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Theorem ogrpinv0lt 31348
Description: In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
Hypotheses
Ref Expression
ogrpinvlt.0 𝐵 = (Base‘𝐺)
ogrpinvlt.1 < = (lt‘𝐺)
ogrpinvlt.2 𝐼 = (invg𝐺)
ogrpinv0lt.3 0 = (0g𝐺)
Assertion
Ref Expression
ogrpinv0lt ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ (𝐼𝑋) < 0 ))

Proof of Theorem ogrpinv0lt
StepHypRef Expression
1 simpll 764 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 𝐺 ∈ oGrp)
2 ogrpgrp 31329 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
31, 2syl 17 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 𝐺 ∈ Grp)
4 ogrpinvlt.0 . . . . . 6 𝐵 = (Base‘𝐺)
5 ogrpinv0lt.3 . . . . . 6 0 = (0g𝐺)
64, 5grpidcl 18607 . . . . 5 (𝐺 ∈ Grp → 0𝐵)
73, 6syl 17 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 0𝐵)
8 simplr 766 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 𝑋𝐵)
9 ogrpinvlt.2 . . . . . 6 𝐼 = (invg𝐺)
104, 9grpinvcl 18627 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
113, 8, 10syl2anc 584 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → (𝐼𝑋) ∈ 𝐵)
12 simpr 485 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 0 < 𝑋)
13 ogrpinvlt.1 . . . . 5 < = (lt‘𝐺)
14 eqid 2738 . . . . 5 (+g𝐺) = (+g𝐺)
154, 13, 14ogrpaddlt 31343 . . . 4 ((𝐺 ∈ oGrp ∧ ( 0𝐵𝑋𝐵 ∧ (𝐼𝑋) ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) < (𝑋(+g𝐺)(𝐼𝑋)))
161, 7, 8, 11, 12, 15syl131anc 1382 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) < (𝑋(+g𝐺)(𝐼𝑋)))
174, 14, 5grplid 18609 . . . 4 ((𝐺 ∈ Grp ∧ (𝐼𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
183, 11, 17syl2anc 584 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
194, 14, 5, 9grprinv 18629 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
203, 8, 19syl2anc 584 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2116, 18, 203brtr3d 5105 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → (𝐼𝑋) < 0 )
22 simpll 764 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝐺 ∈ oGrp)
2322, 2syl 17 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝐺 ∈ Grp)
24 simplr 766 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝑋𝐵)
2523, 24, 10syl2anc 584 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → (𝐼𝑋) ∈ 𝐵)
2622, 2, 63syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 0𝐵)
27 simpr 485 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → (𝐼𝑋) < 0 )
284, 13, 14ogrpaddlt 31343 . . . 4 ((𝐺 ∈ oGrp ∧ ((𝐼𝑋) ∈ 𝐵0𝐵𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) < ( 0 (+g𝐺)𝑋))
2922, 25, 26, 24, 27, 28syl131anc 1382 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) < ( 0 (+g𝐺)𝑋))
304, 14, 5, 9grplinv 18628 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
3123, 24, 30syl2anc 584 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
324, 14, 5grplid 18609 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 (+g𝐺)𝑋) = 𝑋)
3323, 24, 32syl2anc 584 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ( 0 (+g𝐺)𝑋) = 𝑋)
3429, 31, 333brtr3d 5105 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 0 < 𝑋)
3521, 34impbida 798 1 ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ (𝐼𝑋) < 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106   class class class wbr 5074  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  0gc0g 17150  ltcplt 18026  Grpcgrp 18577  invgcminusg 18578  oGrpcogrp 31324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-riota 7232  df-ov 7278  df-0g 17152  df-plt 18048  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581  df-omnd 31325  df-ogrp 31326
This theorem is referenced by:  archirngz  31443
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