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Theorem tgbtwnconn1lem2 23802
Description: Lemma for tgbtwnconn1 23804 (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
tgbtwnconn1lem2  |-  ( ph  ->  ( E  .-  F
)  =  ( C 
.-  D ) )

Proof of Theorem tgbtwnconn1lem2
StepHypRef Expression
1 tgbtwnconn1.p . . . . 5  |-  P  =  ( Base `  G
)
2 tgbtwnconn1.m . . . . 5  |-  .-  =  ( dist `  G )
3 tgbtwnconn1.i . . . . 5  |-  I  =  (Itv `  G )
4 tgbtwnconn1.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
5 tgbtwnconn1.e . . . . 5  |-  ( ph  ->  E  e.  P )
6 tgbtwnconn1.f . . . . 5  |-  ( ph  ->  F  e.  P )
71, 2, 3, 4, 5, 6axtgcgrrflx 23702 . . . 4  |-  ( ph  ->  ( E  .-  F
)  =  ( F 
.-  E ) )
87adantr 465 . . 3  |-  ( (
ph  /\  B  =  C )  ->  ( E  .-  F )  =  ( F  .-  E
) )
94adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  =  C )  ->  G  e. TarskiG )
105adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  =  C )  ->  E  e.  P )
11 tgbtwnconn1.h . . . . . . . 8  |-  ( ph  ->  H  e.  P )
1211adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  =  C )  ->  H  e.  P )
13 tgbtwnconn1.c . . . . . . . 8  |-  ( ph  ->  C  e.  P )
1413adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  =  C )  ->  C  e.  P )
15 tgbtwnconn1.10 . . . . . . . . 9  |-  ( ph  ->  ( E  .-  H
)  =  ( B 
.-  C ) )
1615adantr 465 . . . . . . . 8  |-  ( (
ph  /\  B  =  C )  ->  ( E  .-  H )  =  ( B  .-  C
) )
17 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  B  =  C )  ->  B  =  C )
1817oveq1d 6309 . . . . . . . 8  |-  ( (
ph  /\  B  =  C )  ->  ( B  .-  C )  =  ( C  .-  C
) )
1916, 18eqtrd 2508 . . . . . . 7  |-  ( (
ph  /\  B  =  C )  ->  ( E  .-  H )  =  ( C  .-  C
) )
201, 2, 3, 9, 10, 12, 14, 19axtgcgrid 23703 . . . . . 6  |-  ( (
ph  /\  B  =  C )  ->  E  =  H )
21 tgbtwnconn1.a . . . . . . . 8  |-  ( ph  ->  A  e.  P )
22 tgbtwnconn1.b . . . . . . . 8  |-  ( ph  ->  B  e.  P )
23 tgbtwnconn1.d . . . . . . . 8  |-  ( ph  ->  D  e.  P )
24 tgbtwnconn1.1 . . . . . . . 8  |-  ( ph  ->  A  =/=  B )
25 tgbtwnconn1.2 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A I C ) )
26 tgbtwnconn1.3 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A I D ) )
27 tgbtwnconn1.j . . . . . . . 8  |-  ( ph  ->  J  e.  P )
28 tgbtwnconn1.4 . . . . . . . 8  |-  ( ph  ->  D  e.  ( A I E ) )
29 tgbtwnconn1.5 . . . . . . . 8  |-  ( ph  ->  C  e.  ( A I F ) )
30 tgbtwnconn1.6 . . . . . . . 8  |-  ( ph  ->  E  e.  ( A I H ) )
31 tgbtwnconn1.7 . . . . . . . 8  |-  ( ph  ->  F  e.  ( A I J ) )
32 tgbtwnconn1.8 . . . . . . . 8  |-  ( ph  ->  ( E  .-  D
)  =  ( C 
.-  D ) )
33 tgbtwnconn1.9 . . . . . . . 8  |-  ( ph  ->  ( C  .-  F
)  =  ( C 
.-  D ) )
34 tgbtwnconn1.11 . . . . . . . 8  |-  ( ph  ->  ( F  .-  J
)  =  ( B 
.-  D ) )
351, 3, 4, 21, 22, 13, 23, 24, 25, 26, 2, 5, 6, 11, 27, 28, 29, 30, 31, 32, 33, 15, 34tgbtwnconn1lem1 23801 . . . . . . 7  |-  ( ph  ->  H  =  J )
3635adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =  C )  ->  H  =  J )
3720, 36eqtrd 2508 . . . . 5  |-  ( (
ph  /\  B  =  C )  ->  E  =  J )
3837oveq2d 6310 . . . 4  |-  ( (
ph  /\  B  =  C )  ->  ( F  .-  E )  =  ( F  .-  J
) )
3934adantr 465 . . . 4  |-  ( (
ph  /\  B  =  C )  ->  ( F  .-  J )  =  ( B  .-  D
) )
4017oveq1d 6309 . . . 4  |-  ( (
ph  /\  B  =  C )  ->  ( B  .-  D )  =  ( C  .-  D
) )
4138, 39, 403eqtrd 2512 . . 3  |-  ( (
ph  /\  B  =  C )  ->  ( F  .-  E )  =  ( C  .-  D
) )
428, 41eqtrd 2508 . 2  |-  ( (
ph  /\  B  =  C )  ->  ( E  .-  F )  =  ( C  .-  D
) )
434adantr 465 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  G  e. TarskiG )
446adantr 465 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  F  e.  P )
455adantr 465 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  E  e.  P )
