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Theorem midex 24771
Description: Existence of the midpoint, part Theorem 8.22 of [Schwabhauser] p. 64. Note that this proof requires a construction in 2 dimensions or more, i.e. it does not prove the existence of a midpoint in dimension 1, for a geometry restricted to a line. (Contributed by Thierry Arnoux, 25-Nov-2019.)
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 )
mideu.3  |-  ( ph  ->  GDimTarskiG 2 )
Assertion
Ref Expression
midex  |-  ( ph  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A ) )
Distinct variable groups:    x,  .-    x, A   
x, B    x, G    x, I    x, L    x, P    x, S    ph, x

Proof of Theorem midex
Dummy variables  p  q  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mideu.1 . . . 4  |-  ( ph  ->  A  e.  P )
21adantr 467 . . 3  |-  ( (
ph  /\  A  =  B )  ->  A  e.  P )
3 colperpex.p . . . . 5  |-  P  =  ( Base `  G
)
4 colperpex.d . . . . 5  |-  .-  =  ( dist `  G )
5 colperpex.i . . . . 5  |-  I  =  (Itv `  G )
6 colperpex.l . . . . 5  |-  L  =  (LineG `  G )
7 mideu.s . . . . 5  |-  S  =  (pInvG `  G )
8 colperpex.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
98adantr 467 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  G  e. TarskiG )
10 eqid 2423 . . . . 5  |-  ( S `
 A )  =  ( S `  A
)
113, 4, 5, 6, 7, 9, 2, 10mircinv 24705 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  (
( S `  A
) `  A )  =  A )
12 simpr 463 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  A  =  B )
1311, 12eqtr2d 2465 . . 3  |-  ( (
ph  /\  A  =  B )  ->  B  =  ( ( S `
 A ) `  A ) )
14 fveq2 5879 . . . . . 6  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
1514fveq1d 5881 . . . . 5  |-  ( x  =  A  ->  (
( S `  x
) `  A )  =  ( ( S `
 A ) `  A ) )
1615eqeq2d 2437 . . . 4  |-  ( x  =  A  ->  ( B  =  ( ( S `  x ) `  A )  <->  B  =  ( ( S `  A ) `  A
) ) )
1716rspcev 3183 . . 3  |-  ( ( A  e.  P  /\  B  =  ( ( S `  A ) `  A ) )  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A ) )
182, 13, 17syl2anc 666 . 2  |-  ( (
ph  /\  A  =  B )  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A
) )
198adantr 467 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  B )  ->  G  e. TarskiG )
2019ad2antrr 731 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  ->  G  e. TarskiG )
2120ad4antr 737 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  G  e. TarskiG )
221adantr 467 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  B )  ->  A  e.  P )
2322ad2antrr 731 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  ->  A  e.  P )
2423ad4antr 737 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  A  e.  P
)
25 mideu.2 . . . . . . . . 9  |-  ( ph  ->  B  e.  P )
2625adantr 467 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  B )  ->  B  e.  P )
2726ad2antrr 731 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  ->  B  e.  P )
2827ad4antr 737 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  B  e.  P
)
29 simpr 463 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  B )  ->  A  =/=  B )
3029ad2antrr 731 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  ->  A  =/=  B )
3130ad4antr 737 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  A  =/=  B
)
32 simplr 761 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  -> 
q  e.  P )
3332ad4antr 737 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  q  e.  P
)
34 simp-4r 776 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  p  e.  P
)
35 simpllr 768 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  t  e.  P
)
36 simp-5r 778 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( B L q ) (⟂G `  G
) ( A L B ) )
376, 21, 36perpln1 24747 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( B L q )  e.  ran  L )
383, 5, 6, 21, 24, 28, 31tgelrnln 24667 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( A L B )  e.  ran  L )
393, 4, 5, 6, 21, 37, 38, 36perpcom 24750 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( A L B ) (⟂G `  G
) ( B L q ) )
