Theorem ogrpsub 20112

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 20099	. . . . 5 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simprbi 497	. . . 4 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
3	2	3ad2ant1 1134	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝐺 ∈ oMnd)
4		simp21 1208	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝐵)
5		simp22 1209	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝐵)
6		ogrpgrp 20100	. . . . 5 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
7	6	3ad2ant1 1134	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝐺 ∈ Grp)
8		simp23 1210	. . . 4 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑍 ∈ 𝐵)
9		ogrpsub.0	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
10		eqid 2736	. . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺)
11	9, 10	grpinvcl 18963	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
12	7, 8, 11	syl2anc 585	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
13		simp3 1139	. . 3 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≤ 𝑌)
14		ogrpsub.1	. . . 4 ⊢ ≤ = (le‘𝐺)
15		eqid 2736	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
16	9, 14, 15	omndadd 20103	. . 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 1386	. 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 18961	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 − 𝑍) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑍)))
20	4, 8, 19	syl2anc 585	. 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 18961	. . 3 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) = (𝑌(+g‘𝐺)((invg‘𝐺)‘𝑍)))
22	5, 8, 21	syl2anc 585	. 2 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑌 − 𝑍) = (𝑌(+g‘𝐺)((invg‘𝐺)‘𝑍)))
23	17, 20, 22	3brtr4d 5117	1 ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 − 𝑍) ≤ (𝑌 − 𝑍))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  lecple 17227  Grpcgrp 18909  inv_gcminusg 18910  -_gcsg 18911  oMndcomnd 20094  oGrpcogrp 20095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-minusg 18913  df-sbg 18914  df-omnd 20096  df-ogrp 20097

This theorem is referenced by:  ogrpsublt  20117  ornglmulle  20844  orngrmulle  20845  archiabllem1a  33252  archiabllem2c  33256