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Theorem ogrpinv0le 31386
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 31373 . . . . . 6 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 498 . . . . 5 (𝐺 ∈ oGrp β†’ 𝐺 ∈ oMnd)
32ad2antrr 724 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ 𝐺 ∈ oMnd)
4 omndmnd 31375 . . . . 5 (𝐺 ∈ oMnd β†’ 𝐺 ∈ Mnd)
5 ogrpsub.0 . . . . . 6 𝐡 = (Baseβ€˜πΊ)
6 ogrpinv.3 . . . . . 6 0 = (0gβ€˜πΊ)
75, 6mndidcl 18445 . . . . 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 31374 . . . . . 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 18672 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐡) β†’ (πΌβ€˜π‘‹) ∈ 𝐡)
1411, 9, 13syl2anc 585 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ (πΌβ€˜π‘‹) ∈ 𝐡)
15 simpr 486 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ 0 ≀ 𝑋)
16 ogrpsub.1 . . . . 5 ≀ = (leβ€˜πΊ)
17 eqid 2736 . . . . 5 (+gβ€˜πΊ) = (+gβ€˜πΊ)
185, 16, 17omndadd 31377 . . . 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 18654 . . . 4 ((𝐺 ∈ Grp ∧ (πΌβ€˜π‘‹) ∈ 𝐡) β†’ ( 0 (+gβ€˜πΊ)(πΌβ€˜π‘‹)) = (πΌβ€˜π‘‹))
2111, 14, 20syl2anc 585 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ ( 0 (+gβ€˜πΊ)(πΌβ€˜π‘‹)) = (πΌβ€˜π‘‹))
225, 17, 6, 12grprinv 18674 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐡) β†’ (𝑋(+gβ€˜πΊ)(πΌβ€˜π‘‹)) = 0 )
2311, 9, 22syl2anc 585 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ 0 ≀ 𝑋) β†’ (𝑋(+gβ€˜πΊ)(πΌβ€˜π‘‹)) = 0 )
2419, 21, 233brtr3d 5112 . 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 585 . . . 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 486 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ (πΌβ€˜π‘‹) ≀ 0 )
315, 16, 17omndadd 31377 . . . 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 18673 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐡) β†’ ((πΌβ€˜π‘‹)(+gβ€˜πΊ)𝑋) = 0 )
3426, 27, 33syl2anc 585 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ ((πΌβ€˜π‘‹)(+gβ€˜πΊ)𝑋) = 0 )
355, 17, 6grplid 18654 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐡) β†’ ( 0 (+gβ€˜πΊ)𝑋) = 𝑋)
3626, 27, 35syl2anc 585 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) ∧ (πΌβ€˜π‘‹) ≀ 0 ) β†’ ( 0 (+gβ€˜πΊ)𝑋) = 𝑋)
3732, 34, 363brtr3d 5112 . 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 397   = wceq 1539   ∈ wcel 2104   class class class wbr 5081  β€˜cfv 6458  (class class class)co 7307  Basecbs 16957  +gcplusg 17007  lecple 17014  0gc0g 17195  Mndcmnd 18430  Grpcgrp 18622  invgcminusg 18623  oMndcomnd 31368  oGrpcogrp 31369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fv 6466  df-riota 7264  df-ov 7310  df-0g 17197  df-mgm 18371  df-sgrp 18420  df-mnd 18431  df-grp 18625  df-minusg 18626  df-omnd 31370  df-ogrp 31371
This theorem is referenced by: (None)
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