| Step |
Hyp |
Ref |
Expression |
| 1 |
|
zsum.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
| 2 |
|
zsum.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
| 3 |
|
zsum.3 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝑍 ) |
| 4 |
|
zsum.4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) |
| 5 |
|
zsum.5 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
| 6 |
|
eleq1w |
⊢ ( 𝑛 = 𝑖 → ( 𝑛 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴 ) ) |
| 7 |
|
csbeq1 |
⊢ ( 𝑛 = 𝑖 → ⦋ 𝑛 / 𝑘 ⦌ 𝐵 = ⦋ 𝑖 / 𝑘 ⦌ 𝐵 ) |
| 8 |
6 7
|
ifbieq1d |
⊢ ( 𝑛 = 𝑖 → if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) = if ( 𝑖 ∈ 𝐴 , ⦋ 𝑖 / 𝑘 ⦌ 𝐵 , 0 ) ) |
| 9 |
8
|
cbvmptv |
⊢ ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) = ( 𝑖 ∈ ℤ ↦ if ( 𝑖 ∈ 𝐴 , ⦋ 𝑖 / 𝑘 ⦌ 𝐵 , 0 ) ) |
| 10 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℤ ) ∧ 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ) → 𝜑 ) |
| 11 |
5
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℂ ) |
| 12 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑖 / 𝑘 ⦌ 𝐵 |
| 13 |
12
|
nfel1 |
⊢ Ⅎ 𝑘 ⦋ 𝑖 / 𝑘 ⦌ 𝐵 ∈ ℂ |
| 14 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑖 → 𝐵 = ⦋ 𝑖 / 𝑘 ⦌ 𝐵 ) |
| 15 |
14
|
eleq1d |
⊢ ( 𝑘 = 𝑖 → ( 𝐵 ∈ ℂ ↔ ⦋ 𝑖 / 𝑘 ⦌ 𝐵 ∈ ℂ ) ) |
| 16 |
13 15
|
rspc |
⊢ ( 𝑖 ∈ 𝐴 → ( ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋ 𝑖 / 𝑘 ⦌ 𝐵 ∈ ℂ ) ) |
| 17 |
11 16
|
syl5 |
⊢ ( 𝑖 ∈ 𝐴 → ( 𝜑 → ⦋ 𝑖 / 𝑘 ⦌ 𝐵 ∈ ℂ ) ) |
| 18 |
10 17
|
mpan9 |
⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℤ ) ∧ 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ) ∧ 𝑖 ∈ 𝐴 ) → ⦋ 𝑖 / 𝑘 ⦌ 𝐵 ∈ ℂ ) |
| 19 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℤ ) ∧ 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ) → 𝑚 ∈ ℤ ) |
| 20 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℤ ) ∧ 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ) → 𝑀 ∈ ℤ ) |
| 21 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℤ ) ∧ 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ) → 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ) |
| 22 |
3 1
|
sseqtrdi |
⊢ ( 𝜑 → 𝐴 ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
| 23 |
22
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℤ ) ∧ 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ) → 𝐴 ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
| 24 |
9 18 19 20 21 23
|
sumrb |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℤ ) ∧ 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ) → ( seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ↔ seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 25 |
24
|
biimpd |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℤ ) ∧ 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ) → ( seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 26 |
25
|
expimpd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℤ ) → ( ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 27 |
26
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 28 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → 𝐴 ⊆ 𝑍 ) |
| 29 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
| 30 |
1 29
|
eqsstri |
⊢ 𝑍 ⊆ ℤ |
| 31 |
|
zssre |
⊢ ℤ ⊆ ℝ |
| 32 |
30 31
|
sstri |
⊢ 𝑍 ⊆ ℝ |
| 33 |
|
ltso |
⊢ < Or ℝ |
| 34 |
|
soss |
⊢ ( 𝑍 ⊆ ℝ → ( < Or ℝ → < Or 𝑍 ) ) |
| 35 |
32 33 34
|
mp2 |
⊢ < Or 𝑍 |
| 36 |
|
soss |
⊢ ( 𝐴 ⊆ 𝑍 → ( < Or 𝑍 → < Or 𝐴 ) ) |
| 37 |
28 35 36
|
mpisyl |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → < Or 𝐴 ) |
| 38 |
|
fzfi |
⊢ ( 1 ... 𝑚 ) ∈ Fin |
| 39 |
|
ovex |
⊢ ( 1 ... 𝑚 ) ∈ V |
| 40 |
39
|
f1oen |
⊢ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 → ( 1 ... 𝑚 ) ≈ 𝐴 ) |
| 41 |
40
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → ( 1 ... 𝑚 ) ≈ 𝐴 ) |
| 42 |
41
|
ensymd |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → 𝐴 ≈ ( 1 ... 𝑚 ) ) |
| 43 |
|
enfii |
⊢ ( ( ( 1 ... 𝑚 ) ∈ Fin ∧ 𝐴 ≈ ( 1 ... 𝑚 ) ) → 𝐴 ∈ Fin ) |
| 44 |
38 42 43
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → 𝐴 ∈ Fin ) |
| 45 |
|
fz1iso |
