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Theorem ogrpaddltbi 29546
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 29545 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
543expa 1262 . 2 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
6 simpll 789 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ oGrp)
7 ogrpgrp 29530 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
86, 7syl 17 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ Grp)
9 simplr1 1101 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋𝐵)
10 simplr3 1103 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑍𝐵)
111, 3grpcl 17370 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 + 𝑍) ∈ 𝐵)
128, 9, 10, 11syl3anc 1323 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) ∈ 𝐵)
13 simplr2 1102 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑌𝐵)
141, 3grpcl 17370 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) ∈ 𝐵)
158, 13, 10, 14syl3anc 1323 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + 𝑍) ∈ 𝐵)
16 eqid 2621 . . . . . 6 (invg𝐺) = (invg𝐺)
171, 16grpinvcl 17407 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
188, 10, 17syl2anc 692 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
19 simpr 477 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
201, 2, 3ogrpaddlt 29545 . . . 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 1336 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) < ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)))
221, 3grpass 17371 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))))
238, 9, 10, 18, 22syl13anc 1325 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))))
24 eqid 2621 . . . . . . 7 (0g𝐺) = (0g𝐺)
251, 3, 24, 16grprinv 17409 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (𝑍 + ((invg𝐺)‘𝑍)) = (0g𝐺))
268, 10, 25syl2anc 692 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑍 + ((invg𝐺)‘𝑍)) = (0g𝐺))
2726oveq2d 6631 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))) = (𝑋 + (0g𝐺)))
281, 3, 24grprid 17393 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
298, 9, 28syl2anc 692 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (0g𝐺)) = 𝑋)
3023, 27, 293eqtrd 2659 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = 𝑋)
311, 3grpass 17371 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))))
328, 13, 10, 18, 31syl13anc 1325 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))))
3326oveq2d 6631 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))) = (𝑌 + (0g𝐺)))
341, 3, 24grprid 17393 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (0g𝐺)) = 𝑌)
358, 13, 34syl2anc 692 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (0g𝐺)) = 𝑌)
3632, 33, 353eqtrd 2659 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = 𝑌)
3721, 30, 363brtr3d 4654 . 2 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 < 𝑌)
385, 37impbida 876 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   class class class wbr 4623  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  0gc0g 16040  ltcplt 16881  Grpcgrp 17362  invgcminusg 17363  oGrpcogrp 29525
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-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
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 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-0g 16042  df-plt 16898  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-minusg 17366  df-omnd 29526  df-ogrp 29527
This theorem is referenced by:  ogrpinvlt  29551
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