Theorem omndadd 20069

Description: In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)

Hypotheses

Ref	Expression
omndadd.0	⊢ 𝐵 = (Base‘𝑀)
omndadd.1	⊢ ≤ = (le‘𝑀)
omndadd.2	⊢ + = (+g‘𝑀)

Assertion

Ref	Expression
omndadd	⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))

Proof of Theorem omndadd

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omndadd.0	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
2		omndadd.2	. . . . 5 ⊢ + = (+g‘𝑀)
3		omndadd.1	. . . . 5 ⊢ ≤ = (le‘𝑀)
4	1, 2, 3	isomnd 20064	. . . 4 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))
5	4	simp3bi 1148	. . 3 ⊢ (𝑀 ∈ oMnd → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))
6		breq1 5103	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 ≤ 𝑏 ↔ 𝑋 ≤ 𝑏))
7		oveq1 7375	. . . . . 6 ⊢ (𝑎 = 𝑋 → (𝑎 + 𝑐) = (𝑋 + 𝑐))
8	7	breq1d 5110	. . . . 5 ⊢ (𝑎 = 𝑋 → ((𝑎 + 𝑐) ≤ (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) ≤ (𝑏 + 𝑐)))
9	6, 8	imbi12d 344	. . . 4 ⊢ (𝑎 = 𝑋 → ((𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) ↔ (𝑋 ≤ 𝑏 → (𝑋 + 𝑐) ≤ (𝑏 + 𝑐))))
10		breq2 5104	. . . . 5 ⊢ (𝑏 = 𝑌 → (𝑋 ≤ 𝑏 ↔ 𝑋 ≤ 𝑌))
11		oveq1 7375	. . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑏 + 𝑐) = (𝑌 + 𝑐))
12	11	breq2d 5112	. . . . 5 ⊢ (𝑏 = 𝑌 → ((𝑋 + 𝑐) ≤ (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) ≤ (𝑌 + 𝑐)))
13	10, 12	imbi12d 344	. . . 4 ⊢ (𝑏 = 𝑌 → ((𝑋 ≤ 𝑏 → (𝑋 + 𝑐) ≤ (𝑏 + 𝑐)) ↔ (𝑋 ≤ 𝑌 → (𝑋 + 𝑐) ≤ (𝑌 + 𝑐))))
14		oveq2 7376	. . . . . 6 ⊢ (𝑐 = 𝑍 → (𝑋 + 𝑐) = (𝑋 + 𝑍))
15		oveq2 7376	. . . . . 6 ⊢ (𝑐 = 𝑍 → (𝑌 + 𝑐) = (𝑌 + 𝑍))
16	14, 15	breq12d 5113	. . . . 5 ⊢ (𝑐 = 𝑍 → ((𝑋 + 𝑐) ≤ (𝑌 + 𝑐) ↔ (𝑋 + 𝑍) ≤ (𝑌 + 𝑍)))
17	16	imbi2d 340	. . . 4 ⊢ (𝑐 = 𝑍 → ((𝑋 ≤ 𝑌 → (𝑋 + 𝑐) ≤ (𝑌 + 𝑐)) ↔ (𝑋 ≤ 𝑌 → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))))
18	9, 13, 17	rspc3v 3594	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) → (𝑋 ≤ 𝑌 → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))))
19	5, 18	mpan9 506	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍)))
20	19	3impia 1118	1 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  lecple 17196  Tosetctos 18349  Mndcmnd 18671  oMndcomnd 20060

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-ext 2709  ax-nul 5253

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-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-omnd 20062

This theorem is referenced by:  omndaddr  20070  omndadd2d  20071  omndadd2rd  20072  submomnd  20073  omndmul2  20074  omndmul3  20075  ogrpinv0le  20077  ogrpsub  20078  ogrpaddlt  20079  orngsqr  20811  ornglmulle  20812  orngrmulle  20813