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Theorem tgbtwnconn1lem1 23124
Description: Lemma for tgbtwnconn1 23127 (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 23069 . . 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 23069 . 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 23069 . . 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 23069 . 2  |-  ( ph  ->  B  e.  ( A I J ) )
241, 2, 3, 4, 7, 5, 10, 8, 14, 15tgbtwnexch3 23065 . . 3  |-  ( ph  ->  E  e.  ( B I H ) )
251, 2, 3, 4, 7, 17, 6, 22tgbtwncom 23059 . . . . . . 7  |-  ( ph  ->  F  e.  ( J I A ) )
261, 2, 3, 4, 7, 18, 17, 20tgbtwncom 23059 . . . . . . 7  |-  ( ph  ->  C  e.  ( F I A ) )
271, 2, 3, 4, 6, 17, 18, 7, 25, 26tgbtwnexch2 23067 . . . . . 6  |-  ( ph  ->  C  e.  ( J I A ) )
281, 2, 3, 4, 6, 18, 7, 27tgbtwncom 23059 . . . . 5  |-  ( ph  ->  C  e.  ( A I J ) )
291, 2, 3, 4, 7, 5, 18, 6, 19, 28tgbtwnexch3 23065 . . . 4  |-  ( ph  ->  C  e.  ( B I J ) )
301, 2, 3, 4, 5, 18, 6, 29tgbtwncom 23059 . . 3  |-  ( ph  ->  C  e.  ( J I B ) )
311, 2, 3, 4, 7, 5, 11, 10, 12, 13tgbtwnexch3 23065 . . . 4  |-  ( ph  ->  D  e.  ( B I E ) )
321, 2, 3, 4, 7, 18, 17, 6, 20, 22tgbtwnexch3 23065 . . . . 5  |-  ( ph  ->  F  e.  ( C I J ) )
331, 2, 3, 4, 18, 17, 6, 32tgbtwncom 23059 . . . 4  |-  ( ph  ->  F  e.  ( J I C ) )
341, 2, 3, 4, 6, 17axtgcgrrflx 23039 . . . . 5  |-  ( ph  ->  ( J  .-  F
)  =  ( F 
.-  J ) )
35 tgbtwnconn1.11 . . . . 5  |-  ( ph  ->  ( F  .-  J
)  =  ( B 
.-  D ) )
3634, 35eqtr2d 2493 . . . 4  |-  ( ph  ->  ( B  .-  D
)  =  ( J 
.-  F ) )
37 tgbtwnconn1.8 . . . . . 6  |-  ( ph  ->  ( E  .-  D
)  =  ( C 
.-  D ) )
38 tgbtwnconn1.9 . . . . . 6  |-  ( ph  ->  ( C  .-  F
)  =  ( C 
.-  D ) )
3937, 38eqtr4d 2495 . . . . 5  |-  ( ph  ->  ( E  .-  D
)  =  ( C 
.-  F ) )
401, 2, 3, 4, 10, 11, 18, 17, 39tgcgrcomlr 23051 . . . 4  |-  ( ph  ->  ( D  .-  E
)  =  ( F 
.-  C ) )
411, 2, 3, 4, 5, 11, 10, 6, 17, 18, 31, 33, 36, 40tgcgrextend 23056 . . 3  |-  ( ph  ->  ( B  .-  E
)  =  ( J 
.-  C ) )
42 tgbtwnconn1.10 . . . 4  |-  ( ph  ->  ( E  .-  H
)  =  ( B 
.-  C ) )
431, 2, 3, 4, 18, 5axtgcgrrflx 23039 . . . 4  |-  ( ph  ->  ( C  .-  B
)  =  ( B 
.-  C ) )
4442, 43eqtr4d 2495 . . 3  |-  ( ph  ->  ( E  .-  H
)  =  ( C 
.-  B ) )
451, 2, 3, 4, 5, 10, 8, 6, 18, 5, 24, 30, 41, 44tgcgrextend 23056 . 2  |-  ( ph  ->  ( B  .-  H
)  =  ( J 
.-  B ) )
461, 2, 3, 4, 5, 6axtgcgrrflx 23039 . 2  |-  ( ph  ->  ( B  .-  J
)  =  ( J 
.-  B ) )
471, 2, 3, 4, 5, 6, 5, 7, 8, 6, 9, 16, 23, 45, 46tgsegconeq 23057 1  |-  ( ph  ->  H  =  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = 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-pw 3960  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-trkgb 23025  df-trkgcb 23026  df-trkg 23030
This theorem is referenced by:  tgbtwnconn1lem2  23125  tgbtwnconn1lem3  23126
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