Theorem ogrpinv0le 29716

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

Hypotheses

Ref	Expression
ogrpsub.0	|- B = ( Base ` G )
ogrpsub.1	|- .<_ = ( le ` G )
ogrpinv.2	|- I = ( invg ` G )
ogrpinv.3	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
ogrpinv0le	|- ( ( G e. oGrp /\ X e. B ) -> ( .0. .<_ X <-> ( I ` X ) .<_ .0. ) )

Proof of Theorem ogrpinv0le

Step	Hyp	Ref	Expression
1		isogrp 29702	. . . . . 6 |- ( G e. oGrp <-> ( G e. Grp /\ G e. oMnd ) )
2	1	simprbi 480	. . . . 5 |- ( G e. oGrp -> G e. oMnd )
3	2	ad2antrr 762	. . . 4 |- ( ( ( G e. oGrp /\ X e. B ) /\ .0. .<_ X ) -> G e. oMnd )
4		omndmnd 29704	. . . . 5 |- ( G e. oMnd -> G e. Mnd )
5		ogrpsub.0	. . . . . 6 |- B = ( Base ` G )
6		ogrpinv.3	. . . . . 6 |- .0. = ( 0g ` G )
7	5, 6	mndidcl 17308	. . . . 5 |- ( G e. Mnd -> .0. e. B )
8	3, 4, 7	3syl 18	. . . 4 |- ( ( ( G e. oGrp /\ X e. B ) /\ .0. .<_ X ) -> .0. e. B )
9		simplr 792	. . . 4 |- ( ( ( G e. oGrp /\ X e. B ) /\ .0. .<_ X ) -> X e. B )
10		ogrpgrp 29703	. . . . . 6 |- ( G e. oGrp -> G e. Grp )
11	10	ad2antrr 762	. . . . 5 |- ( ( ( G e. oGrp /\ X e. B ) /\ .0. .<_ X ) -> G e. Grp )
12		ogrpinv.2	. . . . . 6 |- I = ( invg ` G )
13	5, 12	grpinvcl 17467	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( I ` X ) e. B )
14	11, 9, 13	syl2anc 693	. . . 4 |- ( ( ( G e. oGrp /\ X e. B ) /\ .0. .<_ X ) -> ( I ` X ) e. B )
15		simpr 477	. . . 4 |- ( ( ( G e. oGrp /\ X e. B ) /\ .0. .<_ X ) -> .0. .<_ X )
16		ogrpsub.1	. . . . 5 |- .<_ = ( le ` G )
17		eqid 2622	. . . . 5 |- ( +g ` G ) = ( +g ` G )
18	5, 16, 17	omndadd 29706	. . . 4 |- ( ( G e. oMnd /\ ( .0. e. B /\ X e. B /\ ( I ` X ) e. B ) /\ .0. .<_ X ) -> ( .0. ( +g ` G ) ( I ` X ) ) .<_ ( X ( +g ` G ) ( I ` X ) ) )
19	3, 8, 9, 14, 15, 18	syl131anc 1339	. . 3 |- ( ( ( G e. oGrp /\ X e. B ) /\ .0. .<_ X ) -> ( .0. ( +g ` G ) ( I ` X ) ) .<_ ( X ( +g ` G ) ( I ` X ) ) )
20	5, 17, 6	grplid 17452	. . . 4 |- ( ( G e. Grp /\ ( I ` X ) e. B ) -> ( .0. ( +g ` G ) ( I ` X ) ) = ( I ` X ) )
21	11, 14, 20	syl2anc 693	. . 3 |- ( ( ( G e. oGrp /\ X e. B ) /\ .0. .<_ X ) -> ( .0. ( +g ` G ) ( I ` X ) ) = ( I ` X ) )
22	5, 17, 6, 12	grprinv 17469	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( X ( +g ` G ) ( I ` X ) ) = .0. )
23	11, 9, 22	syl2anc 693	. . 3 |- ( ( ( G e. oGrp /\ X e. B ) /\ .0. .<_ X ) -> ( X ( +g ` G ) ( I ` X ) ) = .0. )
24	19, 21, 23	3brtr3d 4684	. 2 |- ( ( ( G e. oGrp /\ X e. B ) /\ .0. .<_ X ) -> ( I ` X ) .<_ .0. )
25	2	ad2antrr 762	. . . 4 |- ( ( ( G e. oGrp /\ X e. B ) /\ ( I ` X ) .<_ .0. ) -> G e. oMnd )
26	10	ad2antrr 762	. . . . 5 |- ( ( ( G e. oGrp /\ X e. B ) /\ ( I ` X ) .<_ .0. ) -> G e. Grp )
27		simplr 792	. . . . 5 |- ( ( ( G e. oGrp /\ X e. B ) /\ ( I ` X ) .<_ .0. ) -> X e. B )
28	26, 27, 13	syl2anc 693	. . . 4 |- ( ( ( G e. oGrp /\ X e. B ) /\ ( I ` X ) .<_ .0. ) -> ( I ` X ) e. B )
29	25, 4, 7	3syl 18	. . . 4 |- ( ( ( G e. oGrp /\ X e. B ) /\ ( I ` X ) .<_ .0. ) -> .0. e. B )
30		simpr 477	. . . 4 |- ( ( ( G e. oGrp /\ X e. B ) /\ ( I ` X ) .<_ .0. ) -> ( I ` X ) .<_ .0. )
31	5, 16, 17	omndadd 29706	. . . 4 |- ( ( G e. oMnd /\ ( ( I ` X ) e. B /\ .0. e. B /\ X e. B ) /\ ( I ` X ) .<_ .0. ) -> ( ( I ` X ) ( +g ` G ) X ) .<_ ( .0. ( +g ` G ) X ) )
32	25, 28, 29, 27, 30, 31	syl131anc 1339	. . 3 |- ( ( ( G e. oGrp /\ X e. B ) /\ ( I ` X ) .<_ .0. ) -> ( ( I ` X ) ( +g ` G ) X ) .<_ ( .0. ( +g ` G ) X ) )
33	5, 17, 6, 12	grplinv 17468	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( ( I ` X ) ( +g ` G ) X ) = .0. )
34	26, 27, 33	syl2anc 693	. . 3 |- ( ( ( G e. oGrp /\ X e. B ) /\ ( I ` X ) .<_ .0. ) -> ( ( I ` X ) ( +g ` G ) X ) = .0. )
35	5, 17, 6	grplid 17452	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( .0. ( +g ` G ) X ) = X )
36	26, 27, 35	syl2anc 693	. . 3 |- ( ( ( G e. oGrp /\ X e. B ) /\ ( I ` X ) .<_ .0. ) -> ( .0. ( +g ` G ) X ) = X )
37	32, 34, 36	3brtr3d 4684	. 2 |- ( ( ( G e. oGrp /\ X e. B ) /\ ( I ` X ) .<_ .0. ) -> .0. .<_ X )
38	24, 37	impbida 877	1 |- ( ( G e. oGrp /\ X e. B ) -> ( .0. .<_ X <-> ( I ` X ) .<_ .0. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   lecple 15948   0gc0g 16100   Mndcmnd 17294   Grpcgrp 17422   invgcminusg 17423  oMndcomnd 29697  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

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-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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  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-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-omnd 29699  df-ogrp 29700

This theorem is referenced by: (None)