Theorem ogrpinvOLD 30259

Description: In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 30-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

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
ogrpinvOLD	⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ≤ 0 )

Proof of Theorem ogrpinvOLD

Step	Hyp	Ref	Expression
1		isogrp 30246	. . . . 5 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simprbi 492	. . . 4 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
3	2	3ad2ant1 1169	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → 𝐺 ∈ oMnd)
4	1	simplbi 493	. . . . 5 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
5	4	3ad2ant1 1169	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → 𝐺 ∈ Grp)
6		ogrpsub.0	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
7		ogrpinv.3	. . . . 5 ⊢ 0 = (0g‘𝐺)
8	6, 7	grpidcl 17803	. . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
9	5, 8	syl 17	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → 0 ∈ 𝐵)
10		simp2 1173	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → 𝑋 ∈ 𝐵)
11		ogrpinv.2	. . . . 5 ⊢ 𝐼 = (invg‘𝐺)
12	6, 11	grpinvcl 17820	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵)
13	5, 10, 12	syl2anc 581	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ∈ 𝐵)
14		simp3 1174	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → 0 ≤ 𝑋)
15		ogrpsub.1	. . . 4 ⊢ ≤ = (le‘𝐺)
16		eqid 2824	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
17	6, 15, 16	omndadd 30250	. . 3 ⊢ ((𝐺 ∈ oMnd ∧ ( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝐼‘𝑋) ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) ≤ (𝑋(+g‘𝐺)(𝐼‘𝑋)))
18	3, 9, 10, 13, 14, 17	syl131anc 1508	. 2 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) ≤ (𝑋(+g‘𝐺)(𝐼‘𝑋)))
19	6, 16, 7	grplid 17805	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
20	5, 13, 19	syl2anc 581	. 2 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋))
21	6, 16, 7, 11	grprinv 17822	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
22	5, 10, 21	syl2anc 581	. 2 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 )
23	18, 20, 22	3brtr3d 4903	1 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ≤ 0 )

Syntax hints:   → wi 4   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166   class class class wbr 4872  ‘cfv 6122  (class class class)co 6904  Basecbs 16221  +_gcplusg 16304  lecple 16311  0_gc0g 16452  Grpcgrp 17775  inv_gcminusg 17776  oMndcomnd 30241  oGrpcogrp 30242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-0g 16454  df-mgm 17594  df-sgrp 17636  df-mnd 17647  df-grp 17778  df-minusg 17779  df-omnd 30243  df-ogrp 30244

This theorem is referenced by: (None)