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Metamath Proof Explorer

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 ) )