Theorem fsumdifsnconst

Description: The sum of constant terms ( k is not free in C ) over an index set excluding a singleton. (Contributed by AV, 7-Jan-2022)

	Ref	Expression
Assertion	fsumdifsnconst	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ℂ ) → Σ 𝑘 ∈ ( 𝐴 ∖ { 𝐵 } ) 𝐶 = ( ( ( ♯ ‘ 𝐴 ) − 1 ) · 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		diffi	⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∖ { 𝐵 } ) ∈ Fin )
2	1	anim1i	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 ∖ { 𝐵 } ) ∈ Fin ∧ 𝐶 ∈ ℂ ) )
3	2	3adant2	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 ∖ { 𝐵 } ) ∈ Fin ∧ 𝐶 ∈ ℂ ) )
4		fsumconst	⊢ ( ( ( 𝐴 ∖ { 𝐵 } ) ∈ Fin ∧ 𝐶 ∈ ℂ ) → Σ 𝑘 ∈ ( 𝐴 ∖ { 𝐵 } ) 𝐶 = ( ( ♯ ‘ ( 𝐴 ∖ { 𝐵 } ) ) · 𝐶 ) )
5	3 4	syl	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ℂ ) → Σ 𝑘 ∈ ( 𝐴 ∖ { 𝐵 } ) 𝐶 = ( ( ♯ ‘ ( 𝐴 ∖ { 𝐵 } ) ) · 𝐶 ) )
6		hashdifsn	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ) → ( ♯ ‘ ( 𝐴 ∖ { 𝐵 } ) ) = ( ( ♯ ‘ 𝐴 ) − 1 ) )
7	6	3adant3	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ℂ ) → ( ♯ ‘ ( 𝐴 ∖ { 𝐵 } ) ) = ( ( ♯ ‘ 𝐴 ) − 1 ) )
8	7	oveq1d	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ℂ ) → ( ( ♯ ‘ ( 𝐴 ∖ { 𝐵 } ) ) · 𝐶 ) = ( ( ( ♯ ‘ 𝐴 ) − 1 ) · 𝐶 ) )
9	5 8	eqtrd	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ℂ ) → Σ 𝑘 ∈ ( 𝐴 ∖ { 𝐵 } ) 𝐶 = ( ( ( ♯ ‘ 𝐴 ) − 1 ) · 𝐶 ) )