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Theorem ogrpinvlt 29501
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)
Assertion
Ref Expression
ogrpinvlt (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝐼𝑌) < (𝐼𝑋)))

Proof of Theorem ogrpinvlt
StepHypRef Expression
1 simp1l 1083 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ oGrp)
2 simp2 1060 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 simp3 1061 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
4 ogrpgrp 29480 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
51, 4syl 17 . . . . 5 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ Grp)
6 ogrpinvlt.0 . . . . . 6 𝐵 = (Base‘𝐺)
7 ogrpinvlt.2 . . . . . 6 𝐼 = (invg𝐺)
86, 7grpinvcl 17383 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝐼𝑌) ∈ 𝐵)
95, 3, 8syl2anc 692 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝐼𝑌) ∈ 𝐵)
10 ogrpinvlt.1 . . . . 5 < = (lt‘𝐺)
11 eqid 2626 . . . . 5 (+g𝐺) = (+g𝐺)
126, 10, 11ogrpaddltbi 29496 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵 ∧ (𝐼𝑌) ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋(+g𝐺)(𝐼𝑌)) < (𝑌(+g𝐺)(𝐼𝑌))))
131, 2, 3, 9, 12syl13anc 1325 . . 3 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋(+g𝐺)(𝐼𝑌)) < (𝑌(+g𝐺)(𝐼𝑌))))
14 eqid 2626 . . . . . 6 (0g𝐺) = (0g𝐺)
156, 11, 14, 7grprinv 17385 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌(+g𝐺)(𝐼𝑌)) = (0g𝐺))
165, 3, 15syl2anc 692 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝑌(+g𝐺)(𝐼𝑌)) = (0g𝐺))
1716breq2d 4630 . . 3 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(+g𝐺)(𝐼𝑌)) < (𝑌(+g𝐺)(𝐼𝑌)) ↔ (𝑋(+g𝐺)(𝐼𝑌)) < (0g𝐺)))
18 simp1r 1084 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (oppg𝐺) ∈ oGrp)
196, 11grpcl 17346 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ (𝐼𝑌) ∈ 𝐵) → (𝑋(+g𝐺)(𝐼𝑌)) ∈ 𝐵)
205, 2, 9, 19syl3anc 1323 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝑋(+g𝐺)(𝐼𝑌)) ∈ 𝐵)
216, 14grpidcl 17366 . . . . 5 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
221, 4, 213syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (0g𝐺) ∈ 𝐵)
236, 7grpinvcl 17383 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
245, 2, 23syl2anc 692 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝐼𝑋) ∈ 𝐵)
256, 10, 11, 1, 18, 20, 22, 24ogrpaddltrbid 29498 . . 3 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(+g𝐺)(𝐼𝑌)) < (0g𝐺) ↔ ((𝐼𝑋)(+g𝐺)(𝑋(+g𝐺)(𝐼𝑌))) < ((𝐼𝑋)(+g𝐺)(0g𝐺))))
2613, 17, 253bitrd 294 . 2 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ((𝐼𝑋)(+g𝐺)(𝑋(+g𝐺)(𝐼𝑌))) < ((𝐼𝑋)(+g𝐺)(0g𝐺))))
276, 11, 14, 7grplinv 17384 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝐼𝑋)(+g𝐺)𝑋) = (0g𝐺))
285, 2, 27syl2anc 692 . . . . 5 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → ((𝐼𝑋)(+g𝐺)𝑋) = (0g𝐺))
2928oveq1d 6620 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (((𝐼𝑋)(+g𝐺)𝑋)(+g𝐺)(𝐼𝑌)) = ((0g𝐺)(+g𝐺)(𝐼𝑌)))
306, 11grpass 17347 . . . . 5 ((𝐺 ∈ Grp ∧ ((𝐼𝑋) ∈ 𝐵𝑋𝐵 ∧ (𝐼𝑌) ∈ 𝐵)) → (((𝐼𝑋)(+g𝐺)𝑋)(+g𝐺)(𝐼𝑌)) = ((𝐼𝑋)(+g𝐺)(𝑋(+g𝐺)(𝐼𝑌))))
315, 24, 2, 9, 30syl13anc 1325 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (((𝐼𝑋)(+g𝐺)𝑋)(+g𝐺)(𝐼𝑌)) = ((𝐼𝑋)(+g𝐺)(𝑋(+g𝐺)(𝐼𝑌))))
326, 11, 14grplid 17368 . . . . 5 ((𝐺 ∈ Grp ∧ (𝐼𝑌) ∈ 𝐵) → ((0g𝐺)(+g𝐺)(𝐼𝑌)) = (𝐼𝑌))
335, 9, 32syl2anc 692 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺)(+g𝐺)(𝐼𝑌)) = (𝐼𝑌))
3429, 31, 333eqtr3d 2668 . . 3 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → ((𝐼𝑋)(+g𝐺)(𝑋(+g𝐺)(𝐼𝑌))) = (𝐼𝑌))
356, 11, 14grprid 17369 . . . 4 ((𝐺 ∈ Grp ∧ (𝐼𝑋) ∈ 𝐵) → ((𝐼𝑋)(+g𝐺)(0g𝐺)) = (𝐼𝑋))
365, 24, 35syl2anc 692 . . 3 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → ((𝐼𝑋)(+g𝐺)(0g𝐺)) = (𝐼𝑋))
3734, 36breq12d 4631 . 2 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (((𝐼𝑋)(+g𝐺)(𝑋(+g𝐺)(𝐼𝑌))) < ((𝐼𝑋)(+g𝐺)(0g𝐺)) ↔ (𝐼𝑌) < (𝐼𝑋)))
3826, 37bitrd 268 1 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝐼𝑌) < (𝐼𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992   class class class wbr 4618  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  0gc0g 16016  ltcplt 16857  Grpcgrp 17338  invgcminusg 17339  oppgcoppg 17691  oGrpcogrp 29475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-tpos 7298  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-dec 11438  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-plusg 15870  df-ple 15877  df-0g 16018  df-plt 16874  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-minusg 17342  df-oppg 17692  df-omnd 29476  df-ogrp 29477
This theorem is referenced by:  archirngz  29520  archiabllem2c  29526
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