403, 5, 6, 21, 28, 33, 37tglnne 24665 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  B  =/=  q
)
413, 5, 6, 21, 28, 33, 40tglinecom 24672 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( B L q )  =  ( q L B ) )
4239, 41breqtrd 4446 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( A L B ) (⟂G `  G
) ( q L B ) )
43 simplr 761 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )
4443simpld 461 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( A L p ) (⟂G `  G
) ( A L B ) )
456, 21, 44perpln1 24747 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( A L p )  e.  ran  L )
463, 4, 5, 6, 21, 45, 38, 44perpcom 24750 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( A L B ) (⟂G `  G
) ( A L p ) )
4731neneqd 2626 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  -.  A  =  B )
4843simprd 465 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) )
4948simpld 461 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( t  e.  ( A L B )  \/  A  =  B ) )
5049orcomd 390 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( A  =  B  \/  t  e.  ( A L B ) ) )
5150ord 379 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( -.  A  =  B  ->  t  e.  ( A L B ) ) )
5247, 51mpd 15 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  t  e.  ( A L B ) )
5348simprd 465 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  t  e.  ( q I p ) )
54 simpr 463 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )
553, 4, 5, 6, 21, 7, 24, 28, 31, 33, 34, 35, 42, 46, 52, 53, 54mideulem 24770 . . . . 5  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( A  .-  p ) (≤G `  G ) ( B 
.-  q ) )  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A ) )
5620ad4antr 737 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  G  e. TarskiG )
5756adantr 467 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P
)  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  /\  ( x  e.  P  /\  A  =  ( ( S `  x ) `  B
) ) )  ->  G  e. TarskiG )
58 simprl 763 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P
)  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  /\  ( x  e.  P  /\  A  =  ( ( S `  x ) `  B
) ) )  ->  x  e.  P )
59 eqid 2423 . . . . . . . 8  |-  ( S `
 x )  =  ( S `  x
)
6027ad4antr 737 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  B  e.  P
)
6160adantr 467 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P
)  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  /\  ( x  e.  P  /\  A  =  ( ( S `  x ) `  B
) ) )  ->  B  e.  P )
62 simprr 765 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P
)  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  /\  ( x  e.  P  /\  A  =  ( ( S `  x ) `  B
) ) )  ->  A  =  ( ( S `  x ) `  B ) )
6362eqcomd 2431 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P
)  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  /\  ( x  e.  P  /\  A  =  ( ( S `  x ) `  B
) ) )  -> 
( ( S `  x ) `  B
)  =  A )
643, 4, 5, 6, 7, 57, 58, 59, 61, 63mircom 24700 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P
)  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  /\  ( x  e.  P  /\  A  =  ( ( S `  x ) `  B
) ) )  -> 
( ( S `  x ) `  A
)  =  B )
6564eqcomd 2431 . . . . . 6  |-  ( ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P
)  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  /\  ( x  e.  P  /\  A  =  ( ( S `  x ) `  B
) ) )  ->  B  =  ( ( S `  x ) `  A ) )
6623ad4antr 737 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  A  e.  P
)
6730ad4antr 737 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  A  =/=  B
)
6867necomd 2696 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  B  =/=  A
)
69 simp-4r 776 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  p  e.  P
)
7032ad4antr 737 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  q  e.  P
)
71 simpllr 768 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  t  e.  P
)
72 simplr 761 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )
7372simpld 461 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( A L p ) (⟂G `  G
