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Theorem ogrpsublt 33098
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 1139 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
2 simp1 1137 . . . . . 6 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ oGrp)
3 simp21 1207 . . . . . 6 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝐵)
4 simp22 1208 . . . . . 6 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑌𝐵)
5 eqid 2737 . . . . . . 7 (le‘𝐺) = (le‘𝐺)
6 ogrpsublt.1 . . . . . . 7 < = (lt‘𝐺)
75, 6pltval 18377 . . . . . 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 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 33093 . . 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 495 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝑌)
16 ogrpgrp 33080 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
172, 16syl 17 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ Grp)
18 simp23 1209 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑍𝐵)
1911, 12grpsubrcan 19039 . . . . 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 257 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) ≠ (𝑌 𝑍))
2311, 12grpsubcl 19038 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
2417, 3, 18, 23syl3anc 1373 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) ∈ 𝐵)
2511, 12grpsubcl 19038 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
2617, 4, 18, 25syl3anc 1373 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑌 𝑍) ∈ 𝐵)
275, 6pltval 18377 . . 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  ltcplt 18354  Grpcgrp 18951  -gcsg 18953  oGrpcogrp 33075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-0g 17486  df-plt 18375  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-omnd 33076  df-ogrp 33077
This theorem is referenced by:  archiabllem1a  33198  archiabllem2a  33201  archiabllem2c  33202
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