Theorem fsummsndifre

Description: A finite sum with one of its integer summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018)

	Ref	Expression
Assertion	fsummsndifre	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → Σ 𝑘 ∈ ( 𝐴 ∖ { 𝑋 } ) 𝐵 ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		nfcv	⊢ Ⅎ 𝑥 𝐵
2		nfcsb1v	⊢ Ⅎ 𝑘 ⦋ 𝑥 / 𝑘 ⦌ 𝐵
3		csbeq1a	⊢ ( 𝑘 = 𝑥 → 𝐵 = ⦋ 𝑥 / 𝑘 ⦌ 𝐵 )
4	1 2 3	cbvsumi	⊢ Σ 𝑘 ∈ ( 𝐴 ∖ { 𝑋 } ) 𝐵 = Σ 𝑥 ∈ ( 𝐴 ∖ { 𝑋 } ) ⦋ 𝑥 / 𝑘 ⦌ 𝐵
5		diffi	⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∖ { 𝑋 } ) ∈ Fin )
6	5	adantr	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ( 𝐴 ∖ { 𝑋 } ) ∈ Fin )
7		eldifi	⊢ ( 𝑥 ∈ ( 𝐴 ∖ { 𝑋 } ) → 𝑥 ∈ 𝐴 )
8		rspcsbela	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ )
9	7 8	sylan	⊢ ( ( 𝑥 ∈ ( 𝐴 ∖ { 𝑋 } ) ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ )
10	9	zred	⊢ ( ( 𝑥 ∈ ( 𝐴 ∖ { 𝑋 } ) ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℝ )
11	10	expcom	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ → ( 𝑥 ∈ ( 𝐴 ∖ { 𝑋 } ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℝ ) )
12	11	adantl	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ( 𝑥 ∈ ( 𝐴 ∖ { 𝑋 } ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℝ ) )
13	12	imp	⊢ ( ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) ∧ 𝑥 ∈ ( 𝐴 ∖ { 𝑋 } ) ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℝ )
14	6 13	fsumrecl	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → Σ 𝑥 ∈ ( 𝐴 ∖ { 𝑋 } ) ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℝ )
15	4 14	eqeltrid	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → Σ 𝑘 ∈ ( 𝐴 ∖ { 𝑋 } ) 𝐵 ∈ ℝ )