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Theorem tgcgrextend 24080
Description: Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) (Shortened by David A. Wheeler and Thierry Arnoux, 22-Apr-2020.)
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 463 . . 3  |-  ( (
ph  /\  A  =  B )  ->  ( B  .-  C )  =  ( E  .-  F
) )
3 simpr 459 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  A  =  B )
43oveq1d 6285 . . 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 463 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  G  e. TarskiG )
10 tgcgrextend.a . . . . . 6  |-  ( ph  ->  A  e.  P )
1110adantr 463 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  A  e.  P )
12 tgcgrextend.b . . . . . 6  |-  ( ph  ->  B  e.  P )
1312adantr 463 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  B  e.  P )
14 tgcgrextend.d . . . . . 6  |-  ( ph  ->  D  e.  P )
1514adantr 463 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  D  e.  P )
16 tgcgrextend.e . . . . . 6  |-  ( ph  ->  E  e.  P )
1716adantr 463 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  E  e.  P )
18 tgcgrextend.3 . . . . . 6  |-  ( ph  ->  ( A  .-  B
)  =  ( D 
.-  E ) )
1918adantr 463 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  ( A  .-  B )  =  ( D  .-  E
) )
205, 6, 7, 9, 11, 13, 15, 17, 19, 3tgcgreq 24077 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  D  =  E )
2120oveq1d 6285 . . 3  |-  ( (
ph  /\  A  =  B )  ->  ( D  .-  F )  =  ( E  .-  F
) )
222, 4, 213eqtr4d 2505 . 2  |-  ( (
ph  /\  A  =  B )  ->  ( A  .-  C )  =  ( D  .-  F
) )
238adantr 463 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  G  e. TarskiG )
24 tgcgrextend.c . . . 4  |-  ( ph  ->  C  e.  P )
2524adantr 463 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  C  e.  P )
2610adantr 463 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  A  e.  P )
27 tgcgrextend.f . . . 4  |-  ( ph  ->  F  e.  P )
2827adantr 463 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  F  e.  P )
2914adantr 463 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  D  e.  P )
3012adantr 463 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  B  e.  P )
3116adantr 463 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  E  e.  P )
32 simpr 459 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  A  =/=  B )
33 tgcgrextend.1 . . . . 5  |-  ( ph  ->  B  e.  ( A I C ) )
3433adantr 463 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  B  e.  ( A I C ) )
35 tgcgrextend.2 . . . . 5  |-  ( ph  ->  E  e.  ( D I F ) )
3635adantr 463 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  E  e.  ( D I F ) )
3718adantr 463 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( A  .-  B )  =  ( D  .-  E ) )
381adantr 463 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( B  .-  C )  =  ( E  .-  F ) )
395, 6, 7, 23, 26, 29tgcgrtriv 24079 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( A  .-  A )  =  ( D  .-  D ) )
405, 6, 7, 23, 26, 30, 29, 31, 37tgcgrcomlr 24075 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( B  .-  A )  =  ( E  .-  D ) )
415, 6, 7, 23, 26, 30, 25, 29, 31, 28, 26, 29, 32, 34, 36, 37, 38, 39, 40axtg5seg 24063 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  ( C  .-  A )  =  ( F  .-  D ) )
425, 6, 7, 23, 25, 26, 28, 29, 41tgcgrcomlr 24075 . 2  |-  ( (
ph  /\  A  =/=  B )  ->  ( A  .-  C )  =  ( D  .-  F ) )
4322, 42pm2.61dane 2772 1  |-  ( ph  ->  ( A  .-  C
)  =  ( D 
.-  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   ` cfv 5570  (class class class)co 6270   Basecbs 14719   distcds 14796  TarskiGcstrkg 24026  Itvcitv 24033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-trkgc 24045  df-trkgcb 24047  df-trkg 24051
This theorem is referenced by:  tgsegconeq  24081  tgcgrxfr  24113  lnext  24158  tgbtwnconn1lem1  24163  tgbtwnconn1lem2  24164  tgbtwnconn1lem3  24165  miriso  24254  midexlem  24273  opphllem  24313
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