) ( A L B ) )
746, 56, 73perpln1 24747 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( A L p )  e.  ran  L )
753, 5, 6, 56, 66, 69, 74tglnne 24665 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  A  =/=  p
)
763, 5, 6, 56, 66, 69, 75tglinecom 24672 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( A L p )  =  ( p L A ) )
7776eqcomd 2431 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( p L A )  =  ( A L p ) )
7877, 74eqeltrd 2511 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( p L A )  e.  ran  L )
793, 5, 6, 56, 60, 66, 68tgelrnln 24667 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( B L A )  e.  ran  L )
803, 5, 6, 56, 66, 60, 67tglinecom 24672 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( A L B )  =  ( B L A ) )
8173, 76, 803brtr3d 4451 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( p L A ) (⟂G `  G
) ( B L A ) )
823, 4, 5, 6, 56, 78, 79, 81perpcom 24750 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( B L A ) (⟂G `  G
) ( p L A ) )
83 simp-5r 778 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( B L q ) (⟂G `  G
) ( A L B ) )
846, 56, 83perpln1 24747 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( B L q )  e.  ran  L )
8583, 80breqtrd 4446 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( B L q ) (⟂G `  G
) ( B L A ) )
863, 4, 5, 6, 56, 84, 79, 85perpcom 24750 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( B L A ) (⟂G `  G
) ( B L q ) )
8767neneqd 2626 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  -.  A  =  B )
8872simprd 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) )
8988simpld 461 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( t  e.  ( A L B )  \/  A  =  B ) )
9089orcomd 390 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( A  =  B  \/  t  e.  ( A L B ) ) )
9190ord 379 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( -.  A  =  B  ->  t  e.  ( A L B ) ) )
9287, 91mpd 15 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  t  e.  ( A L B ) )
9392, 80eleqtrd 2513 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  t  e.  ( B L A ) )
9488simprd 465 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  t  e.  ( q I p ) )
953, 4, 5, 56, 70, 71, 69, 94tgbtwncom 24524 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  t  e.  ( p I q ) )
96 simpr 463 . . . . . . 7  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )
973, 4, 5, 6, 56, 7, 60, 66, 68, 69, 70, 71, 82, 86, 93, 95, 96mideulem 24770 . . . . . 6  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  E. x  e.  P  A  =  ( ( S `  x ) `  B ) )
9865, 97reximddv 2902 . . . . 5  |-  ( ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  /\  p  e.  P )  /\  t  e.  P
)  /\  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  /\  ( B  .-  q ) (≤G `  G ) ( A 
.-  p ) )  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A ) )
99 eqid 2423 . . . . . 6  |-  (≤G `  G )  =  (≤G `  G )
10020ad3antrrr 735 . . . . . 6  |-  ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  ->  G  e. TarskiG )
10123ad3antrrr 735 . . . . . 6  |-  ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  ->  A  e.  P
)
102 simpllr 768 . . . . . 6  |-  ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  ->  p  e.  P
)
10327ad3antrrr 735 . . . . . 6  |-  ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  ->  B  e.  P
)
10432ad3antrrr 735 . . . . . 6  |-  ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  ->  q  e.  P
)
1053, 4, 5, 99, 100, 101, 102, 103, 104legtrid 24628 . . . . 5  |-  ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  ->  ( ( A 
.-  p ) (≤G `  G ) ( B 
.-  q )  \/  ( B  .-  q
) (≤G `  G
) ( A  .-  p ) ) )
10655, 98, 105mpjaodan 794 . . . 4  |-  ( ( ( ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G ) ( A L B ) )  /\  p  e.  P
)  /\  t  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A ) )
107 mideu.3 . . . . . . . 8  |-  ( ph  ->  GDimTarskiG 2 )
108107adantr 467 . . . . . . 7  |-  ( (
