Theorem tgcgrextend 25380

Description: Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) (Shortened by David A. Wheeler and Thierry Arnoux, 22-Apr-2020.)

Hypotheses

Ref	Expression
tkgeom.p	|- P = ( Base ` G )
tkgeom.d	|- .- = ( dist ` G )
tkgeom.i	|- I = (Itv ` G )
tkgeom.g	|- ( ph -> G e. TarskiG )
tgcgrextend.a	|- ( ph -> A e. P )
tgcgrextend.b	|- ( ph -> B e. P )
tgcgrextend.c	|- ( ph -> C e. P )
tgcgrextend.d	|- ( ph -> D e. P )
tgcgrextend.e	|- ( ph -> E e. P )
tgcgrextend.f	|- ( ph -> F e. P )
tgcgrextend.1	|- ( ph -> B e. ( A I C ) )
tgcgrextend.2	|- ( ph -> E e. ( D I F ) )
tgcgrextend.3	|- ( ph -> ( A .- B ) = ( D .- E ) )
tgcgrextend.4	|- ( ph -> ( B .- C ) = ( E .- F ) )

Assertion

Ref	Expression
tgcgrextend	|- ( ph -> ( A .- C ) = ( D .- F ) )

Proof of Theorem tgcgrextend

Step	Hyp	Ref	Expression
1		tgcgrextend.4	. . . 4 |- ( ph -> ( B .- C ) = ( E .- F ) )
2	1	adantr 481	. . 3 |- ( ( ph /\ A = B ) -> ( B .- C ) = ( E .- F ) )
3		simpr 477	. . . 4 |- ( ( ph /\ A = B ) -> A = B )
4	3	oveq1d 6665	. . 3 |- ( ( ph /\ A = B ) -> ( A .- C ) = ( B .- C ) )
5		tkgeom.p	. . . . 5 |- P = ( Base ` G )
6		tkgeom.d	. . . . 5 |- .- = ( dist ` G )
7		tkgeom.i	. . . . 5 |- I = (Itv ` G )
8		tkgeom.g	. . . . . 6 |- ( ph -> G e. TarskiG )
9	8	adantr 481	. . . . 5 |- ( ( ph /\ A = B ) -> G e. TarskiG )
10		tgcgrextend.a	. . . . . 6 |- ( ph -> A e. P )
11	10	adantr 481	. . . . 5 |- ( ( ph /\ A = B ) -> A e. P )
12		tgcgrextend.b	. . . . . 6 |- ( ph -> B e. P )
13	12	adantr 481	. . . . 5 |- ( ( ph /\ A = B ) -> B e. P )
14		tgcgrextend.d	. . . . . 6 |- ( ph -> D e. P )
15	14	adantr 481	. . . . 5 |- ( ( ph /\ A = B ) -> D e. P )
16		tgcgrextend.e	. . . . . 6 |- ( ph -> E e. P )
17	16	adantr 481	. . . . 5 |- ( ( ph /\ A = B ) -> E e. P )
18		tgcgrextend.3	. . . . . 6 |- ( ph -> ( A .- B ) = ( D .- E ) )
19	18	adantr 481	. . . . 5 |- ( ( ph /\ A = B ) -> ( A .- B ) = ( D .- E ) )
20	5, 6, 7, 9, 11, 13, 15, 17, 19, 3	tgcgreq 25377	. . . 4 |- ( ( ph /\ A = B ) -> D = E )
21	20	oveq1d 6665	. . 3 |- ( ( ph /\ A = B ) -> ( D .- F ) = ( E .- F ) )
22	2, 4, 21	3eqtr4d 2666	. 2 |- ( ( ph /\ A = B ) -> ( A .- C ) = ( D .- F ) )
23	8	adantr 481	. . 3 |- ( ( ph /\ A =/= B ) -> G e. TarskiG )
24		tgcgrextend.c	. . . 4 |- ( ph -> C e. P )
25	24	adantr 481	. . 3 |- ( ( ph /\ A =/= B ) -> C e. P )
26	10	adantr 481	. . 3 |- ( ( ph /\ A =/= B ) -> A e. P )
27		tgcgrextend.f	. . . 4 |- ( ph -> F e. P )
28	27	adantr 481	. . 3 |- ( ( ph /\ A =/= B ) -> F e. P )
29	14	adantr 481	. . 3 |- ( ( ph /\ A =/= B ) -> D e. P )
30	12	adantr 481	. . . 4 |- ( ( ph /\ A =/= B ) -> B e. P )
31	16	adantr 481	. . . 4 |- ( ( ph /\ A =/= B ) -> E e. P )
32		simpr 477	. . . 4 |- ( ( ph /\ A =/= B ) -> A =/= B )
33		tgcgrextend.1	. . . . 5 |- ( ph -> B e. ( A I C ) )
34	33	adantr 481	. . . 4 |- ( ( ph /\ A =/= B ) -> B e. ( A I C ) )
35		tgcgrextend.2	. . . . 5 |- ( ph -> E e. ( D I F ) )
36	35	adantr 481	. . . 4 |- ( ( ph /\ A =/= B ) -> E e. ( D I F ) )
37	18	adantr 481	. . . 4 |- ( ( ph /\ A =/= B ) -> ( A .- B ) = ( D .- E ) )
38	1	adantr 481	. . . 4 |- ( ( ph /\ A =/= B ) -> ( B .- C ) = ( E .- F ) )
39	5, 6, 7, 23, 26, 29	tgcgrtriv 25379	. . . 4 |- ( ( ph /\ A =/= B ) -> ( A .- A ) = ( D .- D ) )
40	5, 6, 7, 23, 26, 30, 29, 31, 37	tgcgrcomlr 25375	. . . 4 |- ( ( ph /\ A =/= B ) -> ( B .- A ) = ( E .- D ) )
41	5, 6, 7, 23, 26, 30, 25, 29, 31, 28, 26, 29, 32, 34, 36, 37, 38, 39, 40	axtg5seg 25364	. . 3 |- ( ( ph /\ A =/= B ) -> ( C .- A ) = ( F .- D ) )
42	5, 6, 7, 23, 25, 26, 28, 29, 41	tgcgrcomlr 25375	. 2 |- ( ( ph /\ A =/= B ) -> ( A .- C ) = ( D .- F ) )
43	22, 42	pm2.61dane 2881	1 |- ( ph -> ( A .- C ) = ( D .- F ) )

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-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-trkgcb 25349  df-trkg 25352

This theorem is referenced by:  tgsegconeq  25381  tgcgrxfr  25413  lnext  25462  tgbtwnconn1lem1  25467  tgbtwnconn1lem2  25468  tgbtwnconn1lem3  25469  miriso  25565  mircgrextend  25577  midexlem  25587  opphllem  25627  dfcgra2  25721