Theorem omndadd 33027

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 33022	. . . 4 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))
5	4	simp3bi 1147	. . 3 ⊢ (𝑀 ∈ oMnd → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))
6		breq1 5113	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 ≤ 𝑏 ↔ 𝑋 ≤ 𝑏))
7		oveq1 7397	. . . . . 6 ⊢ (𝑎 = 𝑋 → (𝑎 + 𝑐) = (𝑋 + 𝑐))
8	7	breq1d 5120	. . . . 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 5114	. . . . 5 ⊢ (𝑏 = 𝑌 → (𝑋 ≤ 𝑏 ↔ 𝑋 ≤ 𝑌))
11		oveq1 7397	. . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑏 + 𝑐) = (𝑌 + 𝑐))
12	11	breq2d 5122	. . . . 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 7398	. . . . . 6 ⊢ (𝑐 = 𝑍 → (𝑋 + 𝑐) = (𝑋 + 𝑍))
15		oveq2 7398	. . . . . 6 ⊢ (𝑐 = 𝑍 → (𝑌 + 𝑐) = (𝑌 + 𝑍))
16	14, 15	breq12d 5123	. . . . 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 3607	. . 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 1117	1 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  lecple 17234  Tosetctos 18382  Mndcmnd 18668  oMndcomnd 33018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-omnd 33020

This theorem is referenced by:  omndaddr  33028  omndadd2d  33029  omndadd2rd  33030  submomnd  33031  omndmul2  33033  omndmul3  33034  ogrpinv0le  33036  ogrpsub  33037  ogrpaddlt  33038  orngsqr  33289  ornglmulle  33290  orngrmulle  33291