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 ) ) )