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Theorem ogrpinv0le 30250
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 30236 . . . . . 6 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 492 . . . . 5 (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
32ad2antrr 717 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝐺 ∈ oMnd)
4 omndmnd 30238 . . . . 5 (𝐺 ∈ oMnd → 𝐺 ∈ Mnd)
5 ogrpsub.0 . . . . . 6 𝐵 = (Base‘𝐺)
6 ogrpinv.3 . . . . . 6 0 = (0g𝐺)
75, 6mndidcl 17661 . . . . 5 (𝐺 ∈ Mnd → 0𝐵)
83, 4, 73syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 0𝐵)
9 simplr 785 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝑋𝐵)
10 ogrpgrp 30237 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
1110ad2antrr 717 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝐺 ∈ Grp)
12 ogrpinv.2 . . . . . 6 𝐼 = (invg𝐺)
135, 12grpinvcl 17821 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
1411, 9, 13syl2anc 579 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝐼𝑋) ∈ 𝐵)
15 simpr 479 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → 0 𝑋)
16 ogrpsub.1 . . . . 5 = (le‘𝐺)
17 eqid 2825 . . . . 5 (+g𝐺) = (+g𝐺)
185, 16, 17omndadd 30240 . . . 4 ((𝐺 ∈ oMnd ∧ ( 0𝐵𝑋𝐵 ∧ (𝐼𝑋) ∈ 𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
193, 8, 9, 14, 15, 18syl131anc 1506 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
205, 17, 6grplid 17806 . . . 4 ((𝐺 ∈ Grp ∧ (𝐼𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
2111, 14, 20syl2anc 579 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
225, 17, 6, 12grprinv 17823 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2311, 9, 22syl2anc 579 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2419, 21, 233brtr3d 4904 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 𝑋) → (𝐼𝑋) 0 )
252ad2antrr 717 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 𝐺 ∈ oMnd)
2610ad2antrr 717 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 𝐺 ∈ Grp)
27 simplr 785 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 𝑋𝐵)
2826, 27, 13syl2anc 579 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → (𝐼𝑋) ∈ 𝐵)
2925, 4, 73syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 0𝐵)
30 simpr 479 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → (𝐼𝑋) 0 )
315, 16, 17omndadd 30240 . . . 4 ((𝐺 ∈ oMnd ∧ ((𝐼𝑋) ∈ 𝐵0𝐵𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) ( 0 (+g𝐺)𝑋))
3225, 28, 29, 27, 30, 31syl131anc 1506 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) ( 0 (+g𝐺)𝑋))
335, 17, 6, 12grplinv 17822 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
3426, 27, 33syl2anc 579 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
355, 17, 6grplid 17806 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 (+g𝐺)𝑋) = 𝑋)
3626, 27, 35syl2anc 579 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → ( 0 (+g𝐺)𝑋) = 𝑋)
3732, 34, 363brtr3d 4904 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) 0 ) → 0 𝑋)
3824, 37impbida 835 1 ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 𝑋 ↔ (𝐼𝑋) 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164   class class class wbr 4873  cfv 6123  (class class class)co 6905  Basecbs 16222  +gcplusg 16305  lecple 16312  0gc0g 16453  Mndcmnd 17647  Grpcgrp 17776  invgcminusg 17777  oMndcomnd 30231  oGrpcogrp 30232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-0g 16455  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-grp 17779  df-minusg 17780  df-omnd 30233  df-ogrp 30234
This theorem is referenced by: (None)
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