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Theorem colperpexlem2 24083
Description: Lemma for colperpex 24085. 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 461 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  Q )  ->  A  =  Q )
32fveq2d 5860 . . . . . . . . 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 5858 . . . . . . 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 24021 . . . . . . . 8  |-  ( ph  ->  ( M `  ( M `  C )
)  =  C )
1716adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  =  Q )  ->  ( M `  ( M `  C ) )  =  C )
18 colperpexlem.2 . . . . . . . 8  |-  ( ph  ->  ( K `  ( M `  C )
)  =  ( N `
 C ) )
1918adantr 465 . . . . . . 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 24025 . . . . . . 7  |-  ( ph  ->  ( ( N `  C )  =  C  <-> 
B  =  C ) )
2423adantr 465 . . . . . 6  |-  ( (
ph  /\  A  =  Q )  ->  (
( N `  C
)  =  C  <->  B  =  C ) )
2520, 24mpbid 210 . . . . 5  |-  ( (
ph  /\  A  =  Q )  ->  B  =  C )
2625ex 434 . . . 4  |-  ( ph  ->  ( A  =  Q  ->  B  =  C ) )
2726necon3ad 2653 . . 3  |-  ( ph  ->  ( B  =/=  C  ->  -.  A  =  Q ) )
281, 27mpd 15 . 2  |-  ( ph  ->  -.  A  =  Q )
2928neqned 2646 1  |-  ( ph  ->  A  =/=  Q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   ` cfv 5578   <"cs3 12789   Basecbs 14614   distcds 14688  TarskiGcstrkg 23803  Itvcitv 23810  LineGclng 23811  pInvGcmir 24011  ∟Gcrag 24048
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-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-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-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-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  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-trkgc 23822  df-trkgb 23823  df-trkgcb 23824  df-trkg 23828  df-mir 24012
This theorem is referenced by:  colperpexlem3  24084
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