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Theorem ogrpaddltbi 31975
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 31974 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
543expa 1119 . 2 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
6 simpll 766 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ oGrp)
7 ogrpgrp 31960 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
86, 7syl 17 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ Grp)
9 simplr1 1216 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋𝐵)
10 simplr3 1218 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑍𝐵)
111, 3grpcl 18761 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 + 𝑍) ∈ 𝐵)
128, 9, 10, 11syl3anc 1372 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) ∈ 𝐵)
13 simplr2 1217 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑌𝐵)
141, 3grpcl 18761 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) ∈ 𝐵)
158, 13, 10, 14syl3anc 1372 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + 𝑍) ∈ 𝐵)
16 eqid 2733 . . . . . 6 (invg𝐺) = (invg𝐺)
171, 16grpinvcl 18803 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
188, 10, 17syl2anc 585 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
19 simpr 486 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
201, 2, 3ogrpaddlt 31974 . . . 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 1384 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) < ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)))
221, 3grpass 18762 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))))
238, 9, 10, 18, 22syl13anc 1373 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))))
24 eqid 2733 . . . . . . 7 (0g𝐺) = (0g𝐺)
251, 3, 24, 16grprinv 18806 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (𝑍 + ((invg𝐺)‘𝑍)) = (0g𝐺))
268, 10, 25syl2anc 585 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑍 + ((invg𝐺)‘𝑍)) = (0g𝐺))
2726oveq2d 7374 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))) = (𝑋 + (0g𝐺)))
281, 3, 24grprid 18786 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
298, 9, 28syl2anc 585 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (0g𝐺)) = 𝑋)
3023, 27, 293eqtrd 2777 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = 𝑋)
311, 3grpass 18762 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))))
328, 13, 10, 18, 31syl13anc 1373 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))))
3326oveq2d 7374 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))) = (𝑌 + (0g𝐺)))
341, 3, 24grprid 18786 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (0g𝐺)) = 𝑌)
358, 13, 34syl2anc 585 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (0g𝐺)) = 𝑌)
3632, 33, 353eqtrd 2777 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = 𝑌)
3721, 30, 363brtr3d 5137 . 2 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 < 𝑌)
385, 37impbida 800 1 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107   class class class wbr 5106  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  ltcplt 18202  Grpcgrp 18753  invgcminusg 18754  oGrpcogrp 31955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-riota 7314  df-ov 7361  df-0g 17328  df-plt 18224  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-minusg 18757  df-omnd 31956  df-ogrp 31957
This theorem is referenced by:  ogrpinvlt  31980
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