MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgcgrsub Structured version   Unicode version

Theorem tgcgrsub 23083
Description: Removing identical parts from the end of a line segment preserves congruence. Theorem 4.3 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
tgbtwncgr.p  |-  P  =  ( Base `  G
)
tgbtwncgr.m  |-  .-  =  ( dist `  G )
tgbtwncgr.i  |-  I  =  (Itv `  G )
tgbtwncgr.g  |-  ( ph  ->  G  e. TarskiG )
tgbtwncgr.a  |-  ( ph  ->  A  e.  P )
tgbtwncgr.b  |-  ( ph  ->  B  e.  P )
tgbtwncgr.c  |-  ( ph  ->  C  e.  P )
tgbtwncgr.d  |-  ( ph  ->  D  e.  P )
tgcgrsub.e  |-  ( ph  ->  E  e.  P )
tgcgrsub.f  |-  ( ph  ->  F  e.  P )
tgcgrsub.1  |-  ( ph  ->  B  e.  ( A I C ) )
tgcgrsub.2  |-  ( ph  ->  E  e.  ( D I F ) )
tgcgrsub.3  |-  ( ph  ->  ( A  .-  C
)  =  ( D 
.-  F ) )
tgcgrsub.4  |-  ( ph  ->  ( B  .-  C
)  =  ( E 
.-  F ) )
Assertion
Ref Expression
tgcgrsub  |-  ( ph  ->  ( A  .-  B
)  =  ( D 
.-  E ) )

Proof of Theorem tgcgrsub
StepHypRef Expression
1 tgbtwncgr.p . 2  |-  P  =  ( Base `  G
)
2 tgbtwncgr.m . 2  |-  .-  =  ( dist `  G )
3 tgbtwncgr.i . 2  |-  I  =  (Itv `  G )
4 tgbtwncgr.g . 2  |-  ( ph  ->  G  e. TarskiG )
5 tgbtwncgr.b . 2  |-  ( ph  ->  B  e.  P )
6 tgbtwncgr.a . 2  |-  ( ph  ->  A  e.  P )
7 tgcgrsub.e . 2  |-  ( ph  ->  E  e.  P )
8 tgbtwncgr.d . 2  |-  ( ph  ->  D  e.  P )
9 tgbtwncgr.c . . 3  |-  ( ph  ->  C  e.  P )
10 tgcgrsub.f . . 3  |-  ( ph  ->  F  e.  P )
11 tgcgrsub.1 . . 3  |-  ( ph  ->  B  e.  ( A I C ) )
12 tgcgrsub.2 . . 3  |-  ( ph  ->  E  e.  ( D I F ) )
13 tgcgrsub.3 . . 3  |-  ( ph  ->  ( A  .-  C
)  =  ( D 
.-  F ) )
14 tgcgrsub.4 . . 3  |-  ( ph  ->  ( B  .-  C
)  =  ( E 
.-  F ) )
151, 2, 3, 4, 6, 8tgcgrtriv 23057 . . 3  |-  ( ph  ->  ( A  .-  A
)  =  ( D 
.-  D ) )
161, 2, 3, 4, 6, 9, 8, 10, 13tgcgrcomlr 23053 . . 3  |-  ( ph  ->  ( C  .-  A
)  =  ( F 
.-  D ) )
171, 2, 3, 4, 6, 5, 9, 6, 8, 7, 10, 8, 11, 12, 13, 14, 15, 16tgifscgr 23082 . 2  |-  ( ph  ->  ( B  .-  A
)  =  ( E 
.-  D ) )
181, 2, 3, 4, 5, 6, 7, 8, 17tgcgrcomlr 23053 1  |-  ( ph  ->  ( A  .-  B
)  =  ( D 
.-  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   ` cfv 5518  (class class class)co 6192   Basecbs 14278   distcds 14351  TarskiGcstrkg 23007  Itvcitv 23014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212  df-cda 8440  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-hash 12207  df-trkgc 23026  df-trkgb 23027  df-trkgcb 23028  df-trkg 23032
This theorem is referenced by:  legtri3  23144  legbtwn  23148  colmid  23210  midexlem  23214
  Copyright terms: Public domain W3C validator