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Theorem tgbtwnconn1lem1 24079
Description: Lemma for tgbtwnconn1 24082 (Contributed by Thierry Arnoux, 30-Apr-2019.)
Hypotheses
Ref Expression
tgbtwnconn1.p  |-  P  =  ( Base `  G
)
tgbtwnconn1.i  |-  I  =  (Itv `  G )
tgbtwnconn1.g  |-  ( ph  ->  G  e. TarskiG )
tgbtwnconn1.a  |-  ( ph  ->  A  e.  P )
tgbtwnconn1.b  |-  ( ph  ->  B  e.  P )
tgbtwnconn1.c  |-  ( ph  ->  C  e.  P )
tgbtwnconn1.d  |-  ( ph  ->  D  e.  P )
tgbtwnconn1.1  |-  ( ph  ->  A  =/=  B )
tgbtwnconn1.2  |-  ( ph  ->  B  e.  ( A I C ) )
tgbtwnconn1.3  |-  ( ph  ->  B  e.  ( A I D ) )
tgbtwnconn1.m  |-  .-  =  ( dist `  G )
tgbtwnconn1.e  |-  ( ph  ->  E  e.  P )
tgbtwnconn1.f  |-  ( ph  ->  F  e.  P )
tgbtwnconn1.h  |-  ( ph  ->  H  e.  P )
tgbtwnconn1.j  |-  ( ph  ->  J  e.  P )
tgbtwnconn1.4  |-  ( ph  ->  D  e.  ( A I E ) )
tgbtwnconn1.5  |-  ( ph  ->  C  e.  ( A I F ) )
tgbtwnconn1.6  |-  ( ph  ->  E  e.  ( A I H ) )
tgbtwnconn1.7  |-  ( ph  ->  F  e.  ( A I J ) )
tgbtwnconn1.8  |-  ( ph  ->  ( E  .-  D
)  =  ( C 
.-  D ) )
tgbtwnconn1.9  |-  ( ph  ->  ( C  .-  F
)  =  ( C 
.-  D ) )
tgbtwnconn1.10  |-  ( ph  ->  ( E  .-  H
)  =  ( B 
.-  C ) )
tgbtwnconn1.11  |-  ( ph  ->  ( F  .-  J
)  =  ( B 
.-  D ) )
Assertion
Ref Expression
tgbtwnconn1lem1  |-  ( ph  ->  H  =  J )

Proof of Theorem tgbtwnconn1lem1
StepHypRef Expression
1 tgbtwnconn1.p . 2  |-  P  =  ( Base `  G
)
2 tgbtwnconn1.m . 2  |-  .-  =  ( dist `  G )
3 tgbtwnconn1.i . 2  |-  I  =  (Itv `  G )
4 tgbtwnconn1.g . 2  |-  ( ph  ->  G  e. TarskiG )
5 tgbtwnconn1.b . 2  |-  ( ph  ->  B  e.  P )
6 tgbtwnconn1.j . 2  |-  ( ph  ->  J  e.  P )
7 tgbtwnconn1.a . 2  |-  ( ph  ->  A  e.  P )
8 tgbtwnconn1.h . 2  |-  ( ph  ->  H  e.  P )
9 tgbtwnconn1.1 . 2  |-  ( ph  ->  A  =/=  B )
10 tgbtwnconn1.e . . 3  |-  ( ph  ->  E  e.  P )
11 tgbtwnconn1.d . . . 4  |-  ( ph  ->  D  e.  P )
12 tgbtwnconn1.3 . . . 4  |-  ( ph  ->  B  e.  ( A I D ) )
13 tgbtwnconn1.4 . . . 4  |-  ( ph  ->  D  e.  ( A I E ) )
141, 2, 3, 4, 7, 5, 11, 10, 12, 13tgbtwnexch 24009 . . 3  |-  ( ph  ->  B  e.  ( A I E ) )
15 tgbtwnconn1.6 . . 3  |-  ( ph  ->  E  e.  ( A I H ) )
161, 2, 3, 4, 7, 5, 10, 8, 14, 15tgbtwnexch 24009 . 2  |-  ( ph  ->  B  e.  ( A I H ) )
17 tgbtwnconn1.f . . 3  |-  ( ph  ->  F  e.  P )
18 tgbtwnconn1.c . . . 4  |-  ( ph  ->  C  e.  P )
19 tgbtwnconn1.2 . . . 4  |-  ( ph  ->  B  e.  ( A I C ) )
20 tgbtwnconn1.5 . . . 4  |-  ( ph  ->  C  e.  ( A I F ) )
211, 2, 3, 4, 7, 5, 18, 17, 19, 20tgbtwnexch 24009 . . 3  |-  ( ph  ->  B  e.  ( A I F ) )
