Theorem tgbtwnconn1lem2 25468

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
tgbtwnconn1lem2	|- ( ph -> ( E .- F ) = ( C .- D ) )

Proof of Theorem tgbtwnconn1lem2

Step	Hyp	Ref	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 )
7	1, 2, 3, 4, 5, 6	axtgcgrrflx 25361	. . . 4 |- ( ph -> ( E .- F ) = ( F .- E ) )
8	7	adantr 481	. . 3 |- ( ( ph /\ B = C ) -> ( E .- F ) = ( F .- E ) )
9	4	adantr 481	. . . . . . 7 |- ( ( ph /\ B = C ) -> G e. TarskiG )
10	5	adantr 481	. . . . . . 7 |- ( ( ph /\ B = C ) -> E e. P )
11		tgbtwnconn1.h	. . . . . . . 8 |- ( ph -> H e. P )
12	11	adantr 481	. . . . . . 7 |- ( ( ph /\ B = C ) -> H e. P )
13		tgbtwnconn1.c	. . . . . . . 8 |- ( ph -> C e. P )
14	13	adantr 481	. . . . . . 7 |- ( ( ph /\ B = C ) -> C e. P )
15		tgbtwnconn1.10	. . . . . . . . 9 |- ( ph -> ( E .- H ) = ( B .- C ) )
16	15	adantr 481	. . . . . . . 8 |- ( ( ph /\ B = C ) -> ( E .- H ) = ( B .- C ) )
17		simpr 477	. . . . . . . . 9 |- ( ( ph /\ B = C ) -> B = C )
18	17	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ B = C ) -> ( B .- C ) = ( C .- C ) )
19	16, 18	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ B = C ) -> ( E .- H ) = ( C .- C ) )
20	1, 2, 3, 9, 10, 12, 14, 19	axtgcgrid 25362	. . . . . 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 ) )
35	1, 3, 4, 21, 22, 13, 23, 24, 25, 26, 2, 5, 6, 11, 27, 28, 29, 30, 31, 32, 33, 15, 34	tgbtwnconn1lem1 25467	. . . . . . 7 |- ( ph -> H = J )
36	35	adantr 481	. . . . . 6 |- ( ( ph /\ B = C ) -> H = J )
37	20, 36	eqtrd 2656	. . . . 5 |- ( ( ph /\ B = C ) -> E = J )
38	37	oveq2d 6666	. . . 4 |- ( ( ph /\ B = C ) -> ( F .- E ) = ( F .- J ) )
39	34	adantr 481	. . . 4 |- ( ( ph /\ B = C ) -> ( F .- J ) = ( B .- D ) )
40	17	oveq1d 6665	. . . 4 |- ( ( ph /\ B = C ) -> ( B .- D ) = ( C .- D ) )
41	38, 39, 40	3eqtrd 2660	. . 3 |- ( ( ph /\ B = C ) -> ( F .- E ) = ( C .- D ) )
42	8, 41	eqtrd 2656	. 2 |- ( ( ph /\ B = C ) -> ( E .- F ) = ( C .- D ) )
43	4	adantr 481	. . 3 |- ( ( ph /\ B =/= C ) -> G e. TarskiG )
44	6	adantr 481	. . 3 |- ( ( ph /\ B =/= C ) -> F e. P )
45	5	adantr 481	. . 3 |- ( ( ph /\ B =/= C ) -> E e. P )
46	23	adantr 481	. . 3 |- ( ( ph /\ B =/= C ) -> D e. P )
47	13	adantr 481	. . 3 |- ( ( ph /\ B =/= C ) -> C e. P )
48	22	adantr 481	. . . 4 |- ( ( ph /\ B =/= C ) -> B e. P )
49	27	adantr 481	. . . 4 |- ( ( ph /\ B =/= C ) -> J e. P )
50		simpr 477	. . . 4 |- ( ( ph /\ B =/= C ) -> B =/= C )
51	1, 2, 3, 4, 21, 22, 13, 6, 25, 29	tgbtwnexch3 25389	. . . . 5 |- ( ph -> C e. ( B I F ) )
52	51	adantr 481	. . . 4 |- ( ( ph /\ B =/= C ) -> C e. ( B I F ) )
