Theorem ogrpinv0le 31243

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 31230	. . . . . 6 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simprbi 496	. . . . 5 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
3	2	ad2antrr 722	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝐺 ∈ oMnd)
4		omndmnd 31232	. . . . 5 ⊢ (𝐺 ∈ oMnd → 𝐺 ∈ Mnd)
5		ogrpsub.0	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
6		ogrpinv.3	. . . . . 6 ⊢ 0 = (0g‘𝐺)
7	5, 6	mndidcl 18315	. . . . 5 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
8	3, 4, 7	3syl 18	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 0 ∈ 𝐵)
9		simplr 765	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝑋 ∈ 𝐵)
10		ogrpgrp 31231	. . . . . 6 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
11	10	ad2antrr 722	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝐺 ∈ Grp)
12		ogrpinv.2	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
13	5, 12	grpinvcl 18542	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵)
14	11, 9, 13	syl2anc 583	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ∈ 𝐵)
15		simpr 484	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 0 ≤ 𝑋)
16		ogrpsub.1	. . . . 5 ⊢ ≤ = (le‘𝐺)
17		eqid 2738	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
18	5, 16, 17	omndadd 31234	. . . 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 1381	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) ≤ (𝑋(+g‘𝐺)(𝐼‘𝑋)))
20	5, 17, 6	grplid 18524	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
21	11, 14, 20	syl2anc 583	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
22	5, 17, 6, 12	grprinv 18544	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
23	11, 9, 22	syl2anc 583	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
24	19, 21, 23	3brtr3d 5101	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ≤ 0 )
25	2	ad2antrr 722	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝐺 ∈ oMnd)
26	10	ad2antrr 722	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝐺 ∈ Grp)
27		simplr 765	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝑋 ∈ 𝐵)
28	26, 27, 13	syl2anc 583	. . . 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 484	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → (𝐼‘𝑋) ≤ 0 )
31	5, 16, 17	omndadd 31234	. . . 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 1381	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) ≤ ( 0 (+g‘𝐺)𝑋))
33	5, 17, 6, 12	grplinv 18543	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
34	26, 27, 33	syl2anc 583	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
35	5, 17, 6	grplid 18524	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
36	26, 27, 35	syl2anc 583	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
37	32, 34, 36	3brtr3d 5101	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 0 ≤ 𝑋)
38	24, 37	impbida 797	1 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 ≤ 𝑋 ↔ (𝐼‘𝑋) ≤ 0 ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  lecple 16895  0_gc0g 17067  Mndcmnd 18300  Grpcgrp 18492  inv_gcminusg 18493  oMndcomnd 31225  oGrpcogrp 31226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-riota 7212  df-ov 7258  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-omnd 31227  df-ogrp 31228

This theorem is referenced by: (None)