22 tgbtwnconn1.7 . . 3  |-  ( ph  ->  F  e.  ( A I J ) )
231, 2, 3, 4, 7, 5, 17, 6, 21, 22tgbtwnexch 24009 . 2  |-  ( ph  ->  B  e.  ( A I J ) )
241, 2, 3, 4, 7, 5, 10, 8, 14, 15tgbtwnexch3 24005 . . 3  |-  ( ph  ->  E  e.  ( B I H ) )
251, 2, 3, 4, 7, 18, 17, 6, 20, 22tgbtwnexch 24009 . . . . 5  |-  ( ph  ->  C  e.  ( A I J ) )
261, 2, 3, 4, 7, 5, 18, 6, 19, 25tgbtwnexch3 24005 . . . 4  |-  ( ph  ->  C  e.  ( B I J ) )
271, 2, 3, 4, 5, 18, 6, 26tgbtwncom 23999 . . 3  |-  ( ph  ->  C  e.  ( J I B ) )
281, 2, 3, 4, 7, 5, 11, 10, 12, 13tgbtwnexch3 24005 . . . 4  |-  ( ph  ->  D  e.  ( B I E ) )
291, 2, 3, 4, 7, 18, 17, 6, 20, 22tgbtwnexch3 24005 . . . . 5  |-  ( ph  ->  F  e.  ( C I J ) )
301, 2, 3, 4, 18, 17, 6, 29tgbtwncom 23999 . . . 4  |-  ( ph  ->  F  e.  ( J I C ) )
311, 2, 3, 4, 6, 17axtgcgrrflx 23976 . . . . 5  |-  ( ph  ->  ( J  .-  F
)  =  ( F 
.-  J ) )
32 tgbtwnconn1.11 . . . . 5  |-  ( ph  ->  ( F  .-  J
)  =  ( B 
.-  D ) )
3331, 32eqtr2d 2424 . . . 4  |-  ( ph  ->  ( B  .-  D
)  =  ( J 
.-  F ) )
34 tgbtwnconn1.8 . . . . . 6  |-  ( ph  ->  ( E  .-  D
)  =  ( C 
.-  D ) )
35 tgbtwnconn1.9 . . . . . 6  |-  ( ph  ->  ( C  .-  F
)  =  ( C 
.-  D ) )
3634, 35eqtr4d 2426 . . . . 5  |-  ( ph  ->  ( E  .-  D
)  =  ( C 
.-  F ) )
371, 2, 3, 4, 10, 11, 18, 17, 36tgcgrcomlr 23991 . . . 4  |-  ( ph  ->  ( D  .-  E
)  =  ( F 
.-  C ) )
381, 2, 3, 4, 5, 11, 10, 6, 17, 18, 28, 30, 33, 37tgcgrextend 23996 . . 3  |-  ( ph  ->  ( B  .-  E
)  =  ( J 
.-  C ) )
39 tgbtwnconn1.10 . . . 4  |-  ( ph  ->  ( E  .-  H
)  =  ( B 
.-  C ) )
401, 2, 3, 4, 18, 5axtgcgrrflx 23976 . . . 4  |-  ( ph  ->  ( C  .-  B
)  =  ( B 
.-  C ) )
4139, 40eqtr4d 2426 . . 3  |-  ( ph  ->  ( E  .-  H
)  =  ( C 
.-  B ) )
421, 2, 3, 4, 5, 10, 8, 6, 18, 5, 24, 27, 38, 41tgcgrextend 23996 . 2  |-  ( ph  ->  ( B  .-  H
)  =  ( J 
.-  B ) )
431, 2, 3, 4, 5, 6axtgcgrrflx 23976 . 2  |-  ( ph  ->  ( B  .-  J
)  =  ( J 
.-  B ) )
441, 2, 3, 4, 5, 6, 5, 7, 8, 6, 9, 16, 23, 42, 43tgsegconeq 23997 1  |-  ( ph  ->  H  =  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826    =/= wne 2577   ` cfv 5496  (class class class)co 6196   Basecbs 14634   distcds 14711  TarskiGcstrkg 23942  Itvcitv 23949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-nul 4496
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-iota 5460  df-fv 5504  df-ov 6199  df-trkgc 23961  df-trkgb 23962  df-trkgcb 23963  df-trkg 23967
This theorem is referenced by:  tgbtwnconn1lem2  24080  tgbtwnconn1lem3  24081
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