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Theorem ogrpinv0le 30724
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 30711 . . . . . 6 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 499 . . . . 5 (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
32ad2antrr 724 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝐺 ∈ oMnd)
4 omndmnd 30713 . . . . 5 (𝐺 ∈ oMnd → 𝐺 ∈ Mnd)
5 ogrpsub.0 . . . . . 6 𝐵 = (Base‘𝐺)
6 ogrpinv.3 . . . . . 6 0 = (0g𝐺)
75, 6mndidcl 17905 . . . . 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 30712 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
1110ad2antrr 724 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝐺 ∈ Grp)
12 ogrpinv.2 . . . . . 6 𝐼 = (invg𝐺)
135, 12grpinvcl 18130 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
1411, 9, 13syl2anc 586 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝐼𝑋) ∈ 𝐵)
15 simpr 487 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 0 𝑋)
16 ogrpsub.1 . . . . 5 = (le‘𝐺)
17 eqid 2820 . . . . 5 (+g𝐺) = (+g𝐺)
185, 16, 17omndadd 30715 . . . 4 ((𝐺 ∈ oMnd ∧ ( 0𝐵𝑋𝐵 ∧ (𝐼𝑋) ∈ 𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
193, 8, 9, 14, 15, 18syl131anc 1379 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
205, 17, 6grplid 18112 . . . 4 ((𝐺 ∈ Grp ∧ (𝐼𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
2111, 14, 20syl2anc 586 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
225, 17, 6, 12grprinv 18132 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2311, 9, 22syl2anc 586 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2419, 21, 233brtr3d 5073 . 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 586 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → (𝐼𝑋) ∈ 𝐵)
2925, 4, 73syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 0𝐵)
30 simpr 487 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → (𝐼𝑋) 0 )
315, 16, 17omndadd 30715 . . . 4 ((𝐺 ∈ oMnd ∧ ((𝐼𝑋) ∈ 𝐵0𝐵𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) ( 0 (+g𝐺)𝑋))
3225, 28, 29, 27, 30, 31syl131anc 1379 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) ( 0 (+g𝐺)𝑋))
335, 17, 6, 12grplinv 18131 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
3426, 27, 33syl2anc 586 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
355, 17, 6grplid 18112 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 (+g𝐺)𝑋) = 𝑋)
3626, 27, 35syl2anc 586 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ( 0 (+g𝐺)𝑋) = 𝑋)
3732, 34, 363brtr3d 5073 . 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 208  wa 398   = wceq 1537  wcel 2114   class class class wbr 5042  cfv 6331  (class class class)co 7133  Basecbs 16462  +gcplusg 16544  lecple 16551  0gc0g 16692  Mndcmnd 17890  Grpcgrp 18082  invgcminusg 18083  oMndcomnd 30706  oGrpcogrp 30707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-fv 6339  df-riota 7091  df-ov 7136  df-0g 16694  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-grp 18085  df-minusg 18086  df-omnd 30708  df-ogrp 30709
This theorem is referenced by: (None)
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