ( A L b )  ->  A  =  b ) )
57 idd 25 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  ( A
(midG `  G )
b )  e.  D
)  ->  ( A  =  b  ->  A  =  b ) )
5856, 57jaod 382 . . . . . . 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 625 . . . . . 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 2457 . . . . 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 463 . . . . . . . . 9  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  b  =  A )
6261oveq2d 6306 . . . . . . . 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 24823 . . . . . . . . 9  |-  ( ph  ->  ( A (midG `  G ) A )  =  A )
6463ad3antrrr 736 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  ( A (midG `  G ) A )  =  A )
6562, 64eqtrd 2485 . . . . . . 7  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  ( A (midG `  G )
b )  =  A )
66 simpllr 769 . . . . . . 7  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  A  e.  D )
6765, 66eqeltrd 2529 . . . . . 6  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  ( A (midG `  G )
b )  e.  D
)
6861eqcomd 2457 . . . . . . 7  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  A  =  b )
6968olcd 395 . . . . . 6  |-  ( ( ( ( ph  /\  A  e.  D )  /\  b  e.  P
)  /\  b  =  A )  ->  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) )
7067, 69jca 535 . . . . 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 843 . . . 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 2802 . . 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 3229 . . 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 667 . 2  |-  ( (
ph  /\  A  e.  D )  ->  E! b  e.  P  (
( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( A L b )  \/  A  =  b ) ) )
758adantr 467 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  D )  ->  G  e. TarskiG )
7675ad2antrr 732 . . . . 5  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  G  e. TarskiG )
7721adantr 467 . . . . . . 7  |-  ( (
ph  /\  -.  A  e.  D )  ->  D  e.  ran  L )
7877ad2antrr 732 . . . . . 6  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  D  e.  ran  L )
79 simplr 762 . . . . . 6  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  x  e.  D
)
805, 19, 7, 76, 78, 79tglnpt 24594 . . . . 5  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  x  e.  P
)
81 eqid 2451 . . . . 5  |-  ( (pInvG `  G ) `  x
)  =  ( (pInvG `  G ) `  x
)
821adantr 467 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  D )  ->  A  e.  P )
8382ad2antrr 732 . . . . 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 24706 . . . 4  |-  ( ( ( ( ph  /\  -.  A  e.  D
)  /\  x  e.  D )  /\  ( A L x ) (⟂G `  G ) D )  ->  ( ( (pInvG `  G ) `  x
) `  A )  e.  P )
85 oveq2 6298 . . . . . . . . . 10  |-  ( x  =  ( A (midG `  G ) b )  ->  ( A L x )  =  ( A L ( A (midG `  G )
b ) ) )
8685breq1d 4412 . . . . . . . . 9  |-  ( x  =  ( A (midG `  G ) b )  ->  ( ( A L x ) (⟂G `  G ) D  <->  ( A L ( A (midG `  G ) b ) ) (⟂G `  G
) D ) )
87 simprl 764 . . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  e.  D )  ->  -.  A  e.  D )
895, 6, 7, 19, 75, 77, 82, 88foot 24764 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  A  e.  D )  ->  E! x  e.  D  ( A L x ) (⟂G `  G ) D )
90 reurmo 3010 . . . . . . . . . . 11  |-  ( E! x  e.  D  ( A L x ) (⟂G `  G ) D  ->  E* x  e.  D  ( A L x ) (⟂G `  G
) D )
9189, 90syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  e.  D )  ->  E* x  e.  D  ( A L x ) (⟂G `  G ) D )
9291ad4antr 738 . . . . . . . . 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 732 . . . . . . . . 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 769 . . . . . . . . 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 732 . . . . . . . . . . 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 732 . . . . . . . . . . 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 762 . . . . . . . . . . 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 740 . . . . . . . . . . . . 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 24819 . . . . . . . . . . . 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 24821 . . . . . . . . . . . 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 738 . . . . . . . . . . . . 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 2721 . . . . . . . . . . . . 13  |-  ( ( ( A (midG `  G ) b )  e.  D  /\  -.  A  e.  D )  ->  ( A (midG `  G ) b )  =/=  A )
10387, 101, 102syl2anc 667 . . . . . . . . . . . 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 24534 . . . . . . . . . . 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 24664 . . . . . . . . . . 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 24677 . . . . . . . . . 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 732 . . . . . . . . . . 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 24675 . . . . . . . . . . 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 2629 . . . . . . . . . . . 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 766 . . . . . . . . . . . . . 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 390 . . . . . . . . . . . . 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 379 . . . . . . . . . . . 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 24758 . . . . . . . . . 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 4425 . . . . . . . . 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 3362 . . . . . . . 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 2457 . . . . . . 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 732 . . . . . . . 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 24820 . . . . . . 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 236 . . . . . 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 463 . . . . . . . . 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 732 . . . . . . . . . 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 740 . . . . . . . . . 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 732 . . . . . . . . . 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 762 . . . . . . . . . 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 732 . . . . . . . . . 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 24820 . . . . . . . . 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 214 . . . . . . . 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 732 . . . . . . . 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 2529 . . . . . . 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 467 . . . . . . . . . . . 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 777 . . . . . . . . . . . . 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 24755 . . . . . . . . . . . 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 736 . . . . . . . . . . . 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 24758 . . . . . . . . . . 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 467 . . . . . . . . . . . 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 467 . . . . . . . . . . . 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 24673 . . . . . . . . . . . 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 769 . . . . . . . . . . . 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 463 . . . . . . . . . . . . 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 2679 . . . . . . . . . . . 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 24703 . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . 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 6305 . . . . . . . . . . . . . . 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 2532 . . . . . . . . . . . . . 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 24664 . . . . . . . . . . . . 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 24668 . . . . . . . . . . . 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 24677 . . . . . . . . . . 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 4427 . . . . . . . . . 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 436 . . . . . . . . 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 2642 . . . . . . . 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 380 . . . . . . 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 535 . . . . . 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 843 . . . . 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 2802 . . . 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 3229 . . . 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 667 . . 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 24763 . . 3  |-  ( (
ph  /\  -.  A  e.  D )  ->  E. x  e.  D  ( A L x ) (⟂G `  G ) D )
159157, 158r19.29a 2932 . 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 800 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 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E!wreu 2739   E*wrmo 2740   class class class wbr 4402   ran crn 4835   ` cfv 5582  (class class class)co 6290   2c2 10659   Basecbs 15121   distcds 15199  TarskiGcstrkg 24478  DimTarskiGcstrkgld 24482  Itvcitv 24484  LineGclng 24485  pInvGcmir 24697  ⟂Gcperpg 24740  midGcmid 24814
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-trkgld 24500  df-trkg 24501  df-cgrg 24556  df-leg 24628  df-mir 24698  df-rag 24739  df-perpg 24741  df-mid 24816
This theorem is referenced by:  lmif  24827  islmib  24829
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