Theorem tgbtwnconn1lem1 25467

Description: Lemma for tgbtwnconn1 25470. (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

Step	Hyp	Ref	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 ) )
14	1, 2, 3, 4, 7, 5, 11, 10, 12, 13	tgbtwnexch 25393	. . 3 |- ( ph -> B e. ( A I E ) )
15		tgbtwnconn1.6	. . 3 |- ( ph -> E e. ( A I H ) )
16	1, 2, 3, 4, 7, 5, 10, 8, 14, 15	tgbtwnexch 25393	. 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 ) )
21	1, 2, 3, 4, 7, 5, 18, 17, 19, 20	tgbtwnexch 25393	. . 3 |- ( ph -> B e. ( A I F ) )
22		tgbtwnconn1.7	. . 3 |- ( ph -> F e. ( A I J ) )
23	1, 2, 3, 4, 7, 5, 17, 6, 21, 22	tgbtwnexch 25393	. 2 |- ( ph -> B e. ( A I J ) )
24	1, 2, 3, 4, 7, 5, 10, 8, 14, 15	tgbtwnexch3 25389	. . 3 |- ( ph -> E e. ( B I H ) )
25	1, 2, 3, 4, 7, 18, 17, 6, 20, 22	tgbtwnexch 25393	. . . . 5 |- ( ph -> C e. ( A I J ) )
26	1, 2, 3, 4, 7, 5, 18, 6, 19, 25	tgbtwnexch3 25389	. . . 4 |- ( ph -> C e. ( B I J ) )
27	1, 2, 3, 4, 5, 18, 6, 26	tgbtwncom 25383	. . 3 |- ( ph -> C e. ( J I B ) )
28	1, 2, 3, 4, 7, 5, 11, 10, 12, 13	tgbtwnexch3 25389	. . . 4 |- ( ph -> D e. ( B I E ) )
29	1, 2, 3, 4, 7, 18, 17, 6, 20, 22	tgbtwnexch3 25389	. . . . 5 |- ( ph -> F e. ( C I J ) )
30	1, 2, 3, 4, 18, 17, 6, 29	tgbtwncom 25383	. . . 4 |- ( ph -> F e. ( J I C ) )
31	1, 2, 3, 4, 6, 17	axtgcgrrflx 25361	. . . . 5 |- ( ph -> ( J .- F ) = ( F .- J ) )
32		tgbtwnconn1.11	. . . . 5 |- ( ph -> ( F .- J ) = ( B .- D ) )
33	31, 32	eqtr2d 2657	. . . 4 |- ( ph -> ( B .- D ) = ( J .- F ) )
34		tgbtwnconn1.8	. . . . . 6 |- ( ph -> ( E .- D ) = ( C .- D ) )
35		tgbtwnconn1.9	. . . . . 6 |- ( ph -> ( C .- F ) = ( C .- D ) )
36	34, 35	eqtr4d 2659	. . . . 5 |- ( ph -> ( E .- D ) = ( C .- F ) )
37	1, 2, 3, 4, 10, 11, 18, 17, 36	tgcgrcomlr 25375	. . . 4 |- ( ph -> ( D .- E ) = ( F .- C ) )
38	1, 2, 3, 4, 5, 11, 10, 6, 17, 18, 28, 30, 33, 37	tgcgrextend 25380	. . 3 |- ( ph -> ( B .- E ) = ( J .- C ) )
39		tgbtwnconn1.10	. . . 4 |- ( ph -> ( E .- H ) = ( B .- C ) )
40	1, 2, 3, 4, 10, 8, 5, 18, 39	tgcgrcomr 25373	. . 3 |- ( ph -> ( E .- H ) = ( C .- B ) )
41	1, 2, 3, 4, 5, 10, 8, 6, 18, 5, 24, 27, 38, 40	tgcgrextend 25380	. 2 |- ( ph -> ( B .- H ) = ( J .- B ) )
42	1, 2, 3, 4, 5, 6	axtgcgrrflx 25361	. 2 |- ( ph -> ( B .- J ) = ( J .- B ) )
43	1, 2, 3, 4, 5, 6, 5, 7, 8, 6, 9, 16, 23, 41, 42	tgsegconeq 25381	1 |- ( ph -> H = J )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352

This theorem is referenced by:  tgbtwnconn1lem2  25468  tgbtwnconn1lem3  25469