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Theorem ogrpinvlt 29852
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 1105 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ oGrp)
2 simp2 1082 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 simp3 1083 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
4 ogrpgrp 29831 . . . . . 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 17514 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝐼𝑌) ∈ 𝐵)
95, 3, 8syl2anc 694 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝐼𝑌) ∈ 𝐵)
10 ogrpinvlt.1 . . . . 5 < = (lt‘𝐺)
11 eqid 2651 . . . . 5 (+g𝐺) = (+g𝐺)
126, 10, 11ogrpaddltbi 29847 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵 ∧ (𝐼𝑌) ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋(+g𝐺)(𝐼𝑌)) < (𝑌(+g𝐺)(𝐼𝑌))))
131, 2, 3, 9, 12syl13anc 1368 . . 3 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋(+g𝐺)(𝐼𝑌)) < (𝑌(+g𝐺)(𝐼𝑌))))
14 eqid 2651 . . . . . 6 (0g𝐺) = (0g𝐺)
156, 11, 14, 7grprinv 17516 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌(+g𝐺)(𝐼𝑌)) = (0g𝐺))
165, 3, 15syl2anc 694 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝑌(+g𝐺)(𝐼𝑌)) = (0g𝐺))
1716breq2d 4697 . . 3 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(+g𝐺)(𝐼𝑌)) < (𝑌(+g𝐺)(𝐼𝑌)) ↔ (𝑋(+g𝐺)(𝐼𝑌)) < (0g𝐺)))
18 simp1r 1106 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (oppg𝐺) ∈ oGrp)
196, 11grpcl 17477 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ (𝐼𝑌) ∈ 𝐵) → (𝑋(+g𝐺)(𝐼𝑌)) ∈ 𝐵)
205, 2, 9, 19syl3anc 1366 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝑋(+g𝐺)(𝐼𝑌)) ∈ 𝐵)
216, 14grpidcl 17497 . . . . 5 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
221, 4, 213syl 18 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (0g𝐺) ∈ 𝐵)
236, 7grpinvcl 17514 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
245, 2, 23syl2anc 694 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝐼𝑋) ∈ 𝐵)
256, 10, 11, 1, 18, 20, 22, 24ogrpaddltrbid 29849 . . 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 17515 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝐼𝑋)(+g𝐺)𝑋) = (0g𝐺))
285, 2, 27syl2anc 694 . . . . 5 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → ((𝐼𝑋)(+g𝐺)𝑋) = (0g𝐺))
2928oveq1d 6705 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (((𝐼𝑋)(+g𝐺)𝑋)(+g𝐺)(𝐼𝑌)) = ((0g𝐺)(+g𝐺)(𝐼𝑌)))
306, 11grpass 17478 . . . . 5 ((𝐺 ∈ Grp ∧ ((𝐼𝑋) ∈ 𝐵𝑋𝐵 ∧ (𝐼𝑌) ∈ 𝐵)) → (((𝐼𝑋)(+g𝐺)𝑋)(+g𝐺)(𝐼𝑌)) = ((𝐼𝑋)(+g𝐺)(𝑋(+g𝐺)(𝐼𝑌))))
315, 24, 2, 9, 30syl13anc 1368 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (((𝐼𝑋)(+g𝐺)𝑋)(+g𝐺)(𝐼𝑌)) = ((𝐼𝑋)(+g𝐺)(𝑋(+g𝐺)(𝐼𝑌))))
326, 11, 14grplid 17499 . . . . 5 ((𝐺 ∈ Grp ∧ (𝐼𝑌) ∈ 𝐵) → ((0g𝐺)(+g𝐺)(𝐼𝑌)) = (𝐼𝑌))
335, 9, 32syl2anc 694 . . . 4 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺)(+g𝐺)(𝐼𝑌)) = (𝐼𝑌))
3429, 31, 333eqtr3d 2693 . . 3 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → ((𝐼𝑋)(+g𝐺)(𝑋(+g𝐺)(𝐼𝑌))) = (𝐼𝑌))
356, 11, 14grprid 17500 . . . 4 ((𝐺 ∈ Grp ∧ (𝐼𝑋) ∈ 𝐵) → ((𝐼𝑋)(+g𝐺)(0g𝐺)) = (𝐼𝑋))
365, 24, 35syl2anc 694 . . 3 (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → ((𝐼𝑋)(+g𝐺)(0g𝐺)) = (𝐼𝑋))
3734, 36breq12d 4698 . 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 383  w3a 1054   = wceq 1523  wcel 2030   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  ltcplt 16988  Grpcgrp 17469  invgcminusg 17470  oppgcoppg 17821  oGrpcogrp 29826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-dec 11532  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-ple 16008  df-0g 16149  df-plt 17005  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-oppg 17822  df-omnd 29827  df-ogrp 29828
This theorem is referenced by:  archirngz  29871  archiabllem2c  29877
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