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Theorem colperpexlem2 24225
Description: Lemma for colperpex 24227. 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 459 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  Q )  ->  A  =  Q )
32fveq2d 5778 . . . . . . . . 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 2448 . . . . . . . 8  |-  ( (
ph  /\  A  =  Q )  ->  M  =  K )
76fveq1d 5776 . . . . . . 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 24163 . . . . . . . 8  |-  ( ph  ->  ( M `  ( M `  C )
)  =  C )
1716adantr 463 . . . . . . 7  |-  ( (
ph  /\  A  =  Q )  ->  ( M `  ( M `  C ) )  =  C )
18 colperpexlem.2 . . . . . . . 8  |-  ( ph  ->  ( K `  ( M `  C )
)  =  ( N `
 C ) )
1918adantr 463 . . . . . . 7  |-  ( (
ph  /\  A  =  Q )  ->  ( K `  ( M `  C ) )  =  ( N `  C
) )
207, 17, 193eqtr3rd 2432 . . . . . 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 24167 . . . . . . 7  |-  ( ph  ->  ( ( N `  C )  =  C  <-> 
B  =  C ) )
2423adantr 463 . . . . . 6  |-  ( (
ph  /\  A  =  Q )  ->  (
( N `  C
)  =  C  <->  B  =  C ) )
2520, 24mpbid 210 . . . . 5  |-  ( (
ph  /\  A  =  Q )  ->  B  =  C )
2625ex 432 . . . 4  |-  ( ph  ->  ( A  =  Q  ->  B  =  C ) )
2726necon3ad 2592 . . 3  |-  ( ph  ->  ( B  =/=  C  ->  -.  A  =  Q ) )
281, 27mpd 15 . 2  |-  ( ph  ->  -.  A  =  Q )
2928neqned 2585 1  |-  ( ph  ->  A  =/=  Q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   ` cfv 5496   <"cs3 12718   Basecbs 14634   distcds 14711  TarskiGcstrkg 23942  Itvcitv 23949  LineGclng 23950  pInvGcmir 24153  ∟Gcrag 24190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-trkgc 23961  df-trkgb 23962  df-trkgcb 23963  df-trkg 23967  df-mir 24154
This theorem is referenced by:  colperpexlem3  24226
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