Theorem ogrpinvlt 29724

Description: In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)

Hypotheses

Ref	Expression
ogrpinvlt.0	|- B = ( Base ` G )
ogrpinvlt.1	|- .< = ( lt ` G )
ogrpinvlt.2	|- I = ( invg ` G )

Assertion

Ref	Expression
ogrpinvlt	|- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( I ` Y ) .< ( I ` X ) ) )

Proof of Theorem ogrpinvlt

Step	Hyp	Ref	Expression
1		simp1l 1085	. . . 4 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> G e. oGrp )
2		simp2 1062	. . . 4 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> X e. B )
3		simp3 1063	. . . 4 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> Y e. B )
4		ogrpgrp 29703	. . . . . 6 |- ( G e. oGrp -> G e. Grp )
5	1, 4	syl 17	. . . . 5 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> G e. Grp )
6		ogrpinvlt.0	. . . . . 6 |- B = ( Base ` G )
7		ogrpinvlt.2	. . . . . 6 |- I = ( invg ` G )
8	6, 7	grpinvcl 17467	. . . . 5 |- ( ( G e. Grp /\ Y e. B ) -> ( I ` Y ) e. B )
9	5, 3, 8	syl2anc 693	. . . 4 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( I ` Y ) e. B )
10		ogrpinvlt.1	. . . . 5 |- .< = ( lt ` G )
11		eqid 2622	. . . . 5 |- ( +g ` G ) = ( +g ` G )
12	6, 10, 11	ogrpaddltbi 29719	. . . 4 |- ( ( G e. oGrp /\ ( X e. B /\ Y e. B /\ ( I ` Y ) e. B ) ) -> ( X .< Y <-> ( X ( +g ` G ) ( I ` Y ) ) .< ( Y ( +g ` G ) ( I ` Y ) ) ) )
13	1, 2, 3, 9, 12	syl13anc 1328	. . 3 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( X ( +g ` G ) ( I ` Y ) ) .< ( Y ( +g ` G ) ( I ` Y ) ) ) )
14		eqid 2622	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
15	6, 11, 14, 7	grprinv 17469	. . . . 5 |- ( ( G e. Grp /\ Y e. B ) -> ( Y ( +g ` G ) ( I ` Y ) ) = ( 0g ` G ) )
16	5, 3, 15	syl2anc 693	. . . 4 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( Y ( +g ` G ) ( I ` Y ) ) = ( 0g ` G ) )
17	16	breq2d 4665	. . 3 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( ( X ( +g ` G ) ( I ` Y ) ) .< ( Y ( +g ` G ) ( I ` Y ) ) <-> ( X ( +g ` G ) ( I ` Y ) ) .< ( 0g ` G ) ) )
18		simp1r 1086	. . . 4 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> (oppg ` G ) e. oGrp )
19	6, 11	grpcl 17430	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ ( I ` Y ) e. B ) -> ( X ( +g ` G ) ( I ` Y ) ) e. B )
20	5, 2, 9, 19	syl3anc 1326	. . . 4 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( X ( +g ` G ) ( I ` Y ) ) e. B )
21	6, 14	grpidcl 17450	. . . . 5 |- ( G e. Grp -> ( 0g ` G ) e. B )
22	1, 4, 21	3syl 18	. . . 4 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( 0g ` G ) e. B )
23	6, 7	grpinvcl 17467	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( I ` X ) e. B )
24	5, 2, 23	syl2anc 693	. . . 4 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( I ` X ) e. B )
25	6, 10, 11, 1, 18, 20, 22, 24	ogrpaddltrbid 29721	. . 3 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( ( X ( +g ` G ) ( I ` Y ) ) .< ( 0g ` G ) <-> ( ( I ` X ) ( +g ` G ) ( X ( +g ` G ) ( I ` Y ) ) ) .< ( ( I ` X ) ( +g ` G ) ( 0g ` G ) ) ) )
26	13, 17, 25	3bitrd 294	. 2 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( ( I ` X ) ( +g ` G ) ( X ( +g ` G ) ( I ` Y ) ) ) .< ( ( I ` X ) ( +g ` G ) ( 0g ` G ) ) ) )
27	6, 11, 14, 7	grplinv 17468	. . . . . 6 |- ( ( G e. Grp /\ X e. B ) -> ( ( I ` X ) ( +g ` G ) X ) = ( 0g ` G ) )
28	5, 2, 27	syl2anc 693	. . . . 5 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( ( I ` X ) ( +g ` G ) X ) = ( 0g ` G ) )
29	28	oveq1d 6665	. . . 4 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( ( ( I ` X ) ( +g ` G ) X ) ( +g ` G ) ( I ` Y ) ) = ( ( 0g ` G ) ( +g ` G ) ( I ` Y ) ) )
30	6, 11	grpass 17431	. . . . 5 |- ( ( G e. Grp /\ ( ( I ` X ) e. B /\ X e. B /\ ( I ` Y ) e. B ) ) -> ( ( ( I ` X ) ( +g ` G ) X ) ( +g ` G ) ( I ` Y ) ) = ( ( I ` X ) ( +g ` G ) ( X ( +g ` G ) ( I ` Y ) ) ) )
31	5, 24, 2, 9, 30	syl13anc 1328	. . . 4 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( ( ( I ` X ) ( +g ` G ) X ) ( +g ` G ) ( I ` Y ) ) = ( ( I ` X ) ( +g ` G ) ( X ( +g ` G ) ( I ` Y ) ) ) )
32	6, 11, 14	grplid 17452	. . . . 5 |- ( ( G e. Grp /\ ( I ` Y ) e. B ) -> ( ( 0g ` G ) ( +g ` G ) ( I ` Y ) ) = ( I ` Y ) )
33	5, 9, 32	syl2anc 693	. . . 4 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( ( 0g ` G ) ( +g ` G ) ( I ` Y ) ) = ( I ` Y ) )
34	29, 31, 33	3eqtr3d 2664	. . 3 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( ( I ` X ) ( +g ` G ) ( X ( +g ` G ) ( I ` Y ) ) ) = ( I ` Y ) )
35	6, 11, 14	grprid 17453	. . . 4 |- ( ( G e. Grp /\ ( I ` X ) e. B ) -> ( ( I ` X ) ( +g ` G ) ( 0g ` G ) ) = ( I ` X ) )
36	5, 24, 35	syl2anc 693	. . 3 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( ( I ` X ) ( +g ` G ) ( 0g ` G ) ) = ( I ` X ) )
37	34, 36	breq12d 4666	. 2 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( ( ( I ` X ) ( +g ` G ) ( X ( +g ` G ) ( I ` Y ) ) ) .< ( ( I ` X ) ( +g ` G ) ( 0g ` G ) ) <-> ( I ` Y ) .< ( I ` X ) ) )
38	26, 37	bitrd 268	1 |- ( ( ( G e. oGrp /\ (oppg ` G ) e. oGrp ) /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( I ` Y ) .< ( I ` X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   ltcplt 16941   Grpcgrp 17422   invgcminusg 17423  opp_gcoppg 17775  oGrpcogrp 29698

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-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-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-dec 11494  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-ple 15961  df-0g 16102  df-plt 16958  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-oppg 17776  df-omnd 29699  df-ogrp 29700

This theorem is referenced by:  archirngz  29743  archiabllem2c  29749