⊢ ( ( < Or 𝐴 ∧ 𝐴 ∈ Fin ) → ∃ 𝑔 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) ) |
| 46 |
37 44 45
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → ∃ 𝑔 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) ) |
| 47 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) ) ) → 𝜑 ) |
| 48 |
47 17
|
mpan9 |
⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) ) ) ∧ 𝑖 ∈ 𝐴 ) → ⦋ 𝑖 / 𝑘 ⦌ 𝐵 ∈ ℂ ) |
| 49 |
|
fveq2 |
⊢ ( 𝑛 = 𝑗 → ( 𝑓 ‘ 𝑛 ) = ( 𝑓 ‘ 𝑗 ) ) |
| 50 |
49
|
csbeq1d |
⊢ ( 𝑛 = 𝑗 → ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 = ⦋ ( 𝑓 ‘ 𝑗 ) / 𝑘 ⦌ 𝐵 ) |
| 51 |
|
csbcow |
⊢ ⦋ ( 𝑓 ‘ 𝑗 ) / 𝑖 ⦌ ⦋ 𝑖 / 𝑘 ⦌ 𝐵 = ⦋ ( 𝑓 ‘ 𝑗 ) / 𝑘 ⦌ 𝐵 |
| 52 |
50 51
|
eqtr4di |
⊢ ( 𝑛 = 𝑗 → ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 = ⦋ ( 𝑓 ‘ 𝑗 ) / 𝑖 ⦌ ⦋ 𝑖 / 𝑘 ⦌ 𝐵 ) |
| 53 |
52
|
cbvmptv |
⊢ ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) = ( 𝑗 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑗 ) / 𝑖 ⦌ ⦋ 𝑖 / 𝑘 ⦌ 𝐵 ) |
| 54 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ ↦ ⦋ ( 𝑔 ‘ 𝑗 ) / 𝑖 ⦌ ⦋ 𝑖 / 𝑘 ⦌ 𝐵 ) = ( 𝑗 ∈ ℕ ↦ ⦋ ( 𝑔 ‘ 𝑗 ) / 𝑖 ⦌ ⦋ 𝑖 / 𝑘 ⦌ 𝐵 ) |
| 55 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) ) ) → 𝑚 ∈ ℕ ) |
| 56 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) ) ) → 𝑀 ∈ ℤ ) |
| 57 |
22
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) ) ) → 𝐴 ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
| 58 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) ) ) → 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) |
| 59 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) ) ) → 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) ) |
| 60 |
9 48 53 54 55 56 57 58 59
|
summolem2a |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) ) ) → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) |
| 61 |
60
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → ( 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) |
| 62 |
61
|
exlimdv |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → ( ∃ 𝑔 𝑔 Isom < , < ( ( 1 ... ( ♯ ‘ 𝐴 ) ) , 𝐴 ) → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) |
| 63 |
46 62
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) |
| 64 |
|
breq2 |
⊢ ( 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) → ( seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ↔ seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) |
| 65 |
63 64
|
syl5ibrcom |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → ( 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 66 |
65
|
expimpd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 67 |
66
|
exlimdv |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 68 |
67
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 69 |
27 68
|
jaod |
⊢ ( 𝜑 → ( ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 70 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) → 𝑀 ∈ ℤ ) |
| 71 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) → 𝐴 ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
| 72 |
|
simpr |
⊢ ( ( 𝜑 ∧ seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) |
| 73 |
|
fveq2 |
⊢ ( 𝑚 = 𝑀 → ( ℤ≥ ‘ 𝑚 ) = ( ℤ≥ ‘ 𝑀 ) ) |
| 74 |
73
|
sseq2d |
⊢ ( 𝑚 = 𝑀 → ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ↔ 𝐴 ⊆ ( ℤ≥ ‘ 𝑀 ) ) ) |
| 75 |
|
seqeq1 |
⊢ ( 𝑚 = 𝑀 → seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) = seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ) |
| 76 |
75
|
breq1d |
⊢ ( 𝑚 = 𝑀 → ( seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ↔ seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 77 |
74 76
|
anbi12d |
⊢ ( 𝑚 = 𝑀 → ( ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ↔ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑀 ) ∧ seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) ) |
| 78 |
77
|
rspcev |
⊢ ( ( 𝑀 ∈ ℤ ∧ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑀 ) ∧ seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) → ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 79 |
70 71 72 78
|
syl12anc |
