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