Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ogrpaddlt Structured version   Visualization version   GIF version

 Description: In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Hypotheses
Ref Expression
Assertion
Ref Expression
ogrpaddlt ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))

StepHypRef Expression
1 isogrp 29511 . . . . 5 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 480 . . . 4 (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
323ad2ant1 1080 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ oMnd)
4 simp2 1060 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋𝐵𝑌𝐵𝑍𝐵))
5 simp1 1059 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ oGrp)
6 simp21 1092 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝐵)
7 simp22 1093 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑌𝐵)
8 simp3 1061 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
9 eqid 2621 . . . . . 6 (le‘𝐺) = (le‘𝐺)
10 ogrpaddlt.1 . . . . . 6 < = (lt‘𝐺)
119, 10pltle 16893 . . . . 5 ((𝐺 ∈ oGrp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌𝑋(le‘𝐺)𝑌))
1211imp 445 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋(le‘𝐺)𝑌)
135, 6, 7, 8, 12syl31anc 1326 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋(le‘𝐺)𝑌)
14 ogrpaddlt.0 . . . 4 𝐵 = (Base‘𝐺)
15 ogrpaddlt.2 . . . 4 + = (+g𝐺)
1614, 9, 15omndadd 29515 . . 3 ((𝐺 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋(le‘𝐺)𝑌) → (𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍))
173, 4, 13, 16syl3anc 1323 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍))
1810pltne 16894 . . . . 5 ((𝐺 ∈ oGrp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌𝑋𝑌))
1918imp 445 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝑌)
205, 6, 7, 8, 19syl31anc 1326 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝑌)
21 ogrpgrp 29512 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
2214, 15grprcan 17387 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))
2322biimpd 219 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
2421, 23sylan 488 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
2524necon3d 2811 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝑌 → (𝑋 + 𝑍) ≠ (𝑌 + 𝑍)))
26253impia 1258 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝑌) → (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))
275, 4, 20, 26syl3anc 1323 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))
28 ovex 6638 . . . 4 (𝑋 + 𝑍) ∈ V
29 ovex 6638 . . . 4 (𝑌 + 𝑍) ∈ V
309, 10pltval 16892 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋 + 𝑍) ∈ V ∧ (𝑌 + 𝑍) ∈ V) → ((𝑋 + 𝑍) < (𝑌 + 𝑍) ↔ ((𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍) ∧ (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))))
3128, 29, 30mp3an23 1413 . . 3 (𝐺 ∈ oGrp → ((𝑋 + 𝑍) < (𝑌 + 𝑍) ↔ ((𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍) ∧ (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))))
32313ad2ant1 1080 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → ((𝑋 + 𝑍) < (𝑌 + 𝑍) ↔ ((𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍) ∧ (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))))
3317, 27, 32mpbir2and 956 1 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  Vcvv 3189   class class class wbr 4618  ‘cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  lecple 15880  ltcplt 16873  Grpcgrp 17354  oMndcomnd 29506  oGrpcogrp 29507 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-riota 6571  df-ov 6613  df-0g 16034  df-plt 16890  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-grp 17357  df-omnd 29508  df-ogrp 29509 This theorem is referenced by:  ogrpaddltbi  29528  ogrpaddltrd  29529  ogrpinv0lt  29532  isarchi3  29550  archirngz  29552  archiabllem1b  29555  archiabllem2c  29558  ofldchr  29623
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