Theorem lcmfunsn

Description: The _lcm function for a union of a set of integer and a singleton. (Contributed by AV, 26-Aug-2020)

	Ref	Expression
Assertion	lcmfunsn	⊢ ( ( 𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin ∧ 𝑁 ∈ ℤ ) → ( lcm ‘ ( 𝑌 ∪ { 𝑁 } ) ) = ( ( lcm ‘ 𝑌 ) lcm 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		lcmfunsnlem	⊢ ( ( 𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin ) → ( ∀ 𝑘 ∈ ℤ ( ∀ 𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → ( lcm ‘ 𝑌 ) ∥ 𝑘 ) ∧ ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑌 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑌 ) lcm 𝑛 ) ) )
2		sneq	⊢ ( 𝑛 = 𝑁 → { 𝑛 } = { 𝑁 } )
3	2	uneq2d	⊢ ( 𝑛 = 𝑁 → ( 𝑌 ∪ { 𝑛 } ) = ( 𝑌 ∪ { 𝑁 } ) )
4	3	fveq2d	⊢ ( 𝑛 = 𝑁 → ( lcm ‘ ( 𝑌 ∪ { 𝑛 } ) ) = ( lcm ‘ ( 𝑌 ∪ { 𝑁 } ) ) )
5		oveq2	⊢ ( 𝑛 = 𝑁 → ( ( lcm ‘ 𝑌 ) lcm 𝑛 ) = ( ( lcm ‘ 𝑌 ) lcm 𝑁 ) )
6	4 5	eqeq12d	⊢ ( 𝑛 = 𝑁 → ( ( lcm ‘ ( 𝑌 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑌 ) lcm 𝑛 ) ↔ ( lcm ‘ ( 𝑌 ∪ { 𝑁 } ) ) = ( ( lcm ‘ 𝑌 ) lcm 𝑁 ) ) )
7	6	rspccv	⊢ ( ∀ 𝑛 ∈ ℤ ( lcm ‘ ( 𝑌 ∪ { 𝑛 } ) ) = ( ( lcm ‘ 𝑌 ) lcm 𝑛 ) → ( 𝑁 ∈ ℤ → ( lcm ‘ ( 𝑌 ∪ { 𝑁 } ) ) = ( ( lcm ‘ 𝑌 ) lcm 𝑁 ) ) )
8	1 7	simpl2im	⊢ ( ( 𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin ) → ( 𝑁 ∈ ℤ → ( lcm ‘ ( 𝑌 ∪ { 𝑁 } ) ) = ( ( lcm ‘ 𝑌 ) lcm 𝑁 ) ) )
9	8	3impia	⊢ ( ( 𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin ∧ 𝑁 ∈ ℤ ) → ( lcm ‘ ( 𝑌 ∪ { 𝑁 } ) ) = ( ( lcm ‘ 𝑌 ) lcm 𝑁 ) )