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Theorem ogrpinv0lt 30759
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 766 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 𝐺 ∈ oGrp)
2 ogrpgrp 30740 . . . . . 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 18131 . . . . 5 (𝐺 ∈ Grp → 0𝐵)
73, 6syl 17 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 0𝐵)
8 simplr 768 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 𝑋𝐵)
9 ogrpinvlt.2 . . . . . 6 𝐼 = (invg𝐺)
104, 9grpinvcl 18151 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
113, 8, 10syl2anc 587 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → (𝐼𝑋) ∈ 𝐵)
12 simpr 488 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 0 < 𝑋)
13 ogrpinvlt.1 . . . . 5 < = (lt‘𝐺)
14 eqid 2824 . . . . 5 (+g𝐺) = (+g𝐺)
154, 13, 14ogrpaddlt 30754 . . . 4 ((𝐺 ∈ oGrp ∧ ( 0𝐵𝑋𝐵 ∧ (𝐼𝑋) ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) < (𝑋(+g𝐺)(𝐼𝑋)))
161, 7, 8, 11, 12, 15syl131anc 1380 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) < (𝑋(+g𝐺)(𝐼𝑋)))
174, 14, 5grplid 18133 . . . 4 ((𝐺 ∈ Grp ∧ (𝐼𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
183, 11, 17syl2anc 587 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
194, 14, 5, 9grprinv 18153 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
203, 8, 19syl2anc 587 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2116, 18, 203brtr3d 5083 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → (𝐼𝑋) < 0 )
22 simpll 766 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝐺 ∈ oGrp)
2322, 2syl 17 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝐺 ∈ Grp)
24 simplr 768 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝑋𝐵)
2523, 24, 10syl2anc 587 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → (𝐼𝑋) ∈ 𝐵)
2622, 2, 63syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 0𝐵)
27 simpr 488 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → (𝐼𝑋) < 0 )
284, 13, 14ogrpaddlt 30754 . . . 4 ((𝐺 ∈ oGrp ∧ ((𝐼𝑋) ∈ 𝐵0𝐵𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) < ( 0 (+g𝐺)𝑋))
2922, 25, 26, 24, 27, 28syl131anc 1380 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) < ( 0 (+g𝐺)𝑋))
304, 14, 5, 9grplinv 18152 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
3123, 24, 30syl2anc 587 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
324, 14, 5grplid 18133 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 (+g𝐺)𝑋) = 𝑋)
3323, 24, 32syl2anc 587 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ( 0 (+g𝐺)𝑋) = 𝑋)
3429, 31, 333brtr3d 5083 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 0 < 𝑋)
3521, 34impbida 800 1 ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ (𝐼𝑋) < 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  0gc0g 16713  ltcplt 17551  Grpcgrp 18103  invgcminusg 18104  oGrpcogrp 30735
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-riota 7107  df-ov 7152  df-0g 16715  df-plt 17568  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-omnd 30736  df-ogrp 30737
This theorem is referenced by:  archirngz  30854
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