⊢ ( ( 𝜑 ∧ seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) → ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 80 |
79
|
orcd |
⊢ ( ( 𝜑 ∧ seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) → ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) ) |
| 81 |
80
|
ex |
⊢ ( 𝜑 → ( seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 → ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) ) ) |
| 82 |
69 81
|
impbid |
⊢ ( 𝜑 → ( ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) ↔ seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ) |
| 83 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
| 84 |
29 83
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑗 ∈ ℤ ) |
| 85 |
83 1
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑗 ∈ 𝑍 ) |
| 86 |
4
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝑍 ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) |
| 87 |
86
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ∀ 𝑘 ∈ 𝑍 ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) |
| 88 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) |
| 89 |
88
|
nfeq2 |
⊢ Ⅎ 𝑘 ( 𝐹 ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) |
| 90 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑗 ) ) |
| 91 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑗 → if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) = ⦋ 𝑗 / 𝑘 ⦌ if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) |
| 92 |
90 91
|
eqeq12d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ↔ ( 𝐹 ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) ) |
| 93 |
89 92
|
rspc |
⊢ ( 𝑗 ∈ 𝑍 → ( ∀ 𝑘 ∈ 𝑍 ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) → ( 𝐹 ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) ) |
| 94 |
85 87 93
|
sylc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 𝐹 ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) |
| 95 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑗 ) ∈ V |
| 96 |
94 95
|
eqeltrrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ⦋ 𝑗 / 𝑘 ⦌ if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ∈ V ) |
| 97 |
|
nfcv |
⊢ Ⅎ 𝑛 if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) |
| 98 |
|
nfv |
⊢ Ⅎ 𝑘 𝑛 ∈ 𝐴 |
| 99 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑛 / 𝑘 ⦌ 𝐵 |
| 100 |
|
nfcv |
⊢ Ⅎ 𝑘 0 |
| 101 |
98 99 100
|
nfif |
⊢ Ⅎ 𝑘 if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) |
| 102 |
|
eleq1w |
⊢ ( 𝑘 = 𝑛 → ( 𝑘 ∈ 𝐴 ↔ 𝑛 ∈ 𝐴 ) ) |
| 103 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑛 → 𝐵 = ⦋ 𝑛 / 𝑘 ⦌ 𝐵 ) |
| 104 |
102 103
|
ifbieq1d |
⊢ ( 𝑘 = 𝑛 → if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) = if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) |
| 105 |
97 101 104
|
cbvmpt |
⊢ ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) = ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) |
| 106 |
105
|
eqcomi |
⊢ ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) = ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) |
| 107 |
106
|
fvmpts |
⊢ ( ( 𝑗 ∈ ℤ ∧ ⦋ 𝑗 / 𝑘 ⦌ if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ∈ V ) → ( ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) |
| 108 |
84 96 107
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) |
| 109 |
108 94
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ‘ 𝑗 ) = ( 𝐹 ‘ 𝑗 ) ) |
| 110 |
2 109
|
seqfeq |
⊢ ( 𝜑 → seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) = seq 𝑀 ( + , 𝐹 ) ) |
| 111 |
110
|
breq1d |
⊢ ( 𝜑 → ( seq 𝑀 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ↔ seq 𝑀 ( + , 𝐹 ) ⇝ 𝑥 ) ) |
| 112 |
82 111
|
bitrd |
⊢ ( 𝜑 → ( ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) ↔ seq 𝑀 ( + , 𝐹 ) ⇝ 𝑥 ) ) |
| 113 |
112
|
iotabidv |
⊢ ( 𝜑 → ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) ) = ( ℩ 𝑥 seq 𝑀 ( + , 𝐹 ) ⇝ 𝑥 ) ) |
| 114 |
|
df-sum |
⊢ Σ 𝑘 ∈ 𝐴 𝐵 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) ) |
| 115 |
|
df-fv |
⊢ ( ⇝ ‘ seq 𝑀 ( + , 𝐹 ) ) = ( ℩ 𝑥 seq 𝑀 ( + , 𝐹 ) ⇝ 𝑥 ) |
| 116 |
113 114 115
|
3eqtr4g |
⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘ seq 𝑀 ( + , 𝐹 ) ) ) |