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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 = (invg𝐺)
ogrpinv.3 0 = (0g𝐺)
Assertion
Ref Expression
ogrpinvOLD ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → (𝐼𝑋) 0 )

Proof of Theorem ogrpinvOLD
StepHypRef Expression
1 isogrp 30246 . . . . 5 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 492 . . . 4 (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
323ad2ant1 1169 . . 3 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → 𝐺 ∈ oMnd)
41simplbi 493 . . . . 5 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
543ad2ant1 1169 . . . 4 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → 𝐺 ∈ Grp)
6 ogrpsub.0 . . . . 5 𝐵 = (Base‘𝐺)
7 ogrpinv.3 . . . . 5 0 = (0g𝐺)
86, 7grpidcl 17803 . . . 4 (𝐺 ∈ Grp → 0𝐵)
95, 8syl 17 . . 3 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → 0𝐵)
10 simp2 1173 . . 3 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → 𝑋𝐵)
11 ogrpinv.2 . . . . 5 𝐼 = (invg𝐺)
126, 11grpinvcl 17820 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
135, 10, 12syl2anc 581 . . 3 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → (𝐼𝑋) ∈ 𝐵)
14 simp3 1174 . . 3 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → 0 𝑋)
15 ogrpsub.1 . . . 4 = (le‘𝐺)
16 eqid 2824 . . . 4 (+g𝐺) = (+g𝐺)
176, 15, 16omndadd 30250 . . 3 ((𝐺 ∈ oMnd ∧ ( 0𝐵𝑋𝐵 ∧ (𝐼𝑋) ∈ 𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
183, 9, 10, 13, 14, 17syl131anc 1508 . 2 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
196, 16, 7grplid 17805 . . 3 ((𝐺 ∈ Grp ∧ (𝐼𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
205, 13, 19syl2anc 581 . 2 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
216, 16, 7, 11grprinv 17822 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
225, 10, 21syl2anc 581 . 2 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2318, 20, 223brtr3d 4903 1 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → (𝐼𝑋) 0 )
 Colors of variables: wff setvar class 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  0gc0g 16452  Grpcgrp 17775  invgcminusg 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)
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