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	⊢ − = (-g‘𝐺)

Assertion

Ref	Expression
ogrpsub	⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 − 𝑍) ≤ (𝑌 − 𝑍))

Proof of Theorem ogrpsub

Step	Hyp	Ref	Expression
1		isogrp 32688	. . . . 5 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simprbi 496	. . . 4 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
3	2	3ad2ant1 1130	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝐺 ∈ oMnd)
4		simp21 1203	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝐵)
5		simp22 1204	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝐵)
6		ogrpgrp 32689	. . . . 5 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
7	6	3ad2ant1 1130	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝐺 ∈ Grp)
8		simp23 1205	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑍 ∈ 𝐵)
9		ogrpsub.0	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
10		eqid 2724	. . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺)
11	9, 10	grpinvcl 18907	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
12	7, 8, 11	syl2anc 583	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
13		simp3 1135	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≤ 𝑌)
14		ogrpsub.1	. . . 4 ⊢ ≤ = (le‘𝐺)
15		eqid 2724	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
16	9, 14, 15	omndadd 32692	. . 3 ⊢ ((𝐺 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑍) ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑍)) ≤ (𝑌(+g‘𝐺)((invg‘𝐺)‘𝑍)))
17	3, 4, 5, 12, 13, 16	syl131anc 1380	. 2 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑍)) ≤ (𝑌(+g‘𝐺)((invg‘𝐺)‘𝑍)))
18		ogrpsub.2	. . . 4 ⊢ − = (-g‘𝐺)
19	9, 15, 10, 18	grpsubval 18905	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 − 𝑍) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑍)))
20	4, 8, 19	syl2anc 583	. 2 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 − 𝑍) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑍)))
21	9, 15, 10, 18	grpsubval 18905	. . 3 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) = (𝑌(+g‘𝐺)((invg‘𝐺)‘𝑍)))
22	5, 8, 21	syl2anc 583	. 2 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑌 − 𝑍) = (𝑌(+g‘𝐺)((invg‘𝐺)‘𝑍)))
23	17, 20, 22	3brtr4d 5170	1 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 − 𝑍) ≤ (𝑌 − 𝑍))

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  inv_gcminusg 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