Step |
Hyp |
Ref |
Expression |
1 |
|
nfdisj1 |
⊢ Ⅎ 𝑥 Disj 𝑥 ∈ 𝐴 𝑥 |
2 |
|
nfv |
⊢ Ⅎ 𝑥 𝐴 ∈ Fin |
3 |
|
nfv |
⊢ Ⅎ 𝑥 𝐴 ⊆ Fin |
4 |
1 2 3
|
nf3an |
⊢ Ⅎ 𝑥 ( Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin ) |
5 |
|
simp2 |
⊢ ( ( Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin ) → 𝐴 ∈ Fin ) |
6 |
|
simp3 |
⊢ ( ( Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin ) → 𝐴 ⊆ Fin ) |
7 |
|
simp1 |
⊢ ( ( Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin ) → Disj 𝑥 ∈ 𝐴 𝑥 ) |
8 |
4 5 6 7
|
hashunif |
⊢ ( ( Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = Σ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
9 |
|
simpl |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin ) → 𝐴 ∈ Fin ) |
10 |
|
dfss3 |
⊢ ( 𝐴 ⊆ Fin ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ Fin ) |
11 |
|
hashcl |
⊢ ( 𝑥 ∈ Fin → ( ♯ ‘ 𝑥 ) ∈ ℕ0 ) |
12 |
|
nn0re |
⊢ ( ( ♯ ‘ 𝑥 ) ∈ ℕ0 → ( ♯ ‘ 𝑥 ) ∈ ℝ ) |
13 |
|
nn0ge0 |
⊢ ( ( ♯ ‘ 𝑥 ) ∈ ℕ0 → 0 ≤ ( ♯ ‘ 𝑥 ) ) |
14 |
|
elrege0 |
⊢ ( ( ♯ ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ↔ ( ( ♯ ‘ 𝑥 ) ∈ ℝ ∧ 0 ≤ ( ♯ ‘ 𝑥 ) ) ) |
15 |
12 13 14
|
sylanbrc |
⊢ ( ( ♯ ‘ 𝑥 ) ∈ ℕ0 → ( ♯ ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) |
16 |
11 15
|
syl |
⊢ ( 𝑥 ∈ Fin → ( ♯ ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) |
17 |
16
|
ralimi |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝑥 ∈ Fin → ∀ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) |
18 |
10 17
|
sylbi |
⊢ ( 𝐴 ⊆ Fin → ∀ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) |
19 |
18
|
r19.21bi |
⊢ ( ( 𝐴 ⊆ Fin ∧ 𝑥 ∈ 𝐴 ) → ( ♯ ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) |
20 |
19
|
adantll |
⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin ) ∧ 𝑥 ∈ 𝐴 ) → ( ♯ ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) |
21 |
9 20
|
esumpfinval |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin ) → Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = Σ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
22 |
21
|
3adant1 |
⊢ ( ( Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin ) → Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = Σ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
23 |
8 22
|
eqtr4d |
⊢ ( ( Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
24 |
23
|
3adant1l |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥 ) ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
25 |
24
|
3expa |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥 ) ∧ 𝐴 ∈ Fin ) ∧ 𝐴 ⊆ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
26 |
|
uniexg |
⊢ ( 𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V ) |
27 |
10
|
notbii |
⊢ ( ¬ 𝐴 ⊆ Fin ↔ ¬ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ Fin ) |
28 |
|
rexnal |
⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin ↔ ¬ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ Fin ) |
29 |
27 28
|
bitr4i |
⊢ ( ¬ 𝐴 ⊆ Fin ↔ ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin ) |
30 |
|
elssuni |
⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴 ) |
31 |
|
ssfi |
⊢ ( ( ∪ 𝐴 ∈ Fin ∧ 𝑥 ⊆ ∪ 𝐴 ) → 𝑥 ∈ Fin ) |
32 |
31
|
expcom |
⊢ ( 𝑥 ⊆ ∪ 𝐴 → ( ∪ 𝐴 ∈ Fin → 𝑥 ∈ Fin ) ) |
33 |
32
|
con3d |
⊢ ( 𝑥 ⊆ ∪ 𝐴 → ( ¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin ) ) |
34 |
30 33
|
syl |
