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Theorem ogrpsub 30712
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 30698 . . . . 5 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 499 . . . 4 (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
323ad2ant1 1129 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝐺 ∈ oMnd)
4 simp21 1202 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑋𝐵)
5 simp22 1203 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑌𝐵)
6 ogrpgrp 30699 . . . . 5 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
763ad2ant1 1129 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝐺 ∈ Grp)
8 simp23 1204 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑍𝐵)
9 ogrpsub.0 . . . . 5 𝐵 = (Base‘𝐺)
10 eqid 2821 . . . . 5 (invg𝐺) = (invg𝐺)
119, 10grpinvcl 18145 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
127, 8, 11syl2anc 586 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → ((invg𝐺)‘𝑍) ∈ 𝐵)
13 simp3 1134 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑋 𝑌)
14 ogrpsub.1 . . . 4 = (le‘𝐺)
15 eqid 2821 . . . 4 (+g𝐺) = (+g𝐺)
169, 14, 15omndadd 30702 . . 3 ((𝐺 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵) ∧ 𝑋 𝑌) → (𝑋(+g𝐺)((invg𝐺)‘𝑍)) (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
173, 4, 5, 12, 13, 16syl131anc 1379 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋(+g𝐺)((invg𝐺)‘𝑍)) (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
18 ogrpsub.2 . . . 4 = (-g𝐺)
199, 15, 10, 18grpsubval 18143 . . 3 ((𝑋𝐵𝑍𝐵) → (𝑋 𝑍) = (𝑋(+g𝐺)((invg𝐺)‘𝑍)))
204, 8, 19syl2anc 586 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 𝑍) = (𝑋(+g𝐺)((invg𝐺)‘𝑍)))
219, 15, 10, 18grpsubval 18143 . . 3 ((𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
225, 8, 21syl2anc 586 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑌 𝑍) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
2317, 20, 223brtr4d 5090 1 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 𝑍) (𝑌 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1083   = wceq 1533  wcel 2110   class class class wbr 5058  cfv 6349  (class class class)co 7150  Basecbs 16477  +gcplusg 16559  lecple 16566  Grpcgrp 18097  invgcminusg 18098  -gcsg 18099  oMndcomnd 30693  oGrpcogrp 30694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-minusg 18101  df-sbg 18102  df-omnd 30695  df-ogrp 30696
This theorem is referenced by:  ogrpsublt  30717  archiabllem1a  30815  archiabllem2c  30819  ornglmulle  30873  orngrmulle  30874
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