Metamath Proof Explorer


Theorem sumss2

Description: Change the index set of a sum by adding zeroes. (Contributed by Mario Carneiro, 15-Jul-2013) (Revised by Mario Carneiro, 20-Apr-2014)

Ref Expression
Assertion sumss2 ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ ( 𝐵 ⊆ ( ℤ𝑀 ) ∨ 𝐵 ∈ Fin ) ) → Σ 𝑘𝐴 𝐶 = Σ 𝑘𝐵 if ( 𝑘𝐴 , 𝐶 , 0 ) )

Proof

Step Hyp Ref Expression
1 simpll ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ 𝐵 ⊆ ( ℤ𝑀 ) ) → 𝐴𝐵 )
2 iftrue ( 𝑚𝐴 → if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) = 𝑚 / 𝑘 𝐶 )
3 2 adantl ( ( ∀ 𝑘𝐴 𝐶 ∈ ℂ ∧ 𝑚𝐴 ) → if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) = 𝑚 / 𝑘 𝐶 )
4 nfcsb1v 𝑘 𝑚 / 𝑘 𝐶
5 4 nfel1 𝑘 𝑚 / 𝑘 𝐶 ∈ ℂ
6 csbeq1a ( 𝑘 = 𝑚𝐶 = 𝑚 / 𝑘 𝐶 )
7 6 eleq1d ( 𝑘 = 𝑚 → ( 𝐶 ∈ ℂ ↔ 𝑚 / 𝑘 𝐶 ∈ ℂ ) )
8 5 7 rspc ( 𝑚𝐴 → ( ∀ 𝑘𝐴 𝐶 ∈ ℂ → 𝑚 / 𝑘 𝐶 ∈ ℂ ) )
9 8 impcom ( ( ∀ 𝑘𝐴 𝐶 ∈ ℂ ∧ 𝑚𝐴 ) → 𝑚 / 𝑘 𝐶 ∈ ℂ )
10 3 9 eqeltrd ( ( ∀ 𝑘𝐴 𝐶 ∈ ℂ ∧ 𝑚𝐴 ) → if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) ∈ ℂ )
11 10 ad4ant24 ( ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ 𝐵 ⊆ ( ℤ𝑀 ) ) ∧ 𝑚𝐴 ) → if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) ∈ ℂ )
12 eldifn ( 𝑚 ∈ ( 𝐵𝐴 ) → ¬ 𝑚𝐴 )
13 12 iffalsed ( 𝑚 ∈ ( 𝐵𝐴 ) → if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) = 0 )
14 13 adantl ( ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ 𝐵 ⊆ ( ℤ𝑀 ) ) ∧ 𝑚 ∈ ( 𝐵𝐴 ) ) → if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) = 0 )
15 simpr ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ 𝐵 ⊆ ( ℤ𝑀 ) ) → 𝐵 ⊆ ( ℤ𝑀 ) )
16 1 11 14 15 sumss ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ 𝐵 ⊆ ( ℤ𝑀 ) ) → Σ 𝑚𝐴 if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) = Σ 𝑚𝐵 if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) )
17 simpll ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ 𝐵 ∈ Fin ) → 𝐴𝐵 )
18 10 ad4ant24 ( ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ 𝐵 ∈ Fin ) ∧ 𝑚𝐴 ) → if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) ∈ ℂ )
19 13 adantl ( ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ 𝐵 ∈ Fin ) ∧ 𝑚 ∈ ( 𝐵𝐴 ) ) → if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) = 0 )
20 simpr ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ 𝐵 ∈ Fin ) → 𝐵 ∈ Fin )
21 17 18 19 20 fsumss ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ 𝐵 ∈ Fin ) → Σ 𝑚𝐴 if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) = Σ 𝑚𝐵 if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) )
22 16 21 jaodan ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ ( 𝐵 ⊆ ( ℤ𝑀 ) ∨ 𝐵 ∈ Fin ) ) → Σ 𝑚𝐴 if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) = Σ 𝑚𝐵 if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) )
23 iftrue ( 𝑘𝐴 → if ( 𝑘𝐴 , 𝐶 , 0 ) = 𝐶 )
24 23 sumeq2i Σ 𝑘𝐴 if ( 𝑘𝐴 , 𝐶 , 0 ) = Σ 𝑘𝐴 𝐶
25 nfcv 𝑚 if ( 𝑘𝐴 , 𝐶 , 0 )
26 nfv 𝑘 𝑚𝐴
27 nfcv 𝑘 0
28 26 4 27 nfif 𝑘 if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 )
29 eleq1w ( 𝑘 = 𝑚 → ( 𝑘𝐴𝑚𝐴 ) )
30 29 6 ifbieq1d ( 𝑘 = 𝑚 → if ( 𝑘𝐴 , 𝐶 , 0 ) = if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 ) )
31 25 28 30 cbvsumi Σ 𝑘𝐴 if ( 𝑘𝐴 , 𝐶 , 0 ) = Σ 𝑚𝐴 if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 )
32 24 31 eqtr3i Σ 𝑘𝐴 𝐶 = Σ 𝑚𝐴 if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 )
33 25 28 30 cbvsumi Σ 𝑘𝐵 if ( 𝑘𝐴 , 𝐶 , 0 ) = Σ 𝑚𝐵 if ( 𝑚𝐴 , 𝑚 / 𝑘 𝐶 , 0 )
34 22 32 33 3eqtr4g ( ( ( 𝐴𝐵 ∧ ∀ 𝑘𝐴 𝐶 ∈ ℂ ) ∧ ( 𝐵 ⊆ ( ℤ𝑀 ) ∨ 𝐵 ∈ Fin ) ) → Σ 𝑘𝐴 𝐶 = Σ 𝑘𝐵 if ( 𝑘𝐴 , 𝐶 , 0 ) )