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Theorem ogrpsublt 33081
Description: In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
Hypotheses
Ref Expression
ogrpsublt.0 𝐵 = (Base‘𝐺)
ogrpsublt.1 < = (lt‘𝐺)
ogrpsublt.2 = (-g𝐺)
Assertion
Ref Expression
ogrpsublt ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) < (𝑌 𝑍))

Proof of Theorem ogrpsublt
StepHypRef Expression
1 simp3 1137 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
2 simp1 1135 . . . . . 6 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ oGrp)
3 simp21 1205 . . . . . 6 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝐵)
4 simp22 1206 . . . . . 6 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑌𝐵)
5 eqid 2735 . . . . . . 7 (le‘𝐺) = (le‘𝐺)
6 ogrpsublt.1 . . . . . . 7 < = (lt‘𝐺)
75, 6pltval 18390 . . . . . 6 ((𝐺 ∈ oGrp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐺)𝑌𝑋𝑌)))
82, 3, 4, 7syl3anc 1370 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐺)𝑌𝑋𝑌)))
91, 8mpbid 232 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋(le‘𝐺)𝑌𝑋𝑌))
109simpld 494 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋(le‘𝐺)𝑌)
11 ogrpsublt.0 . . . 4 𝐵 = (Base‘𝐺)
12 ogrpsublt.2 . . . 4 = (-g𝐺)
1311, 5, 12ogrpsub 33076 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋(le‘𝐺)𝑌) → (𝑋 𝑍)(le‘𝐺)(𝑌 𝑍))
1410, 13syld3an3 1408 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍)(le‘𝐺)(𝑌 𝑍))
159simprd 495 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝑌)
16 ogrpgrp 33063 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
172, 16syl 17 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ Grp)
18 simp23 1207 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑍𝐵)
1911, 12grpsubrcan 19052 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ 𝑋 = 𝑌))
2017, 3, 4, 18, 19syl13anc 1371 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ 𝑋 = 𝑌))
2120necon3bid 2983 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → ((𝑋 𝑍) ≠ (𝑌 𝑍) ↔ 𝑋𝑌))
2215, 21mpbird 257 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) ≠ (𝑌 𝑍))
2311, 12grpsubcl 19051 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
2417, 3, 18, 23syl3anc 1370 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) ∈ 𝐵)
2511, 12grpsubcl 19051 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
2617, 4, 18, 25syl3anc 1370 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑌 𝑍) ∈ 𝐵)
275, 6pltval 18390 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋 𝑍) ∈ 𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → ((𝑋 𝑍) < (𝑌 𝑍) ↔ ((𝑋 𝑍)(le‘𝐺)(𝑌 𝑍) ∧ (𝑋 𝑍) ≠ (𝑌 𝑍))))
282, 24, 26, 27syl3anc 1370 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → ((𝑋 𝑍) < (𝑌 𝑍) ↔ ((𝑋 𝑍)(le‘𝐺)(𝑌 𝑍) ∧ (𝑋 𝑍) ≠ (𝑌 𝑍))))
2914, 22, 28mpbir2and 713 1 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) < (𝑌 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  ltcplt 18366  Grpcgrp 18964  -gcsg 18966  oGrpcogrp 33058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-plt 18388  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-omnd 33059  df-ogrp 33060
This theorem is referenced by:  archiabllem1a  33181  archiabllem2a  33184  archiabllem2c  33185
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