Theorem nnmcan

Description: Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995) (Revised by Mario Carneiro, 15-Nov-2014)

	Ref	Expression
Assertion	nnmcan	⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) ∧ ∅ ∈ 𝐴 ) → ( ( 𝐴 ·o 𝐵 ) = ( 𝐴 ·o 𝐶 ) ↔ 𝐵 = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		3anrot	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) ↔ ( 𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ ω ) )
2		nnmword	⊢ ( ( ( 𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ ω ) ∧ ∅ ∈ 𝐴 ) → ( 𝐵 ⊆ 𝐶 ↔ ( 𝐴 ·o 𝐵 ) ⊆ ( 𝐴 ·o 𝐶 ) ) )
3	1 2	sylanb	⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) ∧ ∅ ∈ 𝐴 ) → ( 𝐵 ⊆ 𝐶 ↔ ( 𝐴 ·o 𝐵 ) ⊆ ( 𝐴 ·o 𝐶 ) ) )
4		3anrev	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) ↔ ( 𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω ) )
5		nnmword	⊢ ( ( ( 𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω ) ∧ ∅ ∈ 𝐴 ) → ( 𝐶 ⊆ 𝐵 ↔ ( 𝐴 ·o 𝐶 ) ⊆ ( 𝐴 ·o 𝐵 ) ) )
6	4 5	sylanb	⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) ∧ ∅ ∈ 𝐴 ) → ( 𝐶 ⊆ 𝐵 ↔ ( 𝐴 ·o 𝐶 ) ⊆ ( 𝐴 ·o 𝐵 ) ) )
7	3 6	anbi12d	⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) ∧ ∅ ∈ 𝐴 ) → ( ( 𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵 ) ↔ ( ( 𝐴 ·o 𝐵 ) ⊆ ( 𝐴 ·o 𝐶 ) ∧ ( 𝐴 ·o 𝐶 ) ⊆ ( 𝐴 ·o 𝐵 ) ) ) )
8	7	bicomd	⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) ∧ ∅ ∈ 𝐴 ) → ( ( ( 𝐴 ·o 𝐵 ) ⊆ ( 𝐴 ·o 𝐶 ) ∧ ( 𝐴 ·o 𝐶 ) ⊆ ( 𝐴 ·o 𝐵 ) ) ↔ ( 𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵 ) ) )
9		eqss	⊢ ( ( 𝐴 ·o 𝐵 ) = ( 𝐴 ·o 𝐶 ) ↔ ( ( 𝐴 ·o 𝐵 ) ⊆ ( 𝐴 ·o 𝐶 ) ∧ ( 𝐴 ·o 𝐶 ) ⊆ ( 𝐴 ·o 𝐵 ) ) )
10		eqss	⊢ ( 𝐵 = 𝐶 ↔ ( 𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵 ) )
11	8 9 10	3bitr4g	⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) ∧ ∅ ∈ 𝐴 ) → ( ( 𝐴 ·o 𝐵 ) = ( 𝐴 ·o 𝐶 ) ↔ 𝐵 = 𝐶 ) )