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Theorem ogrpinv0lt 29697
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 789 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 𝐺 ∈ oGrp)
2 ogrpgrp 29677 . . . . . 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 17431 . . . . 5 (𝐺 ∈ Grp → 0𝐵)
73, 6syl 17 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 0𝐵)
8 simplr 791 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 𝑋𝐵)
9 ogrpinvlt.2 . . . . . 6 𝐼 = (invg𝐺)
104, 9grpinvcl 17448 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
113, 8, 10syl2anc 692 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → (𝐼𝑋) ∈ 𝐵)
12 simpr 477 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → 0 < 𝑋)
13 ogrpinvlt.1 . . . . 5 < = (lt‘𝐺)
14 eqid 2620 . . . . 5 (+g𝐺) = (+g𝐺)
154, 13, 14ogrpaddlt 29692 . . . 4 ((𝐺 ∈ oGrp ∧ ( 0𝐵𝑋𝐵 ∧ (𝐼𝑋) ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) < (𝑋(+g𝐺)(𝐼𝑋)))
161, 7, 8, 11, 12, 15syl131anc 1337 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) < (𝑋(+g𝐺)(𝐼𝑋)))
174, 14, 5grplid 17433 . . . 4 ((𝐺 ∈ Grp ∧ (𝐼𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
183, 11, 17syl2anc 692 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
194, 14, 5, 9grprinv 17450 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
203, 8, 19syl2anc 692 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2116, 18, 203brtr3d 4675 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → (𝐼𝑋) < 0 )
22 simpll 789 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝐺 ∈ oGrp)
2322, 2syl 17 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝐺 ∈ Grp)
24 simplr 791 . . . . 5 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 𝑋𝐵)
2523, 24, 10syl2anc 692 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → (𝐼𝑋) ∈ 𝐵)
2622, 2, 63syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 0𝐵)
27 simpr 477 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → (𝐼𝑋) < 0 )
284, 13, 14ogrpaddlt 29692 . . . 4 ((𝐺 ∈ oGrp ∧ ((𝐼𝑋) ∈ 𝐵0𝐵𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) < ( 0 (+g𝐺)𝑋))
2922, 25, 26, 24, 27, 28syl131anc 1337 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) < ( 0 (+g𝐺)𝑋))
304, 14, 5, 9grplinv 17449 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
3123, 24, 30syl2anc 692 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ((𝐼𝑋)(+g𝐺)𝑋) = 0 )
324, 14, 5grplid 17433 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 (+g𝐺)𝑋) = 𝑋)
3323, 24, 32syl2anc 692 . . 3 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → ( 0 (+g𝐺)𝑋) = 𝑋)
3429, 31, 333brtr3d 4675 . 2 (((𝐺 ∈ oGrp ∧ 𝑋𝐵) ∧ (𝐼𝑋) < 0 ) → 0 < 𝑋)
3521, 34impbida 876 1 ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ (𝐼𝑋) < 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988   class class class wbr 4644  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  0gc0g 16081  ltcplt 16922  Grpcgrp 17403  invgcminusg 17404  oGrpcogrp 29672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-0g 16083  df-plt 16939  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-minusg 17407  df-omnd 29673  df-ogrp 29674
This theorem is referenced by:  archirngz  29717
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