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Theorem ogrpinv0le 32953
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 = (invg𝐺)
ogrpinv.3 0 = (0g𝐺)
Assertion
Ref Expression
ogrpinv0le ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 𝑋 ↔ (𝐼𝑋) 0 ))

Proof of Theorem ogrpinv0le
StepHypRef Expression
1 isogrp 32940 . . . . . 6 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 495 . . . . 5 (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
32ad2antrr 724 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝐺 ∈ oMnd)
4 omndmnd 32942 . . . . 5 (𝐺 ∈ oMnd → 𝐺 ∈ Mnd)
5 ogrpsub.0 . . . . . 6 𝐵 = (Base‘𝐺)
6 ogrpinv.3 . . . . . 6 0 = (0g𝐺)
75, 6mndidcl 18736 . . . . 5 (𝐺 ∈ Mnd → 0𝐵)
83, 4, 73syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 0𝐵)
9 simplr 767 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝑋𝐵)
10 ogrpgrp 32941 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
1110ad2antrr 724 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝐺 ∈ Grp)
12 ogrpinv.2 . . . . . 6 𝐼 = (invg𝐺)
135, 12grpinvcl 18976 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
1411, 9, 13syl2anc 582 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝐼𝑋) ∈ 𝐵)
15 simpr 483 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 0 𝑋)
16 ogrpsub.1 . . . . 5 = (le‘𝐺)
17 eqid 2726 . . . . 5 (+g𝐺) = (+g𝐺)
185, 16, 17omndadd 32944 . . . 4 ((𝐺 ∈ oMnd ∧ ( 0𝐵𝑋𝐵 ∧ (𝐼𝑋) ∈ 𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
193, 8, 9, 14, 15, 18syl131anc 1380 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
205, 17, 6grplid 18956 . . . 4 ((𝐺 ∈ Grp ∧ (𝐼𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
2111, 14, 20syl2anc 582 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
225, 17, 6, 12grprinv 18979 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2311, 9, 22syl2anc 582 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2419, 21, 233brtr3d 5176 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝐼𝑋) 0 )
252ad2antrr 724 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 𝐺 ∈ oMnd)
2610ad2antrr 724 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 𝐺 ∈ Grp)
27 simplr 767 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 𝑋𝐵)
2826, 27, 13syl2anc 582 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → (𝐼𝑋) ∈ 𝐵)
2925, 4, 73syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 0𝐵)
30 simpr 483 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → (𝐼𝑋) 0 )
315, 16, 17omndadd 32944 . . . 4 ((𝐺 ∈ oMnd ∧ ((𝐼𝑋) ∈ 𝐵0𝐵𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) ( 0 (+g𝐺)𝑋))
3225, 28, 29, 27, 30, 31syl131anc 1380 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) ( 0 (+g𝐺)𝑋))
335, 17, 6, 12grplinv 18978 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
3426, 27, 33syl2anc 582 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
355, 17, 6grplid 18956 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 (+g𝐺)𝑋) = 𝑋)
3626, 27, 35syl2anc 582 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ( 0 (+g𝐺)𝑋) = 𝑋)
3732, 34, 363brtr3d 5176 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 0 𝑋)
3824, 37impbida 799 1 ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 𝑋 ↔ (𝐼𝑋) 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099   class class class wbr 5145  cfv 6545  (class class class)co 7415  Basecbs 17207  +gcplusg 17260  lecple 17267  0gc0g 17448  Mndcmnd 18721  Grpcgrp 18922  invgcminusg 18923  oMndcomnd 32935  oGrpcogrp 32936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-fv 6553  df-riota 7371  df-ov 7418  df-0g 17450  df-mgm 18627  df-sgrp 18706  df-mnd 18722  df-grp 18925  df-minusg 18926  df-omnd 32937  df-ogrp 32938
This theorem is referenced by: (None)
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