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Theorem tgcgreqb 22935
Description: Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
Hypotheses
Ref Expression
tkgeom.p  |-  P  =  ( Base `  G
)
tkgeom.d  |-  .-  =  ( dist `  G )
tkgeom.i  |-  I  =  (Itv `  G )
tkgeom.g  |-  ( ph  ->  G  e. TarskiG )
tgcgrcomlr.a  |-  ( ph  ->  A  e.  P )
tgcgrcomlr.b  |-  ( ph  ->  B  e.  P )
tgcgrcomlr.c  |-  ( ph  ->  C  e.  P )
tgcgrcomlr.d  |-  ( ph  ->  D  e.  P )
tgcgrcomlr.6  |-  ( ph  ->  ( A  .-  B
)  =  ( C 
.-  D ) )
Assertion
Ref Expression
tgcgreqb  |-  ( ph  ->  ( A  =  B  <-> 
C  =  D ) )

Proof of Theorem tgcgreqb
StepHypRef Expression
1 tkgeom.p . . 3  |-  P  =  ( Base `  G
)
2 tkgeom.d . . 3  |-  .-  =  ( dist `  G )
3 tkgeom.i . . 3  |-  I  =  (Itv `  G )
4 tkgeom.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
54adantr 465 . . 3  |-  ( (
ph  /\  A  =  B )  ->  G  e. TarskiG )
6 tgcgrcomlr.c . . . 4  |-  ( ph  ->  C  e.  P )
76adantr 465 . . 3  |-  ( (
ph  /\  A  =  B )  ->  C  e.  P )
8 tgcgrcomlr.d . . . 4  |-  ( ph  ->  D  e.  P )
98adantr 465 . . 3  |-  ( (
ph  /\  A  =  B )  ->  D  e.  P )
10 tgcgrcomlr.b . . . 4  |-  ( ph  ->  B  e.  P )
1110adantr 465 . . 3  |-  ( (
ph  /\  A  =  B )  ->  B  e.  P )
12 tgcgrcomlr.6 . . . . 5  |-  ( ph  ->  ( A  .-  B
)  =  ( C 
.-  D ) )
1312adantr 465 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  ( A  .-  B )  =  ( C  .-  D
) )
14 simpr 461 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  A  =  B )
1514oveq1d 6106 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  ( A  .-  B )  =  ( B  .-  B
) )
1613, 15eqtr3d 2477 . . 3  |-  ( (
ph  /\  A  =  B )  ->  ( C  .-  D )  =  ( B  .-  B
) )
171, 2, 3, 5, 7, 9, 11, 16axtgcgrid 22924 . 2  |-  ( (
ph  /\  A  =  B )  ->  C  =  D )
184adantr 465 . . 3  |-  ( (
ph  /\  C  =  D )  ->  G  e. TarskiG )
19 tgcgrcomlr.a . . . 4  |-  ( ph  ->  A  e.  P )
2019adantr 465 . . 3  |-  ( (
ph  /\  C  =  D )  ->  A  e.  P )
2110adantr 465 . . 3  |-  ( (
ph  /\  C  =  D )  ->  B  e.  P )
228adantr 465 . . 3  |-  ( (
ph  /\  C  =  D )  ->  D  e.  P )
2312adantr 465 . . . 4  |-  ( (
ph  /\  C  =  D )  ->  ( A  .-  B )  =  ( C  .-  D
) )
24 simpr 461 . . . . 5  |-  ( (
ph  /\  C  =  D )  ->  C  =  D )
2524oveq1d 6106 . . . 4  |-  ( (
ph  /\  C  =  D )  ->  ( C  .-  D )  =  ( D  .-  D
) )
2623, 25eqtrd 2475 . . 3  |-  ( (
ph  /\  C  =  D )  ->  ( A  .-  B )  =  ( D  .-  D
) )
271, 2, 3, 18, 20, 21, 22, 26axtgcgrid 22924 . 2  |-  ( (
ph  /\  C  =  D )  ->  A  =  B )
2817, 27impbida 828 1  |-  ( ph  ->  ( A  =  B  <-> 
C  =  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   Basecbs 14174   distcds 14247  TarskiGcstrkg 22889  Itvcitv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-ov 6094  df-trkgc 22909  df-trkg 22916
This theorem is referenced by:  tgcgreq  22936  tgcgrneq  22937
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