Theorem ogrpinv0le 30250

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 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 ≤ 𝑋 ↔ (𝐼‘𝑋) ≤ 0 ))

Proof of Theorem ogrpinv0le

Step	Hyp	Ref	Expression
1		isogrp 30236	. . . . . 6 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simprbi 492	. . . . 5 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
3	2	ad2antrr 717	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝐺 ∈ oMnd)
4		omndmnd 30238	. . . . 5 ⊢ (𝐺 ∈ oMnd → 𝐺 ∈ Mnd)
5		ogrpsub.0	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
6		ogrpinv.3	. . . . . 6 ⊢ 0 = (0g‘𝐺)
7	5, 6	mndidcl 17661	. . . . 5 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
8	3, 4, 7	3syl 18	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 0 ∈ 𝐵)
9		simplr 785	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝑋 ∈ 𝐵)
10		ogrpgrp 30237	. . . . . 6 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
11	10	ad2antrr 717	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝐺 ∈ Grp)
12		ogrpinv.2	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
13	5, 12	grpinvcl 17821	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵)
14	11, 9, 13	syl2anc 579	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ∈ 𝐵)
15		simpr 479	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 0 ≤ 𝑋)
16		ogrpsub.1	. . . . 5 ⊢ ≤ = (le‘𝐺)
17		eqid 2825	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
18	5, 16, 17	omndadd 30240	. . . 4 ⊢ ((𝐺 ∈ oMnd ∧ ( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝐼‘𝑋) ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) ≤ (𝑋(+g‘𝐺)(𝐼‘𝑋)))
19	3, 8, 9, 14, 15, 18	syl131anc 1506	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) ≤ (𝑋(+g‘𝐺)(𝐼‘𝑋)))
20	5, 17, 6	grplid 17806	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
21	11, 14, 20	syl2anc 579	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
22	5, 17, 6, 12	grprinv 17823	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
23	11, 9, 22	syl2anc 579	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
24	19, 21, 23	3brtr3d 4904	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ≤ 0 )
25	2	ad2antrr 717	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝐺 ∈ oMnd)
26	10	ad2antrr 717	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝐺 ∈ Grp)
27		simplr 785	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝑋 ∈ 𝐵)
28	26, 27, 13	syl2anc 579	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → (𝐼‘𝑋) ∈ 𝐵)
29	25, 4, 7	3syl 18	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 0 ∈ 𝐵)
30		simpr 479	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → (𝐼‘𝑋) ≤ 0 )
31	5, 16, 17	omndadd 30240	. . . 4 ⊢ ((𝐺 ∈ oMnd ∧ ((𝐼‘𝑋) ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) ≤ ( 0 (+g‘𝐺)𝑋))
32	25, 28, 29, 27, 30, 31	syl131anc 1506	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) ≤ ( 0 (+g‘𝐺)𝑋))
33	5, 17, 6, 12	grplinv 17822	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
34	26, 27, 33	syl2anc 579	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
35	5, 17, 6	grplid 17806	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
36	26, 27, 35	syl2anc 579	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
37	32, 34, 36	3brtr3d 4904	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 0 ≤ 𝑋)
38	24, 37	impbida 835	1 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 ≤ 𝑋 ↔ (𝐼‘𝑋) ≤ 0 ))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164   class class class wbr 4873  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  +_gcplusg 16305  lecple 16312  0_gc0g 16453  Mndcmnd 17647  Grpcgrp 17776  inv_gcminusg 17777  oMndcomnd 30231  oGrpcogrp 30232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-0g 16455  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-grp 17779  df-minusg 17780  df-omnd 30233  df-ogrp 30234

This theorem is referenced by: (None)