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Theorem ogrpinv0le 30056
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 30042 . . . . . 6 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 484 . . . . 5 (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
32ad2antrr 705 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝐺 ∈ oMnd)
4 omndmnd 30044 . . . . 5 (𝐺 ∈ oMnd → 𝐺 ∈ Mnd)
5 ogrpsub.0 . . . . . 6 𝐵 = (Base‘𝐺)
6 ogrpinv.3 . . . . . 6 0 = (0g𝐺)
75, 6mndidcl 17516 . . . . 5 (𝐺 ∈ Mnd → 0𝐵)
83, 4, 73syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 0𝐵)
9 simplr 752 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝑋𝐵)
10 ogrpgrp 30043 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
1110ad2antrr 705 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝐺 ∈ Grp)
12 ogrpinv.2 . . . . . 6 𝐼 = (invg𝐺)
135, 12grpinvcl 17675 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
1411, 9, 13syl2anc 573 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝐼𝑋) ∈ 𝐵)
15 simpr 471 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 0 𝑋)
16 ogrpsub.1 . . . . 5 = (le‘𝐺)
17 eqid 2771 . . . . 5 (+g𝐺) = (+g𝐺)
185, 16, 17omndadd 30046 . . . 4 ((𝐺 ∈ oMnd ∧ ( 0𝐵𝑋𝐵 ∧ (𝐼𝑋) ∈ 𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
193, 8, 9, 14, 15, 18syl131anc 1489 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
205, 17, 6grplid 17660 . . . 4 ((𝐺 ∈ Grp ∧ (𝐼𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
2111, 14, 20syl2anc 573 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
225, 17, 6, 12grprinv 17677 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2311, 9, 22syl2anc 573 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2419, 21, 233brtr3d 4817 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝐼𝑋) 0 )
252ad2antrr 705 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 𝐺 ∈ oMnd)
2610ad2antrr 705 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 𝐺 ∈ Grp)
27 simplr 752 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 𝑋𝐵)
2826, 27, 13syl2anc 573 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → (𝐼𝑋) ∈ 𝐵)
2925, 4, 73syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 0𝐵)
30 simpr 471 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → (𝐼𝑋) 0 )
315, 16, 17omndadd 30046 . . . 4 ((𝐺 ∈ oMnd ∧ ((𝐼𝑋) ∈ 𝐵0𝐵𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) ( 0 (+g𝐺)𝑋))
3225, 28, 29, 27, 30, 31syl131anc 1489 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) ( 0 (+g𝐺)𝑋))
335, 17, 6, 12grplinv 17676 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
3426, 27, 33syl2anc 573 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
355, 17, 6grplid 17660 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 (+g𝐺)𝑋) = 𝑋)
3626, 27, 35syl2anc 573 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ( 0 (+g𝐺)𝑋) = 𝑋)
3732, 34, 363brtr3d 4817 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 0 𝑋)
3824, 37impbida 802 1 ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 𝑋 ↔ (𝐼𝑋) 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  lecple 16156  0gc0g 16308  Mndcmnd 17502  Grpcgrp 17630  invgcminusg 17631  oMndcomnd 30037  oGrpcogrp 30038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-omnd 30039  df-ogrp 30040
This theorem is referenced by: (None)
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