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Theorem ogrpsub 30258
 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 30243 . . . . 5 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 492 . . . 4 (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
323ad2ant1 1167 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝐺 ∈ oMnd)
4 simp21 1267 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑋𝐵)
5 simp22 1268 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑌𝐵)
6 ogrpgrp 30244 . . . . 5 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
763ad2ant1 1167 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝐺 ∈ Grp)
8 simp23 1269 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑍𝐵)
9 ogrpsub.0 . . . . 5 𝐵 = (Base‘𝐺)
10 eqid 2825 . . . . 5 (invg𝐺) = (invg𝐺)
119, 10grpinvcl 17828 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
127, 8, 11syl2anc 579 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → ((invg𝐺)‘𝑍) ∈ 𝐵)
13 simp3 1172 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → 𝑋 𝑌)
14 ogrpsub.1 . . . 4 = (le‘𝐺)
15 eqid 2825 . . . 4 (+g𝐺) = (+g𝐺)
169, 14, 15omndadd 30247 . . 3 ((𝐺 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵) ∧ 𝑋 𝑌) → (𝑋(+g𝐺)((invg𝐺)‘𝑍)) (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
173, 4, 5, 12, 13, 16syl131anc 1506 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋(+g𝐺)((invg𝐺)‘𝑍)) (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
18 ogrpsub.2 . . . 4 = (-g𝐺)
199, 15, 10, 18grpsubval 17826 . . 3 ((𝑋𝐵𝑍𝐵) → (𝑋 𝑍) = (𝑋(+g𝐺)((invg𝐺)‘𝑍)))
204, 8, 19syl2anc 579 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 𝑍) = (𝑋(+g𝐺)((invg𝐺)‘𝑍)))
219, 15, 10, 18grpsubval 17826 . . 3 ((𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
225, 8, 21syl2anc 579 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑌 𝑍) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
2317, 20, 223brtr4d 4907 1 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 𝑍) (𝑌 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   class class class wbr 4875  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  +gcplusg 16312  lecple 16319  Grpcgrp 17783  invgcminusg 17784  -gcsg 17785  oMndcomnd 30238  oGrpcogrp 30239 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-0g 16462  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-grp 17786  df-minusg 17787  df-sbg 17788  df-omnd 30240  df-ogrp 30241 This theorem is referenced by:  ogrpsublt  30263  archiabllem1a  30286  archiabllem2c  30290  ornglmulle  30346  orngrmulle  30347
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