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Theorem ogrpaddltbi 32957
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 32956 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
543expa 1115 . 2 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
6 simpll 765 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ oGrp)
7 ogrpgrp 32942 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
86, 7syl 17 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ Grp)
9 simplr1 1212 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋𝐵)
10 simplr3 1214 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑍𝐵)
111, 3grpcl 18931 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 + 𝑍) ∈ 𝐵)
128, 9, 10, 11syl3anc 1368 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) ∈ 𝐵)
13 simplr2 1213 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑌𝐵)
141, 3grpcl 18931 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) ∈ 𝐵)
158, 13, 10, 14syl3anc 1368 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + 𝑍) ∈ 𝐵)
16 eqid 2726 . . . . . 6 (invg𝐺) = (invg𝐺)
171, 16grpinvcl 18977 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
188, 10, 17syl2anc 582 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
19 simpr 483 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
201, 2, 3ogrpaddlt 32956 . . . 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 1380 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) < ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)))
221, 3grpass 18932 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))))
238, 9, 10, 18, 22syl13anc 1369 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))))
24 eqid 2726 . . . . . . 7 (0g𝐺) = (0g𝐺)
251, 3, 24, 16grprinv 18980 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (𝑍 + ((invg𝐺)‘𝑍)) = (0g𝐺))
268, 10, 25syl2anc 582 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑍 + ((invg𝐺)‘𝑍)) = (0g𝐺))
2726oveq2d 7432 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))) = (𝑋 + (0g𝐺)))
281, 3, 24grprid 18958 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
298, 9, 28syl2anc 582 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (0g𝐺)) = 𝑋)
3023, 27, 293eqtrd 2770 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = 𝑋)
311, 3grpass 18932 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))))
328, 13, 10, 18, 31syl13anc 1369 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))))
3326oveq2d 7432 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))) = (𝑌 + (0g𝐺)))
341, 3, 24grprid 18958 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (0g𝐺)) = 𝑌)
358, 13, 34syl2anc 582 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (0g𝐺)) = 𝑌)
3632, 33, 353eqtrd 2770 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = 𝑌)
3721, 30, 363brtr3d 5176 . 2 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 < 𝑌)
385, 37impbida 799 1 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099   class class class wbr 5145  cfv 6546  (class class class)co 7416  Basecbs 17208  +gcplusg 17261  0gc0g 17449  ltcplt 18328  Grpcgrp 18923  invgcminusg 18924  oGrpcogrp 32937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-riota 7372  df-ov 7419  df-0g 17451  df-plt 18350  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-grp 18926  df-minusg 18927  df-omnd 32938  df-ogrp 32939
This theorem is referenced by:  ogrpinvlt  32962
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