Theorem tgfscgr 25463

Description: Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)

Hypotheses

Ref	Expression
tglngval.p	|- P = ( Base ` G )
tglngval.l	|- L = (LineG ` G )
tglngval.i	|- I = (Itv ` G )
tglngval.g	|- ( ph -> G e. TarskiG )
tglngval.x	|- ( ph -> X e. P )
tglngval.y	|- ( ph -> Y e. P )
tgcolg.z	|- ( ph -> Z e. P )
lnxfr.r	|- .~ = (cgrG ` G )
lnxfr.a	|- ( ph -> A e. P )
lnxfr.b	|- ( ph -> B e. P )
lnxfr.d	|- .- = ( dist ` G )
tgfscgr.t	|- ( ph -> T e. P )
tgfscgr.c	|- ( ph -> C e. P )
tgfscgr.d	|- ( ph -> D e. P )
tgfscgr.1	|- ( ph -> ( Y e. ( X L Z ) \/ X = Z ) )
tgfscgr.2	|- ( ph -> <" X Y Z "> .~ <" A B C "> )
tgfscgr.3	|- ( ph -> ( X .- T ) = ( A .- D ) )
tgfscgr.4	|- ( ph -> ( Y .- T ) = ( B .- D ) )
tgfscgr.5	|- ( ph -> X =/= Y )

Assertion

Ref	Expression
tgfscgr	|- ( ph -> ( Z .- T ) = ( C .- D ) )