⊢ ( 𝑥 ∈ 𝐴 → ( ¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin ) ) |
35 |
34
|
rexlimiv |
⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin ) |
36 |
29 35
|
sylbi |
⊢ ( ¬ 𝐴 ⊆ Fin → ¬ ∪ 𝐴 ∈ Fin ) |
37 |
|
hashinf |
⊢ ( ( ∪ 𝐴 ∈ V ∧ ¬ ∪ 𝐴 ∈ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = +∞ ) |
38 |
26 36 37
|
syl2an |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = +∞ ) |
39 |
|
vex |
⊢ 𝑥 ∈ V |
40 |
|
hashinf |
⊢ ( ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ Fin ) → ( ♯ ‘ 𝑥 ) = +∞ ) |
41 |
39 40
|
mpan |
⊢ ( ¬ 𝑥 ∈ Fin → ( ♯ ‘ 𝑥 ) = +∞ ) |
42 |
41
|
reximi |
⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ∃ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = +∞ ) |
43 |
29 42
|
sylbi |
⊢ ( ¬ 𝐴 ⊆ Fin → ∃ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = +∞ ) |
44 |
|
nfv |
⊢ Ⅎ 𝑥 𝐴 ∈ 𝑉 |
45 |
|
nfre1 |
⊢ Ⅎ 𝑥 ∃ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = +∞ |
46 |
44 45
|
nfan |
⊢ Ⅎ 𝑥 ( 𝐴 ∈ 𝑉 ∧ ∃ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = +∞ ) |
47 |
|
simpl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∃ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = +∞ ) → 𝐴 ∈ 𝑉 ) |
48 |
|
hashf2 |
⊢ ♯ : V ⟶ ( 0 [,] +∞ ) |
49 |
|
ffvelrn |
⊢ ( ( ♯ : V ⟶ ( 0 [,] +∞ ) ∧ 𝑥 ∈ V ) → ( ♯ ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
50 |
48 39 49
|
mp2an |
⊢ ( ♯ ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) |
51 |
50
|
a1i |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ ∃ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = +∞ ) ∧ 𝑥 ∈ 𝐴 ) → ( ♯ ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
52 |
|
simpr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∃ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = +∞ ) → ∃ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = +∞ ) |
53 |
46 47 51 52
|
esumpinfval |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∃ 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = +∞ ) → Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = +∞ ) |
54 |
43 53
|
sylan2 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin ) → Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = +∞ ) |
55 |
38 54
|
eqtr4d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
56 |
55
|
3adant2 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ Fin ∧ ¬ 𝐴 ⊆ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
57 |
56
|
3adant1r |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥 ) ∧ 𝐴 ∈ Fin ∧ ¬ 𝐴 ⊆ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
58 |
57
|
3expa |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥 ) ∧ 𝐴 ∈ Fin ) ∧ ¬ 𝐴 ⊆ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
59 |
25 58
|
pm2.61dan |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥 ) ∧ 𝐴 ∈ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
60 |
|
pwfi |
⊢ ( ∪ 𝐴 ∈ Fin ↔ 𝒫 ∪ 𝐴 ∈ Fin ) |
61 |
|
pwuni |
⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 |
62 |
|
ssfi |
⊢ ( ( 𝒫 ∪ 𝐴 ∈ Fin ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴 ) → 𝐴 ∈ Fin ) |
63 |
61 62
|
mpan2 |
⊢ ( 𝒫 ∪ 𝐴 ∈ Fin → 𝐴 ∈ Fin ) |
64 |
60 63
|
sylbi |
⊢ ( ∪ 𝐴 ∈ Fin → 𝐴 ∈ Fin ) |
65 |
64
|
con3i |
⊢ ( ¬ 𝐴 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin ) |
66 |
26 65 37
|
syl2an |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = +∞ ) |
