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Theorem ogrpsub 32702
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 32688 . . . . 5 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 496 . . . 4 (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
323ad2ant1 1130 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝐺 ∈ oMnd)
4 simp21 1203 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑋𝐵)
5 simp22 1204 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑌𝐵)
6 ogrpgrp 32689 . . . . 5 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
763ad2ant1 1130 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝐺 ∈ Grp)
8 simp23 1205 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑍𝐵)
9 ogrpsub.0 . . . . 5 𝐵 = (Base‘𝐺)
10 eqid 2724 . . . . 5 (invg𝐺) = (invg𝐺)
119, 10grpinvcl 18907 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
127, 8, 11syl2anc 583 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → ((invg𝐺)‘𝑍) ∈ 𝐵)
13 simp3 1135 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑋 𝑌)
14 ogrpsub.1 . . . 4 = (le‘𝐺)
15 eqid 2724 . . . 4 (+g𝐺) = (+g𝐺)
169, 14, 15omndadd 32692 . . 3 ((𝐺 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵) ∧ 𝑋 𝑌) → (𝑋(+g𝐺)((invg𝐺)‘𝑍)) (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
173, 4, 5, 12, 13, 16syl131anc 1380 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋(+g𝐺)((invg𝐺)‘𝑍)) (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
18 ogrpsub.2 . . . 4 = (-g𝐺)
199, 15, 10, 18grpsubval 18905 . . 3 ((𝑋𝐵𝑍𝐵) → (𝑋 𝑍) = (𝑋(+g𝐺)((invg𝐺)‘𝑍)))
204, 8, 19syl2anc 583 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 𝑍) = (𝑋(+g𝐺)((invg𝐺)‘𝑍)))
219, 15, 10, 18grpsubval 18905 . . 3 ((𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
225, 8, 21syl2anc 583 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑌 𝑍) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
2317, 20, 223brtr4d 5170 1 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 𝑍) (𝑌 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5138  cfv 6533  (class class class)co 7401  Basecbs 17143  +gcplusg 17196  lecple 17203  Grpcgrp 18853  invgcminusg 18854  -gcsg 18855  oMndcomnd 32683  oGrpcogrp 32684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-0g 17386  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-grp 18856  df-minusg 18857  df-sbg 18858  df-omnd 32685  df-ogrp 32686
This theorem is referenced by:  ogrpsublt  32707  archiabllem1a  32805  archiabllem2c  32809  ornglmulle  32889  orngrmulle  32890
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