Proof of Theorem tgfscgr

Step	Hyp	Ref	Expression
1		tglngval.p	. . 3 |- P = ( Base ` G )
2		lnxfr.d	. . 3 |- .- = ( dist ` G )
3		tglngval.i	. . 3 |- I = (Itv ` G )
4		tglngval.g	. . . 4 |- ( ph -> G e. TarskiG )
5	4	adantr 481	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> G e. TarskiG )
6		tglngval.x	. . . 4 |- ( ph -> X e. P )
7	6	adantr 481	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> X e. P )
8		tglngval.y	. . . 4 |- ( ph -> Y e. P )
9	8	adantr 481	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> Y e. P )
10		tgcolg.z	. . . 4 |- ( ph -> Z e. P )
11	10	adantr 481	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> Z e. P )
12		lnxfr.a	. . . 4 |- ( ph -> A e. P )
13	12	adantr 481	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> A e. P )
14		lnxfr.b	. . . 4 |- ( ph -> B e. P )
15	14	adantr 481	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> B e. P )
16		tgfscgr.c	. . . 4 |- ( ph -> C e. P )
17	16	adantr 481	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> C e. P )
18		tgfscgr.t	. . . 4 |- ( ph -> T e. P )
19	18	adantr 481	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> T e. P )
20		tgfscgr.d	. . . 4 |- ( ph -> D e. P )
21	20	adantr 481	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> D e. P )
22		tgfscgr.5	. . . 4 |- ( ph -> X =/= Y )
23	22	adantr 481	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> X =/= Y )
24		simpr 477	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> Y e. ( X I Z ) )
25		lnxfr.r	. . . 4 |- .~ = (cgrG ` G )
26		tgfscgr.2	. . . . 5 |- ( ph -> <" X Y Z "> .~ <" A B C "> )
27	26	adantr 481	. . . 4 |- ( ( ph /\ Y e. ( X I Z ) ) -> <" X Y Z "> .~ <" A B C "> )
28	1, 2, 3, 25, 5, 7, 9, 11, 13, 15, 17, 27, 24	tgbtwnxfr 25425	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> B e. ( A I C ) )
29	1, 2, 3, 25, 5, 7, 9, 11, 13, 15, 17, 27	cgr3simp1 25415	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> ( X .- Y ) = ( A .- B ) )
30	1, 2, 3, 25, 5, 7, 9, 11, 13, 15, 17, 27	cgr3simp2 25416	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> ( Y .- Z ) = ( B .- C ) )
31		tgfscgr.3	. . . 4 |- ( ph -> ( X .- T ) = ( A .- D ) )
32	31	adantr 481	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> ( X .- T ) = ( A .- D ) )
33		tgfscgr.4	. . . 4 |- ( ph -> ( Y .- T ) = ( B .- D ) )
34	33	adantr 481	. . 3 |- ( ( ph /\ Y e. ( X I Z ) ) -> ( Y .- T ) = ( B .- D ) )
35	1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 24, 28, 29, 30, 32, 34	axtg5seg 25364	. 2 |- ( ( ph /\ Y e. ( X I Z ) ) -> ( Z .- T ) = ( C .- D ) )
36	4	adantr 481	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> G e. TarskiG )
37	8	adantr 481	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> Y e. P )
38	6	adantr 481	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> X e. P )
39	10	adantr 481	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> Z e. P )
40	14	adantr 481	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> B e. P )
41	12	adantr 481	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> A e. P )
42	16	adantr 481	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> C e. P )
43	18	adantr 481	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> T e. P )
44	20	adantr 481	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> D e. P )
45	22	necomd 2849	. . . 4 |- ( ph -> Y =/= X )
46	45	adantr 481	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> Y =/= X )
47		simpr 477	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> X e. ( Y I Z ) )
48	26	adantr 481	. . . . 5 |- ( ( ph /\ X e. ( Y I Z ) ) -> <" X Y Z "> .~ <" A B C "> )
49	1, 2, 3, 25, 36, 38, 37, 39, 41, 40, 42, 48	cgr3swap12 25418	. . . 4 |- ( ( ph /\ X e. ( Y I Z ) ) -> <" Y X Z "> .~ <" B A C "> )
50	1, 2, 3, 25, 36, 37, 38, 39, 40, 41, 42, 49, 47	tgbtwnxfr 25425	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> A e. ( B I C ) )
51	1, 2, 3, 25, 36, 37, 38, 39, 40, 41, 42, 49	cgr3simp1 25415	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> ( Y .- X ) = ( B .- A ) )
52	1, 2, 3, 25, 36, 37, 38, 39, 40, 41, 42, 49	cgr3simp2 25416	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> ( X .- Z ) = ( A .- C ) )
53	33	adantr 481	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> ( Y .- T ) = ( B .- D ) )
54	31	adantr 481	. . 3 |- ( ( ph /\ X e. ( Y I Z ) ) -> ( X .- T ) = ( A .- D ) )
55	1, 2, 3, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 50, 51, 52, 53, 54	axtg5seg 25364	. 2 |- ( ( ph /\ X e. ( Y I Z ) ) -> ( Z .- T ) = ( C .- D ) )
56	4	adantr 481	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> G e. TarskiG )
57	6	adantr 481	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> X e. P )
58	10	adantr 481	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> Z e. P )
59	8	adantr 481	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> Y e. P )
60	18	adantr 481	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> T e. P )
61	12	adantr 481	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> A e. P )
62	16	adantr 481	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> C e. P )
63	14	adantr 481	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> B e. P )
64	20	adantr 481	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> D e. P )
65		simpr 477	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> Z e. ( X I Y ) )
66	26	adantr 481	. . . . 5 |- ( ( ph /\ Z e. ( X I Y ) ) -> <" X Y Z "> .~ <" A B C "> )
67	1, 2, 3, 25, 56, 57, 59, 58, 61, 63, 62, 66	cgr3swap23 25419	. . . 4 |- ( ( ph /\ Z e. ( X I Y ) ) -> <" X Z Y "> .~ <" A C B "> )
68	1, 2, 3, 25, 56, 57, 58, 59, 61, 62, 63, 67, 65	tgbtwnxfr 25425	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> C e. ( A I B ) )
69	1, 2, 3, 25, 56, 57, 59, 58, 61, 63, 62, 66	cgr3simp1 25415	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> ( X .- Y ) = ( A .- B ) )
70	1, 2, 3, 25, 56, 57, 58, 59, 61, 62, 63, 67	cgr3simp2 25416	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> ( Z .- Y ) = ( C .- B ) )
71	31	adantr 481	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> ( X .- T ) = ( A .- D ) )
72	33	adantr 481	. . 3 |- ( ( ph /\ Z e. ( X I Y ) ) -> ( Y .- T ) = ( B .- D ) )
73	1, 2, 3, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 68, 69, 70, 71, 72	tgifscgr 25403	. 2 |- ( ( ph /\ Z e. ( X I Y ) ) -> ( Z .- T ) = ( C .- D ) )
74		tgfscgr.1	. . 3 |- ( ph -> ( Y e. ( X L Z ) \/ X = Z ) )
75		tglngval.l	. . . 4 |- L = (LineG ` G )
76	1, 75, 3, 4, 6, 10, 8	tgcolg 25449	. . 3 |- ( ph -> ( ( Y e. ( X L Z ) \/ X = Z ) <-> ( Y e. ( X I Z ) \/ X e. ( Y I Z ) \/ Z e. ( X I Y ) ) ) )
77	74, 76	mpbid 222	. 2 |- ( ph -> ( Y e. ( X I Z ) \/ X e. ( Y I Z ) \/ Z e. ( X I Y ) ) )
78	35, 55, 73, 77	mpjao3dan 1395	1 |- ( ph -> ( Z .- T ) = ( C .- D ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  cgrGccgrg 25405

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406

This theorem is referenced by:  lncgr  25464  mirtrcgr  25578  symquadlem  25584  cgracgr  25710  cgraswap  25712  cgrg3col4  25734