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Theorem ogrpsublt 29053
 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 1056 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
2 simp1 1054 . . . . . 6 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ oGrp)
3 simp21 1087 . . . . . 6 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝐵)
4 simp22 1088 . . . . . 6 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑌𝐵)
5 eqid 2610 . . . . . . 7 (le‘𝐺) = (le‘𝐺)
6 ogrpsublt.1 . . . . . . 7 < = (lt‘𝐺)
75, 6pltval 16783 . . . . . 6 ((𝐺 ∈ oGrp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐺)𝑌𝑋𝑌)))
82, 3, 4, 7syl3anc 1318 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐺)𝑌𝑋𝑌)))
91, 8mpbid 221 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋(le‘𝐺)𝑌𝑋𝑌))
109simpld 474 . . 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 29048 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋(le‘𝐺)𝑌) → (𝑋 𝑍)(le‘𝐺)(𝑌 𝑍))
1410, 13syld3an3 1363 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍)(le‘𝐺)(𝑌 𝑍))
159simprd 478 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝑌)
16 ogrpgrp 29034 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
172, 16syl 17 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ Grp)
18 simp23 1089 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑍𝐵)
1911, 12grpsubrcan 17319 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ 𝑋 = 𝑌))
2017, 3, 4, 18, 19syl13anc 1320 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ 𝑋 = 𝑌))
2120necon3bid 2826 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → ((𝑋 𝑍) ≠ (𝑌 𝑍) ↔ 𝑋𝑌))
2215, 21mpbird 246 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) ≠ (𝑌 𝑍))
2311, 12grpsubcl 17318 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
2417, 3, 18, 23syl3anc 1318 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) ∈ 𝐵)
2511, 12grpsubcl 17318 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
2617, 4, 18, 25syl3anc 1318 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑌 𝑍) ∈ 𝐵)
275, 6pltval 16783 . . 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 1318 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → ((𝑋 𝑍) < (𝑌 𝑍) ↔ ((𝑋 𝑍)(le‘𝐺)(𝑌 𝑍) ∧ (𝑋 𝑍) ≠ (𝑌 𝑍))))
2914, 22, 28mpbir2and 959 1 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) < (𝑌 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  ltcplt 16764  Grpcgrp 17245  -gcsg 17247  oGrpcogrp 29029 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-0g 15925  df-plt 16781  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-omnd 29030  df-ogrp 29031 This theorem is referenced by:  archiabllem1a  29076  archiabllem2a  29079  archiabllem2c  29080
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