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Theorem colperpexlem2 24766
Description: Lemma for colperpex 24768. Second part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 10-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 )
colperpexlem.s  |-  S  =  (pInvG `  G )
colperpexlem.m  |-  M  =  ( S `  A
)
colperpexlem.n  |-  N  =  ( S `  B
)
colperpexlem.k  |-  K  =  ( S `  Q
)
colperpexlem.a  |-  ( ph  ->  A  e.  P )
colperpexlem.b  |-  ( ph  ->  B  e.  P )
colperpexlem.c  |-  ( ph  ->  C  e.  P )
colperpexlem.q  |-  ( ph  ->  Q  e.  P )
colperpexlem.1  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
colperpexlem.2  |-  ( ph  ->  ( K `  ( M `  C )
)  =  ( N `
 C ) )
colperpexlem2.e  |-  ( ph  ->  B  =/=  C )
Assertion
Ref Expression
colperpexlem2  |-  ( ph  ->  A  =/=  Q )

Proof of Theorem colperpexlem2
StepHypRef Expression
1 colperpexlem2.e . . 3  |-  ( ph  ->  B  =/=  C )
2 simpr 463 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  Q )  ->  A  =  Q )
32fveq2d 5867 . . . . . . . . 9  |-  ( (
ph  /\  A  =  Q )  ->  ( S `  A )  =  ( S `  Q ) )
4 colperpexlem.m . . . . . . . . 9  |-  M  =  ( S `  A
)
5 colperpexlem.k . . . . . . . . 9  |-  K  =  ( S `  Q
)
63, 4, 53eqtr4g 2509 . . . . . . . 8  |-  ( (
ph  /\  A  =  Q )  ->  M  =  K )
76fveq1d 5865 . . . . . . 7  |-  ( (
ph  /\  A  =  Q )  ->  ( M `  ( M `  C ) )  =  ( K `  ( M `  C )
) )
8 colperpex.p . . . . . . . . 9  |-  P  =  ( Base `  G
)
9 colperpex.d . . . . . . . . 9  |-  .-  =  ( dist `  G )
10 colperpex.i . . . . . . . . 9  |-  I  =  (Itv `  G )
11 colperpex.l . . . . . . . . 9  |-  L  =  (LineG `  G )
12 colperpexlem.s . . . . . . . . 9  |-  S  =  (pInvG `  G )
13 colperpex.g . . . . . . . . 9  |-  ( ph  ->  G  e. TarskiG )
14 colperpexlem.a . . . . . . . . 9  |-  ( ph  ->  A  e.  P )
15 colperpexlem.c . . . . . . . . 9  |-  ( ph  ->  C  e.  P )
168, 9, 10, 11, 12, 13, 14, 4, 15mirmir 24700 . . . . . . . 8  |-  ( ph  ->  ( M `  ( M `  C )
)  =  C )
1716adantr 467 . . . . . . 7  |-  ( (
ph  /\  A  =  Q )  ->  ( M `  ( M `  C ) )  =  C )
18 colperpexlem.2 . . . . . . . 8  |-  ( ph  ->  ( K `  ( M `  C )
)  =  ( N `
 C ) )
1918adantr 467 . . . . . . 7  |-  ( (
ph  /\  A  =  Q )  ->  ( K `  ( M `  C ) )  =  ( N `  C
) )
207, 17, 193eqtr3rd 2493 . . . . . 6  |-  ( (
ph  /\  A  =  Q )  ->  ( N `  C )  =  C )
21 colperpexlem.b . . . . . . . 8  |-  ( ph  ->  B  e.  P )
22 colperpexlem.n . . . . . . . 8  |-  N  =  ( S `  B
)
238, 9, 10, 11, 12, 13, 21, 22, 15mirinv 24704 . . . . . . 7  |-  ( ph  ->  ( ( N `  C )  =  C  <-> 
B  =  C ) )
2423adantr 467 . . . . . 6  |-  ( (
ph  /\  A  =  Q )  ->  (
( N `  C
)  =  C  <->  B  =  C ) )
2520, 24mpbid 214 . . . . 5  |-  ( (
ph  /\  A  =  Q )  ->  B  =  C )
2625ex 436 . . . 4  |-  ( ph  ->  ( A  =  Q  ->  B  =  C ) )
2726necon3ad 2636 . . 3  |-  ( ph  ->  ( B  =/=  C  ->  -.  A  =  Q ) )
281, 27mpd 15 . 2  |-  ( ph  ->  -.  A  =  Q )
2928neqned 2630 1  |-  ( ph  ->  A  =/=  Q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621   ` cfv 5581   <"cs3 12933   Basecbs 15114   distcds 15192  TarskiGcstrkg 24471  Itvcitv 24477  LineGclng 24478  pInvGcmir 24690  ∟Gcrag 24731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-trkgc 24489  df-trkgb 24490  df-trkgcb 24491  df-trkg 24494  df-mir 24691
This theorem is referenced by:  colperpexlem3  24767
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