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Metamath Proof Explorer

Theorem nnmwordri

Description: Weak ordering property of ordinal multiplication. Proposition 8.21 of TakeutiZaring p. 63, limited to natural numbers. (Contributed by Mario Carneiro, 17-Nov-2014)

Ref Expression
Assertion nnmwordri ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ·o 𝐶 ) ⊆ ( 𝐵 ·o 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 nnmwordi ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐶 ·o 𝐴 ) ⊆ ( 𝐶 ·o 𝐵 ) ) )
2 nnmcom ⊢ ( ( 𝐴 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐴 ·o 𝐶 ) = ( 𝐶 ·o 𝐴 ) )
3 2 3adant2 ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐴 ·o 𝐶 ) = ( 𝐶 ·o 𝐴 ) )
4 nnmcom ⊢ ( ( 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐵 ·o 𝐶 ) = ( 𝐶 ·o 𝐵 ) )
5 4 3adant1 ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐵 ·o 𝐶 ) = ( 𝐶 ·o 𝐵 ) )
6 3 5 sseq12d ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( ( 𝐴 ·o 𝐶 ) ⊆ ( 𝐵 ·o 𝐶 ) ↔ ( 𝐶 ·o 𝐴 ) ⊆ ( 𝐶 ·o 𝐵 ) ) )
7 1 6 sylibrd ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ·o 𝐶 ) ⊆ ( 𝐵 ·o 𝐶 ) ) )