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Theorem ogrpinv0le 32274
Description: In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
Hypotheses
Ref Expression
ogrpsub.0 𝐡 = (Baseβ€˜πΊ)
ogrpsub.1 ≀ = (leβ€˜πΊ)
ogrpinv.2 𝐼 = (invgβ€˜πΊ)
ogrpinv.3 0 = (0gβ€˜πΊ)
Assertion
Ref Expression
ogrpinv0le ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) β†’ ( 0 ≀ 𝑋 ↔ (πΌβ€˜π‘‹) ≀ 0 ))

Proof of Theorem ogrpinv0le
StepHypRef Expression
1 isogrp 32261 . . . . . 6 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 497 . . . . 5 (𝐺 ∈ oGrp β†’ 𝐺 ∈ oMnd)
32ad2antrr 724 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ 𝐺 ∈ oMnd)
4 omndmnd 32263 . . . . 5 (𝐺 ∈ oMnd β†’ 𝐺 ∈ Mnd)
5 ogrpsub.0 . . . . . 6 𝐡 = (Baseβ€˜πΊ)
6 ogrpinv.3 . . . . . 6 0 = (0gβ€˜πΊ)
75, 6mndidcl 18642 . . . . 5 (𝐺 ∈ Mnd β†’ 0 ∈ 𝐡)
83, 4, 73syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ 0 ∈ 𝐡)
9 simplr 767 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ 𝑋 ∈ 𝐡)
10 ogrpgrp 32262 . . . . . 6 (𝐺 ∈ oGrp β†’ 𝐺 ∈ Grp)
1110ad2antrr 724 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ 𝐺 ∈ Grp)
12 ogrpinv.2 . . . . . 6 𝐼 = (invgβ€˜πΊ)
135, 12grpinvcl 18874 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐡) β†’ (πΌβ€˜π‘‹) ∈ 𝐡)
1411, 9, 13syl2anc 584 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ (πΌβ€˜π‘‹) ∈ 𝐡)
15 simpr 485 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ 0 ≀ 𝑋)
16 ogrpsub.1 . . . . 5 ≀ = (leβ€˜πΊ)
17 eqid 2732 . . . . 5 (+gβ€˜πΊ) = (+gβ€˜πΊ)
185, 16, 17omndadd 32265 . . . 4 ((𝐺 ∈ oMnd ∧ ( 0 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡 ∧ (πΌβ€˜π‘‹) ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ ( 0 (+gβ€˜πΊ)(πΌβ€˜π‘‹)) ≀ (𝑋(+gβ€˜πΊ)(πΌβ€˜π‘‹)))
193, 8, 9, 14, 15, 18syl131anc 1383 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ ( 0 (+gβ€˜πΊ)(πΌβ€˜π‘‹)) ≀ (𝑋(+gβ€˜πΊ)(πΌβ€˜π‘‹)))
205, 17, 6grplid 18854 . . . 4 ((𝐺 ∈ Grp ∧ (πΌβ€˜π‘‹) ∈ 𝐡) β†’ ( 0 (+gβ€˜πΊ)(πΌβ€˜π‘‹)) = (πΌβ€˜π‘‹))
2111, 14, 20syl2anc 584 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ ( 0 (+gβ€˜πΊ)(πΌβ€˜π‘‹)) = (πΌβ€˜π‘‹))
225, 17, 6, 12grprinv 18877 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐡) β†’ (𝑋(+gβ€˜πΊ)(πΌβ€˜π‘‹)) = 0 )
2311, 9, 22syl2anc 584 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ (𝑋(+gβ€˜πΊ)(πΌβ€˜π‘‹)) = 0 )
2419, 21, 233brtr3d 5179 . 2 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ (πΌβ€˜π‘‹) ≀ 0 )
252ad2antrr 724 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ 𝐺 ∈ oMnd)
2610ad2antrr 724 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ 𝐺 ∈ Grp)
27 simplr 767 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ 𝑋 ∈ 𝐡)
2826, 27, 13syl2anc 584 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ (πΌβ€˜π‘‹) ∈ 𝐡)
2925, 4, 73syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ 0 ∈ 𝐡)
30 simpr 485 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ (πΌβ€˜π‘‹) ≀ 0 )
315, 16, 17omndadd 32265 . . . 4 ((𝐺 ∈ oMnd ∧ ((πΌβ€˜π‘‹) ∈ 𝐡 ∧ 0 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ ((πΌβ€˜π‘‹)(+gβ€˜πΊ)𝑋) ≀ ( 0 (+gβ€˜πΊ)𝑋))
3225, 28, 29, 27, 30, 31syl131anc 1383 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ ((πΌβ€˜π‘‹)(+gβ€˜πΊ)𝑋) ≀ ( 0 (+gβ€˜πΊ)𝑋))
335, 17, 6, 12grplinv 18876 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐡) β†’ ((πΌβ€˜π‘‹)(+gβ€˜πΊ)𝑋) = 0 )
3426, 27, 33syl2anc 584 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ ((πΌβ€˜π‘‹)(+gβ€˜πΊ)𝑋) = 0 )
355, 17, 6grplid 18854 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐡) β†’ ( 0 (+gβ€˜πΊ)𝑋) = 𝑋)
3626, 27, 35syl2anc 584 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ ( 0 (+gβ€˜πΊ)𝑋) = 𝑋)
3732, 34, 363brtr3d 5179 . 2 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ 0 ≀ 𝑋)
3824, 37impbida 799 1 ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) β†’ ( 0 ≀ 𝑋 ↔ (πΌβ€˜π‘‹) ≀ 0 ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  +gcplusg 17199  lecple 17206  0gc0g 17387  Mndcmnd 18627  Grpcgrp 18821  invgcminusg 18822  oMndcomnd 32256  oGrpcogrp 32257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7367  df-ov 7414  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-grp 18824  df-minusg 18825  df-omnd 32258  df-ogrp 32259
This theorem is referenced by: (None)
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