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Theorem ogrpinv0lt 33082
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 767 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 𝐺 ∈ oGrp)
2 ogrpgrp 33063 . . . . . 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 18996 . . . . 5 (𝐺 ∈ Grp → 0𝐵)
73, 6syl 17 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 0𝐵)
8 simplr 769 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 𝑋𝐵)
9 ogrpinvlt.2 . . . . . 6 𝐼 = (invg𝐺)
104, 9grpinvcl 19018 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
113, 8, 10syl2anc 584 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → (𝐼𝑋) ∈ 𝐵)
12 simpr 484 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 0 < 𝑋)
13 ogrpinvlt.1 . . . . 5 < = (lt‘𝐺)
14 eqid 2735 . . . . 5 (+g𝐺) = (+g𝐺)
154, 13, 14ogrpaddlt 33077 . . . 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 18998 . . . 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 19021 . . . 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 5179 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → (𝐼𝑋) < 0 )
22 simpll 767 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝐺 ∈ oGrp)
2322, 2syl 17 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝐺 ∈ Grp)
24 simplr 769 . . . . 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 484 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → (𝐼𝑋) < 0 )
284, 13, 14ogrpaddlt 33077 . . . 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 19020 . . . 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 18998 . . . 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 5179 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 0 < 𝑋)
3521, 34impbida 801 1 ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ (𝐼𝑋) < 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  ltcplt 18366  Grpcgrp 18964  invgcminusg 18965  oGrpcogrp 33058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-plt 18388  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-omnd 33059  df-ogrp 33060
This theorem is referenced by:  archirngz  33179
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