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Theorem ogrpsub 29544
Description: In an ordered group, the ordering is compatible with group subtraction. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
ogrpsub.0 𝐵 = (Base‘𝐺)
ogrpsub.1 = (le‘𝐺)
ogrpsub.2 = (-g𝐺)
Assertion
Ref Expression
ogrpsub ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 𝑍) (𝑌 𝑍))

Proof of Theorem ogrpsub
StepHypRef Expression
1 isogrp 29529 . . . . 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 simp21 1092 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑋𝐵)
5 simp22 1093 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑌𝐵)
6 ogrpgrp 29530 . . . . 5 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
763ad2ant1 1080 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝐺 ∈ Grp)
8 simp23 1094 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑍𝐵)
9 ogrpsub.0 . . . . 5 𝐵 = (Base‘𝐺)
10 eqid 2621 . . . . 5 (invg𝐺) = (invg𝐺)
119, 10grpinvcl 17407 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
127, 8, 11syl2anc 692 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → ((invg𝐺)‘𝑍) ∈ 𝐵)
13 simp3 1061 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑋 𝑌)
14 ogrpsub.1 . . . 4 = (le‘𝐺)
15 eqid 2621 . . . 4 (+g𝐺) = (+g𝐺)
169, 14, 15omndadd 29533 . . 3 ((𝐺 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵) ∧ 𝑋 𝑌) → (𝑋(+g𝐺)((invg𝐺)‘𝑍)) (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
173, 4, 5, 12, 13, 16syl131anc 1336 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋(+g𝐺)((invg𝐺)‘𝑍)) (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
18 ogrpsub.2 . . . 4 = (-g𝐺)
199, 15, 10, 18grpsubval 17405 . . 3 ((𝑋𝐵𝑍𝐵) → (𝑋 𝑍) = (𝑋(+g𝐺)((invg𝐺)‘𝑍)))
204, 8, 19syl2anc 692 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 𝑍) = (𝑋(+g𝐺)((invg𝐺)‘𝑍)))
219, 15, 10, 18grpsubval 17405 . . 3 ((𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
225, 8, 21syl2anc 692 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑌 𝑍) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
2317, 20, 223brtr4d 4655 1 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 𝑍) (𝑌 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4623  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  lecple 15888  Grpcgrp 17362  invgcminusg 17363  -gcsg 17364  oMndcomnd 29524  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  ax-un 6914
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-pw 4138  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-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-minusg 17366  df-sbg 17367  df-omnd 29526  df-ogrp 29527
This theorem is referenced by:  ogrpsublt  29549  archiabllem1a  29572  archiabllem2c  29576  ornglmulle  29632  orngrmulle  29633
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