53	35	oveq2d 6666	. . . . . . . 8 |- ( ph -> ( A I H ) = ( A I J ) )
54	30, 53	eleqtrd 2703	. . . . . . 7 |- ( ph -> E e. ( A I J ) )
55	1, 2, 3, 4, 21, 23, 5, 27, 28, 54	tgbtwnexch3 25389	. . . . . 6 |- ( ph -> E e. ( D I J ) )
56	1, 2, 3, 4, 23, 5, 27, 55	tgbtwncom 25383	. . . . 5 |- ( ph -> E e. ( J I D ) )
57	56	adantr 481	. . . 4 |- ( ( ph /\ B =/= C ) -> E e. ( J I D ) )
58	35	adantr 481	. . . . . 6 |- ( ( ph /\ B =/= C ) -> H = J )
59	58	oveq2d 6666	. . . . 5 |- ( ( ph /\ B =/= C ) -> ( E .- H ) = ( E .- J ) )
60	15	adantr 481	. . . . 5 |- ( ( ph /\ B =/= C ) -> ( E .- H ) = ( B .- C ) )
61	1, 2, 3, 43, 45, 49	axtgcgrrflx 25361	. . . . 5 |- ( ( ph /\ B =/= C ) -> ( E .- J ) = ( J .- E ) )
62	59, 60, 61	3eqtr3d 2664	. . . 4 |- ( ( ph /\ B =/= C ) -> ( B .- C ) = ( J .- E ) )
63	33, 32	eqtr4d 2659	. . . . 5 |- ( ph -> ( C .- F ) = ( E .- D ) )
64	63	adantr 481	. . . 4 |- ( ( ph /\ B =/= C ) -> ( C .- F ) = ( E .- D ) )
65	1, 2, 3, 4, 21, 22, 23, 5, 26, 28	tgbtwnexch3 25389	. . . . . 6 |- ( ph -> D e. ( B I E ) )
66	65	adantr 481	. . . . 5 |- ( ( ph /\ B =/= C ) -> D e. ( B I E ) )
67	1, 2, 3, 4, 21, 13, 6, 27, 29, 31	tgbtwnexch3 25389	. . . . . . 7 |- ( ph -> F e. ( C I J ) )
68	1, 2, 3, 4, 13, 6, 27, 67	tgbtwncom 25383	. . . . . 6 |- ( ph -> F e. ( J I C ) )
69	68	adantr 481	. . . . 5 |- ( ( ph /\ B =/= C ) -> F e. ( J I C ) )
70	1, 2, 3, 4, 27, 6	axtgcgrrflx 25361	. . . . . . 7 |- ( ph -> ( J .- F ) = ( F .- J ) )
71	70, 34	eqtr2d 2657	. . . . . 6 |- ( ph -> ( B .- D ) = ( J .- F ) )
72	71	adantr 481	. . . . 5 |- ( ( ph /\ B =/= C ) -> ( B .- D ) = ( J .- F ) )
73	1, 2, 3, 4, 13, 6, 5, 23, 63	tgcgrcomlr 25375	. . . . . . 7 |- ( ph -> ( F .- C ) = ( D .- E ) )
74	73	adantr 481	. . . . . 6 |- ( ( ph /\ B =/= C ) -> ( F .- C ) = ( D .- E ) )
75	74	eqcomd 2628	. . . . 5 |- ( ( ph /\ B =/= C ) -> ( D .- E ) = ( F .- C ) )
76	1, 2, 3, 43, 48, 46, 45, 49, 44, 47, 66, 69, 72, 75	tgcgrextend 25380	. . . 4 |- ( ( ph /\ B =/= C ) -> ( B .- E ) = ( J .- C ) )
77	1, 2, 3, 43, 47, 45	axtgcgrrflx 25361	. . . 4 |- ( ( ph /\ B =/= C ) -> ( C .- E ) = ( E .- C ) )
78	1, 2, 3, 43, 48, 47, 44, 49, 45, 46, 45, 47, 50, 52, 57, 62, 64, 76, 77	axtg5seg 25364	. . 3 |- ( ( ph /\ B =/= C ) -> ( F .- E ) = ( D .- C ) )
79	1, 2, 3, 43, 44, 45, 46, 47, 78	tgcgrcomlr 25375	. 2 |- ( ( ph /\ B =/= C ) -> ( E .- F ) = ( C .- D ) )
80	42, 79	pm2.61dane 2881	1 |- ( ph -> ( E .- F ) = ( C .- D ) )

Syntax hints:   -> wi 4   /\ wa 384   = 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:  tgbtwnconn1lem3  25469