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Metamath Proof Explorer

Theorem omndaddr

Description: In a right 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 omndaddr ( ( ( oppg ‘ 𝑀 ) ∈ oMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑍 + 𝑋 ) ≤ ( 𝑍 + 𝑌 ) )

Proof

Step Hyp Ref Expression
1 omndadd.0 ⊢ 𝐵 = ( Base ‘ 𝑀 )
2 omndadd.1 ⊢ ≤ = ( le ‘ 𝑀 )
3 omndadd.2 ⊢ + = ( +g ‘ 𝑀 )
4 eqid ⊢ ( oppg ‘ 𝑀 ) = ( oppg ‘ 𝑀 )
5 4 1 oppgbas ⊢ 𝐵 = ( Base ‘ ( oppg ‘ 𝑀 ) )
6 4 2 oppgle ⊢ ≤ = ( le ‘ ( oppg ‘ 𝑀 ) )
7 eqid ⊢ ( +g ‘ ( oppg ‘ 𝑀 ) ) = ( +g ‘ ( oppg ‘ 𝑀 ) )
8 5 6 7 omndadd ⊢ ( ( ( oppg ‘ 𝑀 ) ∈ oMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ( +g ‘ ( oppg ‘ 𝑀 ) ) 𝑍 ) ≤ ( 𝑌 ( +g ‘ ( oppg ‘ 𝑀 ) ) 𝑍 ) )
9 3 4 7 oppgplus ⊢ ( 𝑋 ( +g ‘ ( oppg ‘ 𝑀 ) ) 𝑍 ) = ( 𝑍 + 𝑋 )
10 3 4 7 oppgplus ⊢ ( 𝑌 ( +g ‘ ( oppg ‘ 𝑀 ) ) 𝑍 ) = ( 𝑍 + 𝑌 )
11 8 9 10 3brtr3g ⊢ ( ( ( oppg ‘ 𝑀 ) ∈ oMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑍 + 𝑋 ) ≤ ( 𝑍 + 𝑌 ) )