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Theorem ogrpinv0lt 31930
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 765 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 𝐺 ∈ oGrp)
2 ogrpgrp 31911 . . . . . 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 18778 . . . . 5 (𝐺 ∈ Grp → 0𝐵)
73, 6syl 17 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 0𝐵)
8 simplr 767 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 𝑋𝐵)
9 ogrpinvlt.2 . . . . . 6 𝐼 = (invg𝐺)
104, 9grpinvcl 18798 . . . . 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 2736 . . . . 5 (+g𝐺) = (+g𝐺)
154, 13, 14ogrpaddlt 31925 . . . 4 ((𝐺 ∈ oGrp ∧ ( 0𝐵𝑋𝐵 ∧ (𝐼𝑋) ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) < (𝑋(+g𝐺)(𝐼𝑋)))
161, 7, 8, 11, 12, 15syl131anc 1383 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) < (𝑋(+g𝐺)(𝐼𝑋)))
174, 14, 5grplid 18780 . . . 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 18801 . . . 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 5136 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → (𝐼𝑋) < 0 )
22 simpll 765 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝐺 ∈ oGrp)
2322, 2syl 17 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝐺 ∈ Grp)
24 simplr 767 . . . . 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 31925 . . . 4 ((𝐺 ∈ oGrp ∧ ((𝐼𝑋) ∈ 𝐵0𝐵𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) < ( 0 (+g𝐺)𝑋))
2922, 25, 26, 24, 27, 28syl131anc 1383 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) < ( 0 (+g𝐺)𝑋))
304, 14, 5, 9grplinv 18800 . . . 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 18780 . . . 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 5136 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 0 < 𝑋)
3521, 34impbida 799 1 ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ (𝐼𝑋) < 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  0gc0g 17321  ltcplt 18197  Grpcgrp 18748  invgcminusg 18749  oGrpcogrp 31906
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-riota 7313  df-ov 7360  df-0g 17323  df-plt 18219  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-omnd 31907  df-ogrp 31908
This theorem is referenced by:  archirngz  32025
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