Theorem ogrpinv0le 30056

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 30042	. . . . . 6 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simprbi 484	. . . . 5 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
3	2	ad2antrr 705	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝐺 ∈ oMnd)
4		omndmnd 30044	. . . . 5 ⊢ (𝐺 ∈ oMnd → 𝐺 ∈ Mnd)
5		ogrpsub.0	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
6		ogrpinv.3	. . . . . 6 ⊢ 0 = (0g‘𝐺)
7	5, 6	mndidcl 17516	. . . . 5 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
8	3, 4, 7	3syl 18	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 0 ∈ 𝐵)
9		simplr 752	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝑋 ∈ 𝐵)
10		ogrpgrp 30043	. . . . . 6 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
11	10	ad2antrr 705	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝐺 ∈ Grp)
12		ogrpinv.2	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
13	5, 12	grpinvcl 17675	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵)
14	11, 9, 13	syl2anc 573	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ∈ 𝐵)
15		simpr 471	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 0 ≤ 𝑋)
16		ogrpsub.1	. . . . 5 ⊢ ≤ = (le‘𝐺)
17		eqid 2771	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
18	5, 16, 17	omndadd 30046	. . . 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 1489	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) ≤ (𝑋(+g‘𝐺)(𝐼‘𝑋)))
20	5, 17, 6	grplid 17660	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
21	11, 14, 20	syl2anc 573	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
22	5, 17, 6, 12	grprinv 17677	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
23	11, 9, 22	syl2anc 573	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
24	19, 21, 23	3brtr3d 4817	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ≤ 0 )
25	2	ad2antrr 705	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝐺 ∈ oMnd)
26	10	ad2antrr 705	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝐺 ∈ Grp)
27		simplr 752	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝑋 ∈ 𝐵)
28	26, 27, 13	syl2anc 573	. . . 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 471	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → (𝐼‘𝑋) ≤ 0 )
31	5, 16, 17	omndadd 30046	. . . 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 1489	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) ≤ ( 0 (+g‘𝐺)𝑋))
33	5, 17, 6, 12	grplinv 17676	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
34	26, 27, 33	syl2anc 573	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
35	5, 17, 6	grplid 17660	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
36	26, 27, 35	syl2anc 573	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
37	32, 34, 36	3brtr3d 4817	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 0 ≤ 𝑋)
38	24, 37	impbida 802	1 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 ≤ 𝑋 ↔ (𝐼‘𝑋) ≤ 0 ))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145   class class class wbr 4786  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  +_gcplusg 16149  lecple 16156  0_gc0g 16308  Mndcmnd 17502  Grpcgrp 17630  inv_gcminusg 17631  oMndcomnd 30037  oGrpcogrp 30038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-omnd 30039  df-ogrp 30040

This theorem is referenced by: (None)