Theorem fsumsplitsndif

Description: Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 31-Aug-2018)

	Ref	Expression
Assertion	fsumsplitsndif	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → Σ 𝑘 ∈ 𝐴 𝐵 = ( Σ 𝑘 ∈ ( 𝐴 ∖ { 𝑋 } ) 𝐵 + ⦋ 𝑋 / 𝑘 ⦌ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		neldifsnd	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ¬ 𝑋 ∈ ( 𝐴 ∖ { 𝑋 } ) )
2		disjsn	⊢ ( ( ( 𝐴 ∖ { 𝑋 } ) ∩ { 𝑋 } ) = ∅ ↔ ¬ 𝑋 ∈ ( 𝐴 ∖ { 𝑋 } ) )
3	1 2	sylibr	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ( ( 𝐴 ∖ { 𝑋 } ) ∩ { 𝑋 } ) = ∅ )
4		uncom	⊢ ( ( 𝐴 ∖ { 𝑋 } ) ∪ { 𝑋 } ) = ( { 𝑋 } ∪ ( 𝐴 ∖ { 𝑋 } ) )
5		simp2	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → 𝑋 ∈ 𝐴 )
6	5	snssd	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → { 𝑋 } ⊆ 𝐴 )
7		undif	⊢ ( { 𝑋 } ⊆ 𝐴 ↔ ( { 𝑋 } ∪ ( 𝐴 ∖ { 𝑋 } ) ) = 𝐴 )
8	6 7	sylib	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ( { 𝑋 } ∪ ( 𝐴 ∖ { 𝑋 } ) ) = 𝐴 )
9	4 8	syl5req	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → 𝐴 = ( ( 𝐴 ∖ { 𝑋 } ) ∪ { 𝑋 } ) )
10		simp1	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → 𝐴 ∈ Fin )
11		rspcsbela	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ )
12	11	zcnd	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℂ )
13	12	expcom	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ → ( 𝑥 ∈ 𝐴 → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℂ ) )
14	13	3ad2ant3	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ( 𝑥 ∈ 𝐴 → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℂ ) )
15	14	imp	⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) ∧ 𝑥 ∈ 𝐴 ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℂ )
16	3 9 10 15	fsumsplit	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → Σ 𝑥 ∈ 𝐴 ⦋ 𝑥 / 𝑘 ⦌ 𝐵 = ( Σ 𝑥 ∈ ( 𝐴 ∖ { 𝑋 } ) ⦋ 𝑥 / 𝑘 ⦌ 𝐵 + Σ 𝑥 ∈ { 𝑋 } ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ) )
17		nfcv	⊢ Ⅎ 𝑥 𝐵
18		nfcsb1v	⊢ Ⅎ 𝑘 ⦋ 𝑥 / 𝑘 ⦌ 𝐵
19		csbeq1a	⊢ ( 𝑘 = 𝑥 → 𝐵 = ⦋ 𝑥 / 𝑘 ⦌ 𝐵 )
20	17 18 19	cbvsumi	⊢ Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑥 ∈ 𝐴 ⦋ 𝑥 / 𝑘 ⦌ 𝐵
21	17 18 19	cbvsumi	⊢ Σ 𝑘 ∈ ( 𝐴 ∖ { 𝑋 } ) 𝐵 = Σ 𝑥 ∈ ( 𝐴 ∖ { 𝑋 } ) ⦋ 𝑥 / 𝑘 ⦌ 𝐵
22	17 18 19	cbvsumi	⊢ Σ 𝑘 ∈ { 𝑋 } 𝐵 = Σ 𝑥 ∈ { 𝑋 } ⦋ 𝑥 / 𝑘 ⦌ 𝐵
23	21 22	oveq12i	⊢ ( Σ 𝑘 ∈ ( 𝐴 ∖ { 𝑋 } ) 𝐵 + Σ 𝑘 ∈ { 𝑋 } 𝐵 ) = ( Σ 𝑥 ∈ ( 𝐴 ∖ { 𝑋 } ) ⦋ 𝑥 / 𝑘 ⦌ 𝐵 + Σ 𝑥 ∈ { 𝑋 } ⦋ 𝑥 / 𝑘 ⦌ 𝐵 )
24	16 20 23	3eqtr4g	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → Σ 𝑘 ∈ 𝐴 𝐵 = ( Σ 𝑘 ∈ ( 𝐴 ∖ { 𝑋 } ) 𝐵 + Σ 𝑘 ∈ { 𝑋 } 𝐵 ) )
25		rspcsbela	⊢ ( ( 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ⦋ 𝑋 / 𝑘 ⦌ 𝐵 ∈ ℤ )
26	25	zcnd	⊢ ( ( 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ⦋ 𝑋 / 𝑘 ⦌ 𝐵 ∈ ℂ )
27	26	3adant1	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ⦋ 𝑋 / 𝑘 ⦌ 𝐵 ∈ ℂ )
28		sumsns	⊢ ( ( 𝑋 ∈ 𝐴 ∧ ⦋ 𝑋 / 𝑘 ⦌ 𝐵 ∈ ℂ ) → Σ 𝑘 ∈ { 𝑋 } 𝐵 = ⦋ 𝑋 / 𝑘 ⦌ 𝐵 )
29	5 27 28	syl2anc	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → Σ 𝑘 ∈ { 𝑋 } 𝐵 = ⦋ 𝑋 / 𝑘 ⦌ 𝐵 )
30	29	oveq2d	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ( Σ 𝑘 ∈ ( 𝐴 ∖ { 𝑋 } ) 𝐵 + Σ 𝑘 ∈ { 𝑋 } 𝐵 ) = ( Σ 𝑘 ∈ ( 𝐴 ∖ { 𝑋 } ) 𝐵 + ⦋ 𝑋 / 𝑘 ⦌ 𝐵 ) )
31	24 30	eqtrd	⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → Σ 𝑘 ∈ 𝐴 𝐵 = ( Σ 𝑘 ∈ ( 𝐴 ∖ { 𝑋 } ) 𝐵 + ⦋ 𝑋 / 𝑘 ⦌ 𝐵 ) )