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

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

Proof of Theorem ogrpaddltbi
StepHypRef Expression
1 ogrpaddlt.0 . . . 4 𝐵 = (Base‘𝐺)
2 ogrpaddlt.1 . . . 4 < = (lt‘𝐺)
3 ogrpaddlt.2 . . . 4 + = (+g𝐺)
41, 2, 3ogrpaddlt 29049 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
543expa 1257 . 2 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
6 simpll 786 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ oGrp)
7 ogrpgrp 29034 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
86, 7syl 17 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ Grp)
9 simplr1 1096 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋𝐵)
10 simplr3 1098 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑍𝐵)
111, 3grpcl 17253 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 + 𝑍) ∈ 𝐵)
128, 9, 10, 11syl3anc 1318 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) ∈ 𝐵)
13 simplr2 1097 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑌𝐵)
141, 3grpcl 17253 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) ∈ 𝐵)
158, 13, 10, 14syl3anc 1318 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + 𝑍) ∈ 𝐵)
16 eqid 2610 . . . . . 6 (invg𝐺) = (invg𝐺)
171, 16grpinvcl 17290 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
188, 10, 17syl2anc 691 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
19 simpr 476 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
201, 2, 3ogrpaddlt 29049 . . . 4 ((𝐺 ∈ oGrp ∧ ((𝑋 + 𝑍) ∈ 𝐵 ∧ (𝑌 + 𝑍) ∈ 𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) < ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)))
216, 12, 15, 18, 19, 20syl131anc 1331 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) < ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)))
221, 3grpass 17254 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))))
238, 9, 10, 18, 22syl13anc 1320 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))))
24 eqid 2610 . . . . . . 7 (0g𝐺) = (0g𝐺)
251, 3, 24, 16grprinv 17292 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (𝑍 + ((invg𝐺)‘𝑍)) = (0g𝐺))
268, 10, 25syl2anc 691 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑍 + ((invg𝐺)‘𝑍)) = (0g𝐺))
2726oveq2d 6565 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))) = (𝑋 + (0g𝐺)))
281, 3, 24grprid 17276 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
298, 9, 28syl2anc 691 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (0g𝐺)) = 𝑋)
3023, 27, 293eqtrd 2648 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = 𝑋)
311, 3grpass 17254 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))))
328, 13, 10, 18, 31syl13anc 1320 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))))
3326oveq2d 6565 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))) = (𝑌 + (0g𝐺)))
341, 3, 24grprid 17276 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (0g𝐺)) = 𝑌)
358, 13, 34syl2anc 691 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (0g𝐺)) = 𝑌)
3632, 33, 353eqtrd 2648 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = 𝑌)
3721, 30, 363brtr3d 4614 . 2 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 < 𝑌)
385, 37impbida 873 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   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  ltcplt 16764  Grpcgrp 17245  invgcminusg 17246  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 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-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-0g 15925  df-plt 16781  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-omnd 29030  df-ogrp 29031 This theorem is referenced by:  ogrpinvlt  29055
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