Theorem ogrpinv0le 30716

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 30703	. . . . . 6 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simprbi 499	. . . . 5 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
3	2	ad2antrr 724	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝐺 ∈ oMnd)
4		omndmnd 30705	. . . . 5 ⊢ (𝐺 ∈ oMnd → 𝐺 ∈ Mnd)
5		ogrpsub.0	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
6		ogrpinv.3	. . . . . 6 ⊢ 0 = (0g‘𝐺)
7	5, 6	mndidcl 17926	. . . . 5 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
8	3, 4, 7	3syl 18	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 0 ∈ 𝐵)
9		simplr 767	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝑋 ∈ 𝐵)
10		ogrpgrp 30704	. . . . . 6 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
11	10	ad2antrr 724	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝐺 ∈ Grp)
12		ogrpinv.2	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
13	5, 12	grpinvcl 18151	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵)
14	11, 9, 13	syl2anc 586	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ∈ 𝐵)
15		simpr 487	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 0 ≤ 𝑋)
16		ogrpsub.1	. . . . 5 ⊢ ≤ = (le‘𝐺)
17		eqid 2821	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
18	5, 16, 17	omndadd 30707	. . . 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 1379	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) ≤ (𝑋(+g‘𝐺)(𝐼‘𝑋)))
20	5, 17, 6	grplid 18133	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
21	11, 14, 20	syl2anc 586	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
22	5, 17, 6, 12	grprinv 18153	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
23	11, 9, 22	syl2anc 586	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
24	19, 21, 23	3brtr3d 5097	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ≤ 0 )
25	2	ad2antrr 724	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝐺 ∈ oMnd)
26	10	ad2antrr 724	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝐺 ∈ Grp)
27		simplr 767	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝑋 ∈ 𝐵)
28	26, 27, 13	syl2anc 586	. . . 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 487	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → (𝐼‘𝑋) ≤ 0 )
31	5, 16, 17	omndadd 30707	. . . 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 1379	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) ≤ ( 0 (+g‘𝐺)𝑋))
33	5, 17, 6, 12	grplinv 18152	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
34	26, 27, 33	syl2anc 586	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
35	5, 17, 6	grplid 18133	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
36	26, 27, 35	syl2anc 586	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
37	32, 34, 36	3brtr3d 5097	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 0 ≤ 𝑋)
38	24, 37	impbida 799	1 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 ≤ 𝑋 ↔ (𝐼‘𝑋) ≤ 0 ))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  +_gcplusg 16565  lecple 16572  0_gc0g 16713  Mndcmnd 17911  Grpcgrp 18103  inv_gcminusg 18104  oMndcomnd 30698  oGrpcogrp 30699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-riota 7114  df-ov 7159  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-omnd 30700  df-ogrp 30701

This theorem is referenced by: (None)