Theorem tg5segofs 30751

Description: Rephrase axtg5seg 25364 using the outer five segment predicate. Theorem 2.10 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.)

Hypotheses

Ref	Expression
tg5segofs.p	|- P = ( Base ` G )
tg5segofs.m	|- .- = ( dist ` G )
tg5segofs.s	|- I = (Itv ` G )
tg5segofs.g	|- ( ph -> G e. TarskiG )
tg5segofs.a	|- ( ph -> A e. P )
tg5segofs.b	|- ( ph -> B e. P )
tg5segofs.c	|- ( ph -> C e. P )
tg5segofs.d	|- ( ph -> D e. P )
tg5segofs.e	|- ( ph -> E e. P )
tg5segofs.f	|- ( ph -> F e. P )
tg5segofs.o	|- O = (AFS ` G )
tg5segofs.h	|- ( ph -> H e. P )
tg5segofs.i	|- ( ph -> I e. P )
tg5segofs.1	|- ( ph -> <. <. A , B >. , <. C , D >. >. O <. <. E , F >. , <. H , I >. >. )
tg5segofs.2	|- ( ph -> A =/= B )

Assertion

Ref	Expression
tg5segofs	|- ( ph -> ( C .- D ) = ( H .- I ) )

Proof of Theorem tg5segofs

Step	Hyp	Ref	Expression
1		tg5segofs.p	. 2 |- P = ( Base ` G )
2		tg5segofs.m	. 2 |- .- = ( dist ` G )
3		tg5segofs.s	. 2 |- I = (Itv ` G )
4		tg5segofs.g	. 2 |- ( ph -> G e. TarskiG )
5		tg5segofs.a	. 2 |- ( ph -> A e. P )
6		tg5segofs.b	. 2 |- ( ph -> B e. P )
7		tg5segofs.c	. 2 |- ( ph -> C e. P )
8		tg5segofs.e	. 2 |- ( ph -> E e. P )
9		tg5segofs.f	. 2 |- ( ph -> F e. P )
10		tg5segofs.h	. 2 |- ( ph -> H e. P )
11		tg5segofs.d	. 2 |- ( ph -> D e. P )
12		tg5segofs.i	. 2 |- ( ph -> I e. P )
13		tg5segofs.2	. 2 |- ( ph -> A =/= B )
14		tg5segofs.1	. . . . 5 |- ( ph -> <. <. A , B >. , <. C , D >. >. O <. <. E , F >. , <. H , I >. >. )
15		tg5segofs.o	. . . . . 6 |- O = (AFS ` G )
16	1, 2, 3, 4, 15, 5, 6, 7, 11, 8, 9, 10, 12	brafs 30750	. . . . 5 |- ( ph -> ( <. <. A , B >. , <. C , D >. >. O <. <. E , F >. , <. H , I >. >. <-> ( ( B e. ( A I C ) /\ F e. ( E I H ) ) /\ ( ( A .- B ) = ( E .- F ) /\ ( B .- C ) = ( F .- H ) ) /\ ( ( A .- D ) = ( E .- I ) /\ ( B .- D ) = ( F .- I ) ) ) ) )
17	14, 16	mpbid 222	. . . 4 |- ( ph -> ( ( B e. ( A I C ) /\ F e. ( E I H ) ) /\ ( ( A .- B ) = ( E .- F ) /\ ( B .- C ) = ( F .- H ) ) /\ ( ( A .- D ) = ( E .- I ) /\ ( B .- D ) = ( F .- I ) ) ) )
18	17	simp1d 1073	. . 3 |- ( ph -> ( B e. ( A I C ) /\ F e. ( E I H ) ) )
19	18	simpld 475	. 2 |- ( ph -> B e. ( A I C ) )
20	18	simprd 479	. 2 |- ( ph -> F e. ( E I H ) )
21	17	simp2d 1074	. . 3 |- ( ph -> ( ( A .- B ) = ( E .- F ) /\ ( B .- C ) = ( F .- H ) ) )
22	21	simpld 475	. 2 |- ( ph -> ( A .- B ) = ( E .- F ) )
23	21	simprd 479	. 2 |- ( ph -> ( B .- C ) = ( F .- H ) )
24	17	simp3d 1075	. . 3 |- ( ph -> ( ( A .- D ) = ( E .- I ) /\ ( B .- D ) = ( F .- I ) ) )
25	24	simpld 475	. 2 |- ( ph -> ( A .- D ) = ( E .- I ) )
26	24	simprd 479	. 2 |- ( ph -> ( B .- D ) = ( F .- I ) )
27	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 19, 20, 22, 23, 25, 26	axtg5seg 25364	1 |- ( ph -> ( C .- D ) = ( H .- I ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  AFScafs 30747

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-mo 2475  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-csb 3534  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-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-trkgcb 25349  df-trkg 25352  df-afs 30748

This theorem is referenced by: (None)