4623adantr 465 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  D  e.  P )
4713adantr 465 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  C  e.  P )
4822adantr 465 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  P )
4927adantr 465 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  J  e.  P )
50 simpr 461 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  B  =/=  C )
511, 2, 3, 4, 21, 22, 13, 6, 25, 29tgbtwnexch3 23728 . . . . 5  |-  ( ph  ->  C  e.  ( B I F ) )
5251adantr 465 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  C  e.  ( B I F ) )
5335oveq2d 6310 . . . . . . . 8  |-  ( ph  ->  ( A I H )  =  ( A I J ) )
5430, 53eleqtrd 2557 . . . . . . 7  |-  ( ph  ->  E  e.  ( A I J ) )
551, 2, 3, 4, 21, 23, 5, 27, 28, 54tgbtwnexch3 23728 . . . . . 6  |-  ( ph  ->  E  e.  ( D I J ) )
561, 2, 3, 4, 23, 5, 27, 55tgbtwncom 23722 . . . . 5  |-  ( ph  ->  E  e.  ( J I D ) )
5756adantr 465 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  E  e.  ( J I D ) )
5835adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  H  =  J )
5958oveq2d 6310 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( E  .-  H )  =  ( E  .-  J ) )
6015adantr 465 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( E  .-  H )  =  ( B  .-  C ) )
611, 2, 3, 43, 45, 49axtgcgrrflx 23702 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( E  .-  J )  =  ( J  .-  E ) )
6259, 60, 613eqtr3d 2516 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( B  .-  C )  =  ( J  .-  E ) )
6333, 32eqtr4d 2511 . . . . 5  |-  ( ph  ->  ( C  .-  F
)  =  ( E 
.-  D ) )
6463adantr 465 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( C  .-  F )  =  ( E  .-  D ) )
651, 2, 3, 4, 21, 22, 23, 5, 26, 28tgbtwnexch3 23728 . . . . . 6  |-  ( ph  ->  D  e.  ( B I E ) )
6665adantr 465 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  D  e.  ( B I E ) )
671, 2, 3, 4, 21, 13, 6, 27, 29, 31tgbtwnexch3 23728 . . . . . . 7  |-  ( ph  ->  F  e.  ( C I J ) )
681, 2, 3, 4, 13, 6, 27, 67tgbtwncom 23722 . . . . . 6  |-  ( ph  ->  F  e.  ( J I C ) )
6968adantr 465 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  F  e.  ( J I C ) )
701, 2, 3, 4, 27, 6axtgcgrrflx 23702 . . . . . . 7  |-  ( ph  ->  ( J  .-  F
)  =  ( F 
.-  J ) )
7170, 34eqtr2d 2509 . . . . . 6  |-  ( ph  ->  ( B  .-  D
)  =  ( J 
.-  F ) )
7271adantr 465 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( B  .-  D )  =  ( J  .-  F ) )
731, 2, 3, 4, 13, 6, 5, 23, 63tgcgrcomlr 23714 . . . . . . 7  |-  ( ph  ->  ( F  .-  C
)  =  ( D 
.-  E ) )
7473adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  ( F  .-  C )  =  ( D  .-  E ) )
7574eqcomd 2475 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( D  .-  E )  =  ( F  .-  C ) )
761, 2, 3, 43, 48, 46, 45, 49, 44, 47, 66, 69, 72, 75tgcgrextend 23719 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( B  .-  E )  =  ( J  .-  C ) )
771, 2, 3, 43, 47, 45axtgcgrrflx 23702 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( C  .-  E )  =  ( E  .-  C ) )
781, 2, 3, 43, 48, 47, 44, 49, 45, 46, 45, 47, 50, 52, 57, 62, 64, 76, 77axtg5seg 23705 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  ( F  .-  E )  =  ( D  .-  C ) )
791, 2, 3, 43, 44, 45, 46, 47, 78tgcgrcomlr 23714 . 2  |-  ( (
ph  /\  B  =/=  C )  ->  ( E  .-  F )  =  ( C  .-  D ) )
8042, 79pm2.61dane 2785 1  |-  ( ph  ->  ( E  .-  F
)  =  ( C 
.-  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5593  (class class class)co 6294   Basecbs 14502   distcds 14576  TarskiGcstrkg 23668  Itvcitv 23675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-iota 5556  df-fv 5601  df-ov 6297  df-trkgc 23687  df-trkgb 23688  df-trkgcb 23689  df-trkg 23693
This theorem is referenced by:  tgbtwnconn1lem3  23803
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