Theorem omndadd

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

	Ref	Expression
Hypotheses	omndadd.0	⊢ 𝐵 = ( Base ‘ 𝑀 )
	omndadd.1	⊢ ≤ = ( le ‘ 𝑀 )
	omndadd.2	⊢ + = ( +g ‘ 𝑀 )
Assertion	omndadd	⊢ ( ( 𝑀 ∈ oMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 + 𝑍 ) ≤ ( 𝑌 + 𝑍 ) )

Proof

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