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Theorem tgcgrextend 23056
Description: Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p  |-  P  =  ( Base `  G
)
tkgeom.d  |-  .-  =  ( dist `  G )
tkgeom.i  |-  I  =  (Itv `  G )
tkgeom.g  |-  ( ph  ->  G  e. TarskiG )
tgcgrextend.a  |-  ( ph  ->  A  e.  P )
tgcgrextend.b  |-  ( ph  ->  B  e.  P )
tgcgrextend.c  |-  ( ph  ->  C  e.  P )
tgcgrextend.d  |-  ( ph  ->  D  e.  P )
tgcgrextend.e  |-  ( ph  ->  E  e.  P )
tgcgrextend.f  |-  ( ph  ->  F  e.  P )
tgcgrextend.1  |-  ( ph  ->  B  e.  ( A I C ) )
tgcgrextend.2  |-  ( ph  ->  E  e.  ( D I F ) )
tgcgrextend.3  |-  ( ph  ->  ( A  .-  B
)  =  ( D 
.-  E ) )
tgcgrextend.4  |-  ( ph  ->  ( B  .-  C
)  =  ( E 
.-  F ) )
Assertion
Ref Expression
tgcgrextend  |-  ( ph  ->  ( A  .-  C
)  =  ( D 
.-  F ) )

Proof of Theorem tgcgrextend
StepHypRef Expression
1 tgcgrextend.4 . . . 4  |-  ( ph  ->  ( B  .-  C
)  =  ( E 
.-  F ) )
21adantr 465 . . 3  |-  ( (
ph  /\  A  =  B )  ->  ( B  .-  C )  =  ( E  .-  F
) )
3 simpr 461 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  A  =  B )
43oveq1d 6205 . . 3  |-  ( (
ph  /\  A  =  B )  ->  ( A  .-  C )  =  ( B  .-  C
) )
5 tkgeom.p . . . . 5  |-  P  =  ( Base `  G
)
6 tkgeom.d . . . . 5  |-  .-  =  ( dist `  G )
7 tkgeom.i . . . . 5  |-  I  =  (Itv `  G )
8 tkgeom.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
98adantr 465 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  G  e. TarskiG )
10 tgcgrextend.d . . . . . 6  |-  ( ph  ->  D  e.  P )
1110adantr 465 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  D  e.  P )
12 tgcgrextend.e . . . . . 6  |-  ( ph  ->  E  e.  P )
1312adantr 465 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  E  e.  P )
14 tgcgrextend.a . . . . . 6  |-  ( ph  ->  A  e.  P )
1514adantr 465 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  A  e.  P )
163oveq2d 6206 . . . . . 6  |-  ( (
ph  /\  A  =  B )  ->  ( A  .-  A )  =  ( A  .-  B
) )
17 tgcgrextend.3 . . . . . . 7  |-  ( ph  ->  ( A  .-  B
)  =  ( D 
.-  E ) )
1817adantr 465 . . . . . 6  |-  ( (
ph  /\  A  =  B )  ->  ( A  .-  B )  =  ( D  .-  E
) )
1916, 18eqtr2d 2493 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  ( D  .-  E )  =  ( A  .-  A
) )
205, 6, 7, 9, 11, 13, 15, 19axtgcgrid 23040 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  D  =  E )
2120oveq1d 6205 . . 3  |-  ( (
ph  /\  A  =  B )  ->  ( D  .-  F )  =  ( E  .-  F
) )
222, 4, 213eqtr4d 2502 . 2  |-  ( (
ph  /\  A  =  B )  ->  ( A  .-  C )  =  ( D  .-  F
) )
238adantr 465 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  G  e. TarskiG )
24 tgcgrextend.c . . . 4  |-  ( ph  ->  C  e.  P )
2524adantr 465 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  C  e.  P )
2614adantr 465 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  A  e.  P )
27 tgcgrextend.f . . . 4  |-  ( ph  ->  F  e.  P )
2827adantr 465 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  F  e.  P )
2910adantr 465 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  D  e.  P )
30 tgcgrextend.b . . . . 5  |-  ( ph  ->  B  e.  P )
3130adantr 465 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  B  e.  P )
3212adantr 465 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  E  e.  P )
33 simpr 461 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  A  =/=  B )
34 tgcgrextend.1 . . . . 5  |-  ( ph  ->  B  e.  ( A I C ) )
3534adantr 465 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  B  e.  ( A I C ) )
36 tgcgrextend.2 . . . . 5  |-  ( ph  ->  E  e.  ( D I F ) )
3736adantr 465 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  E  e.  ( D I F ) )
3817adantr 465 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( A  .-  B )  =  ( D  .-  E ) )
391adantr 465 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( B  .-  C )  =  ( E  .-  F ) )
405, 6, 7, 23, 26, 29tgcgrtriv 23055 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( A  .-  A )  =  ( D  .-  D ) )
415, 6, 7, 23, 26, 31, 29, 32, 38tgcgrcomlr 23051 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( B  .-  A )  =  ( E  .-  D ) )
425, 6, 7, 23, 26, 31, 25, 29, 32, 28, 26, 29, 33, 35, 37, 38, 39, 40, 41axtg5seg 23042 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  ( C  .-  A )  =  ( F  .-  D ) )
435, 6, 7, 23, 25, 26, 28, 29, 42tgcgrcomlr 23051 . 2  |-  ( (
ph  /\  A  =/=  B )  ->  ( A  .-  C )  =  ( D  .-  F ) )
44 exmidne 2654 . . 3  |-  ( A  =  B  \/  A  =/=  B )
4544a1i 11 . 2  |-  ( ph  ->  ( A  =  B  \/  A  =/=  B
) )
4622, 43, 45mpjaodan 784 1  |-  ( ph  ->  ( A  .-  C
)  =  ( D 
.-  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   ` cfv 5516  (class class class)co 6190   Basecbs 14276   distcds 14349  TarskiGcstrkg 23005  Itvcitv 23012
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-iota 5479  df-fv 5524  df-ov 6193  df-trkgc 23024  df-trkgcb 23026  df-trkg 23030
This theorem is referenced by:  tgsegconeq  23057  tgcgrxfr  23089  lnext  23119  tgbtwnconn1lem1  23124  tgbtwnconn1lem2  23125  tgbtwnconn1lem3  23126  miriso  23199  midexlem  23212  mideulem  23244
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