Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ogrpinv0le Structured version   Visualization version   GIF version

Theorem ogrpinv0le 29047
 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 29033 . . . . . 6 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 479 . . . . 5 (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
32ad2antrr 758 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝐺 ∈ oMnd)
4 omndmnd 29035 . . . . 5 (𝐺 ∈ oMnd → 𝐺 ∈ Mnd)
5 ogrpsub.0 . . . . . 6 𝐵 = (Base‘𝐺)
6 ogrpinv.3 . . . . . 6 0 = (0g𝐺)
75, 6mndidcl 17131 . . . . 5 (𝐺 ∈ Mnd → 0𝐵)
83, 4, 73syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 0𝐵)
9 simplr 788 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝑋𝐵)
10 ogrpgrp 29034 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
1110ad2antrr 758 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝐺 ∈ Grp)
12 ogrpinv.2 . . . . . 6 𝐼 = (invg𝐺)
135, 12grpinvcl 17290 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
1411, 9, 13syl2anc 691 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝐼𝑋) ∈ 𝐵)
15 simpr 476 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 0 𝑋)
16 ogrpsub.1 . . . . 5 = (le‘𝐺)
17 eqid 2610 . . . . 5 (+g𝐺) = (+g𝐺)
185, 16, 17omndadd 29037 . . . 4 ((𝐺 ∈ oMnd ∧ ( 0𝐵𝑋𝐵 ∧ (𝐼𝑋) ∈ 𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
193, 8, 9, 14, 15, 18syl131anc 1331 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
205, 17, 6grplid 17275 . . . 4 ((𝐺 ∈ Grp ∧ (𝐼𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
2111, 14, 20syl2anc 691 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
225, 17, 6, 12grprinv 17292 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2311, 9, 22syl2anc 691 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2419, 21, 233brtr3d 4614 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝐼𝑋) 0 )
252ad2antrr 758 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 𝐺 ∈ oMnd)
2610ad2antrr 758 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 𝐺 ∈ Grp)
27 simplr 788 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 𝑋𝐵)
2826, 27, 13syl2anc 691 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → (𝐼𝑋) ∈ 𝐵)
2925, 4, 73syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 0𝐵)
30 simpr 476 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → (𝐼𝑋) 0 )
315, 16, 17omndadd 29037 . . . 4 ((𝐺 ∈ oMnd ∧ ((𝐼𝑋) ∈ 𝐵0𝐵𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) ( 0 (+g𝐺)𝑋))
3225, 28, 29, 27, 30, 31syl131anc 1331 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) ( 0 (+g𝐺)𝑋))
335, 17, 6, 12grplinv 17291 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
3426, 27, 33syl2anc 691 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
355, 17, 6grplid 17275 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 (+g𝐺)𝑋) = 𝑋)
3626, 27, 35syl2anc 691 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ( 0 (+g𝐺)𝑋) = 𝑋)
3732, 34, 363brtr3d 4614 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 0 𝑋)
3824, 37impbida 873 1 ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 𝑋 ↔ (𝐼𝑋) 0 ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  lecple 15775  0gc0g 15923  Mndcmnd 17117  Grpcgrp 17245  invgcminusg 17246  oMndcomnd 29028  oGrpcogrp 29029 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-omnd 29030  df-ogrp 29031 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator