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Theorem ogrpsublt 31066
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 1140 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
2 simp1 1138 . . . . . 6 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ oGrp)
3 simp21 1208 . . . . . 6 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝐵)
4 simp22 1209 . . . . . 6 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑌𝐵)
5 eqid 2737 . . . . . . 7 (le‘𝐺) = (le‘𝐺)
6 ogrpsublt.1 . . . . . . 7 < = (lt‘𝐺)
75, 6pltval 17838 . . . . . 6 ((𝐺 ∈ oGrp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐺)𝑌𝑋𝑌)))
82, 3, 4, 7syl3anc 1373 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐺)𝑌𝑋𝑌)))
91, 8mpbid 235 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋(le‘𝐺)𝑌𝑋𝑌))
109simpld 498 . . 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 31061 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋(le‘𝐺)𝑌) → (𝑋 𝑍)(le‘𝐺)(𝑌 𝑍))
1410, 13syld3an3 1411 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍)(le‘𝐺)(𝑌 𝑍))
159simprd 499 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝑌)
16 ogrpgrp 31048 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
172, 16syl 17 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ Grp)
18 simp23 1210 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑍𝐵)
1911, 12grpsubrcan 18444 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ 𝑋 = 𝑌))
2017, 3, 4, 18, 19syl13anc 1374 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ 𝑋 = 𝑌))
2120necon3bid 2985 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → ((𝑋 𝑍) ≠ (𝑌 𝑍) ↔ 𝑋𝑌))
2215, 21mpbird 260 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) ≠ (𝑌 𝑍))
2311, 12grpsubcl 18443 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
2417, 3, 18, 23syl3anc 1373 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) ∈ 𝐵)
2511, 12grpsubcl 18443 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
2617, 4, 18, 25syl3anc 1373 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑌 𝑍) ∈ 𝐵)
275, 6pltval 17838 . . 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 1373 . 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940   class class class wbr 5053  cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809  ltcplt 17815  Grpcgrp 18365  -gcsg 18367  oGrpcogrp 31043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-0g 16946  df-plt 17836  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-grp 18368  df-minusg 18369  df-sbg 18370  df-omnd 31044  df-ogrp 31045
This theorem is referenced by:  archiabllem1a  31164  archiabllem2a  31167  archiabllem2c  31168
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