ph  /\  A  =/=  B )  ->  GDimTarskiG 2 )
109108ad2antrr 731 . . . . . 6  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  ->  GDimTarskiG 2 )
1103, 4, 5, 6, 20, 23, 27, 32, 30, 109colperpex 24767 . . . . 5  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  ->  E. p  e.  P  ( ( A L p ) (⟂G `  G
) ( A L B )  /\  E. t  e.  P  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )
111 r19.42v 2984 . . . . . 6  |-  ( E. t  e.  P  ( ( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) )  <->  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  E. t  e.  P  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )
112111rexbii 2928 . . . . 5  |-  ( E. p  e.  P  E. t  e.  P  (
( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) )  <->  E. p  e.  P  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  E. t  e.  P  ( ( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )
113110, 112sylibr 216 . . . 4  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  ->  E. p  e.  P  E. t  e.  P  ( ( A L p ) (⟂G `  G
) ( A L B )  /\  (
( t  e.  ( A L B )  \/  A  =  B )  /\  t  e.  ( q I p ) ) ) )
114106, 113r19.29vva 2973 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  q  e.  P )  /\  ( B L q ) (⟂G `  G
) ( A L B ) )  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A ) )
11529necomd 2696 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  B  =/=  A )
1163, 4, 5, 6, 19, 26, 22, 22, 115, 108colperpex 24767 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  E. q  e.  P  ( ( B L q ) (⟂G `  G ) ( B L A )  /\  E. s  e.  P  ( ( s  e.  ( B L A )  \/  B  =  A )  /\  s  e.  ( A I q ) ) ) )
117 simprl 763 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  B )  /\  (
( B L q ) (⟂G `  G
) ( B L A )  /\  E. s  e.  P  (
( s  e.  ( B L A )  \/  B  =  A )  /\  s  e.  ( A I q ) ) ) )  ->  ( B L q ) (⟂G `  G
) ( B L A ) )
1183, 5, 6, 19, 22, 26, 29tglinecom 24672 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  B )  ->  ( A L B )  =  ( B L A ) )
119118adantr 467 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  B )  /\  (
( B L q ) (⟂G `  G
) ( B L A )  /\  E. s  e.  P  (
( s  e.  ( B L A )  \/  B  =  A )  /\  s  e.  ( A I q ) ) ) )  ->  ( A L B )  =  ( B L A ) )
120117, 119breqtrrd 4448 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  B )  /\  (
( B L q ) (⟂G `  G
) ( B L A )  /\  E. s  e.  P  (
( s  e.  ( B L A )  \/  B  =  A )  /\  s  e.  ( A I q ) ) ) )  ->  ( B L q ) (⟂G `  G
) ( A L B ) )
121120ex 436 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  ( (
( B L q ) (⟂G `  G
) ( B L A )  /\  E. s  e.  P  (
( s  e.  ( B L A )  \/  B  =  A )  /\  s  e.  ( A I q ) ) )  -> 
( B L q ) (⟂G `  G
) ( A L B ) ) )
122121reximdv 2900 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( E. q  e.  P  (
( B L q ) (⟂G `  G
) ( B L A )  /\  E. s  e.  P  (
( s  e.  ( B L A )  \/  B  =  A )  /\  s  e.  ( A I q ) ) )  ->  E. q  e.  P  ( B L q ) (⟂G `  G )
( A L B ) ) )
123116, 122mpd 15 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  E. q  e.  P  ( B L q ) (⟂G `  G ) ( A L B ) )
124114, 123r19.29a 2971 . 2  |-  ( (
ph  /\  A  =/=  B )  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A
) )
12518, 124pm2.61dane 2743 1  |-  ( ph  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619   E.wrex 2777   class class class wbr 4421   ran crn 4852   ` cfv 5599  (class class class)co 6303   2c2 10661   Basecbs 15114   distcds 15192  TarskiGcstrkg 24470  DimTarskiGcstrkgld 24474  Itvcitv 24476  LineGclng 24477  ≤Gcleg 24619  pInvGcmir 24689  ⟂Gcperpg 24732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-3 10671  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-fzo 11918  df-hash 12517  df-word 12662  df-concat 12664  df-s1 12665  df-s2 12940  df-s3 12941  df-trkgc 24488  df-trkgb 24489  df-trkgcb 24490  df-trkgld 24492  df-trkg 24493  df-cgrg 24548  df-leg 24620  df-mir 24690  df-rag 24731  df-perpg 24733
This theorem is referenced by:  mideu  24772  opphllem5  24785  opphl  24788
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