Theorem ogrpinv0le 20350

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 20338	. . . . . 6 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simprbi 503	. . . . 5 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
3	2	ad2antrr 739	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝐺 ∈ oMnd)
4		omndmnd 20340	. . . . 5 ⊢ (𝐺 ∈ oMnd → 𝐺 ∈ Mnd)
5		ogrpsub.0	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
6		ogrpinv.3	. . . . . 6 ⊢ 0 = (0g‘𝐺)
7	5, 6	mndidcl 18939	. . . . 5 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
8	3, 4, 7	3syl 19	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 0 ∈ 𝐵)
9		simplr 781	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝑋 ∈ 𝐵)
10		ogrpgrp 20339	. . . . . 6 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
11	10	ad2antrr 739	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 𝐺 ∈ Grp)
12		ogrpinv.2	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
13	5, 12	grpinvcl 19198	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵)
14	11, 9, 13	syl2anc 596	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ∈ 𝐵)
15		simpr 490	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → 0 ≤ 𝑋)
16		ogrpsub.1	. . . . 5 ⊢ ≤ = (le‘𝐺)
17		eqid 2760	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
18	5, 16, 17	omndadd 20342	. . . 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 1410	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) ≤ (𝑋(+g‘𝐺)(𝐼‘𝑋)))
20	5, 17, 6	grplid 19178	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
21	11, 14, 20	syl2anc 596	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
22	5, 17, 6, 12	grprinv 19201	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
23	11, 9, 22	syl2anc 596	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
24	19, 21, 23	3brtr3d 5135	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ≤ 0 )
25	2	ad2antrr 739	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝐺 ∈ oMnd)
26	10	ad2antrr 739	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝐺 ∈ Grp)
27		simplr 781	. . . . 5 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 𝑋 ∈ 𝐵)
28	26, 27, 13	syl2anc 596	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → (𝐼‘𝑋) ∈ 𝐵)
29	25, 4, 7	3syl 19	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 0 ∈ 𝐵)
30		simpr 490	. . . 4 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → (𝐼‘𝑋) ≤ 0 )
31	5, 16, 17	omndadd 20342	. . . 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 1410	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) ≤ ( 0 (+g‘𝐺)𝑋))
33	5, 17, 6, 12	grplinv 19200	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
34	26, 27, 33	syl2anc 596	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ((𝐼‘𝑋)(+g‘𝐺)𝑋) = 0 )
35	5, 17, 6	grplid 19178	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
36	26, 27, 35	syl2anc 596	. . 3 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → ( 0 (+g‘𝐺)𝑋) = 𝑋)
37	32, 34, 36	3brtr3d 5135	. 2 ⊢ (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ (𝐼‘𝑋) ≤ 0 ) → 0 ≤ 𝑋)
38	24, 37	impbida 813	1 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 ≤ 𝑋 ↔ (𝐼‘𝑋) ≤ 0 ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  lecple 17435  0_gc0g 17610  Mndcmnd 18923  Grpcgrp 19144  inv_gcminusg 19145  oMndcomnd 20333  oGrpcogrp 20334

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-riota 7377  df-ov 7423  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-omnd 20335  df-ogrp 20336

This theorem is used by: (None)