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Theorem lmieq 24881
Description: Equality deduction for line mirroring. Theorem 10.7 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-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 )
lmif.m  |-  M  =  ( (lInvG `  G
) `  D )
lmif.l  |-  L  =  (LineG `  G )
lmif.d  |-  ( ph  ->  D  e.  ran  L
)
lmicl.1  |-  ( ph  ->  A  e.  P )
lmieq.c  |-  ( ph  ->  B  e.  P )
lmieq.d  |-  ( ph  ->  ( M `  A
)  =  ( M `
 B ) )
Assertion
Ref Expression
lmieq  |-  ( ph  ->  A  =  B )

Proof of Theorem lmieq
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 fveq2 5887 . . 3  |-  ( b  =  A  ->  ( M `  b )  =  ( M `  A ) )
21eqeq1d 2463 . 2  |-  ( b  =  A  ->  (
( M `  b
)  =  ( M `
 B )  <->  ( M `  A )  =  ( M `  B ) ) )
3 fveq2 5887 . . 3  |-  ( b  =  B  ->  ( M `  b )  =  ( M `  B ) )
43eqeq1d 2463 . 2  |-  ( b  =  B  ->  (
( M `  b
)  =  ( M `
 B )  <->  ( M `  B )  =  ( M `  B ) ) )
5 ismid.p . . 3  |-  P  =  ( Base `  G
)
6 ismid.d . . 3  |-  .-  =  ( dist `  G )
7 ismid.i . . 3  |-  I  =  (Itv `  G )
8 ismid.g . . 3  |-  ( ph  ->  G  e. TarskiG )
9 ismid.1 . . 3  |-  ( ph  ->  GDimTarskiG 2 )
10 lmif.m . . 3  |-  M  =  ( (lInvG `  G
) `  D )
11 lmif.l . . 3  |-  L  =  (LineG `  G )
12 lmif.d . . 3  |-  ( ph  ->  D  e.  ran  L
)
13 lmieq.c . . . 4  |-  ( ph  ->  B  e.  P )
145, 6, 7, 8, 9, 10, 11, 12, 13lmicl 24876 . . 3  |-  ( ph  ->  ( M `  B
)  e.  P )
155, 6, 7, 8, 9, 10, 11, 12, 14lmireu 24880 . 2  |-  ( ph  ->  E! b  e.  P  ( M `  b )  =  ( M `  B ) )
16 lmicl.1 . 2  |-  ( ph  ->  A  e.  P )
17 lmieq.d . 2  |-  ( ph  ->  ( M `  A
)  =  ( M `
 B ) )
18 eqidd 2462 . 2  |-  ( ph  ->  ( M `  B
)  =  ( M `
 B ) )
192, 4, 15, 16, 13, 17, 18reu2eqd 3246 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1454    e. wcel 1897   class class class wbr 4415   ran crn 4853   ` cfv 5600   2c2 10686   Basecbs 15169   distcds 15247  TarskiGcstrkg 24526  DimTarskiGcstrkgld 24530  Itvcitv 24532  LineGclng 24533  lInvGclmi 24863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-pm 7500  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-card 8398  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-3 10696  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-fzo 11946  df-hash 12547  df-word 12696  df-concat 12698  df-s1 12699  df-s2 12980  df-s3 12981  df-trkgc 24544  df-trkgb 24545  df-trkgcb 24546  df-trkgld 24548  df-trkg 24549  df-cgrg 24604  df-leg 24676  df-mir 24746  df-rag 24787  df-perpg 24789  df-mid 24864  df-lmi 24865
This theorem is referenced by:  trgcopyeulem  24895
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