67 |
|
nftru |
⊢ Ⅎ 𝑥 ⊤ |
68 |
|
unrab |
⊢ ( { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∪ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) = { 𝑥 ∈ 𝐴 ∣ ( ( ♯ ‘ 𝑥 ) = 0 ∨ ¬ ( ♯ ‘ 𝑥 ) = 0 ) } |
69 |
|
exmid |
⊢ ( ( ♯ ‘ 𝑥 ) = 0 ∨ ¬ ( ♯ ‘ 𝑥 ) = 0 ) |
70 |
69
|
rgenw |
⊢ ∀ 𝑥 ∈ 𝐴 ( ( ♯ ‘ 𝑥 ) = 0 ∨ ¬ ( ♯ ‘ 𝑥 ) = 0 ) |
71 |
|
rabid2 |
⊢ ( 𝐴 = { 𝑥 ∈ 𝐴 ∣ ( ( ♯ ‘ 𝑥 ) = 0 ∨ ¬ ( ♯ ‘ 𝑥 ) = 0 ) } ↔ ∀ 𝑥 ∈ 𝐴 ( ( ♯ ‘ 𝑥 ) = 0 ∨ ¬ ( ♯ ‘ 𝑥 ) = 0 ) ) |
72 |
70 71
|
mpbir |
⊢ 𝐴 = { 𝑥 ∈ 𝐴 ∣ ( ( ♯ ‘ 𝑥 ) = 0 ∨ ¬ ( ♯ ‘ 𝑥 ) = 0 ) } |
73 |
68 72
|
eqtr4i |
⊢ ( { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∪ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) = 𝐴 |
74 |
73
|
a1i |
⊢ ( ⊤ → ( { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∪ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) = 𝐴 ) |
75 |
67 74
|
esumeq1d |
⊢ ( ⊤ → Σ* 𝑥 ∈ ( { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∪ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) ( ♯ ‘ 𝑥 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
76 |
75
|
mptru |
⊢ Σ* 𝑥 ∈ ( { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∪ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) ( ♯ ‘ 𝑥 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) |
77 |
|
nfrab1 |
⊢ Ⅎ 𝑥 { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } |
78 |
|
nfrab1 |
⊢ Ⅎ 𝑥 { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } |
79 |
|
rabexg |
⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∈ V ) |
80 |
|
rabexg |
⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ∈ V ) |
81 |
|
rabnc |
⊢ ( { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∩ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) = ∅ |
82 |
81
|
a1i |
⊢ ( 𝐴 ∈ 𝑉 → ( { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∩ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) = ∅ ) |
83 |
50
|
a1i |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ) → ( ♯ ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
84 |
50
|
a1i |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) → ( ♯ ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
85 |
44 77 78 79 80 82 83 84
|
esumsplit |
⊢ ( 𝐴 ∈ 𝑉 → Σ* 𝑥 ∈ ( { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∪ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) ( ♯ ‘ 𝑥 ) = ( Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) +𝑒 Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ) ) |
86 |
76 85
|
eqtr3id |
⊢ ( 𝐴 ∈ 𝑉 → Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = ( Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) +𝑒 Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ) ) |
87 |
86
|
adantr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = ( Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) +𝑒 Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ) ) |
88 |
|
nfv |
⊢ Ⅎ 𝑥 ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) |
89 |
80
|
adantr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ∈ V ) |
90 |
|
simpr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ¬ 𝐴 ∈ Fin ) |
91 |
|
dfrab3 |
⊢ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } = ( 𝐴 ∩ { 𝑥 ∣ ( ♯ ‘ 𝑥 ) = 0 } ) |
92 |
|
hasheq0 |
⊢ ( 𝑥 ∈ V → ( ( ♯ ‘ 𝑥 ) = 0 ↔ 𝑥 = ∅ ) ) |
93 |
39 92
|
ax-mp |
⊢ ( ( ♯ ‘ 𝑥 ) = 0 ↔ 𝑥 = ∅ ) |
94 |
93
|
abbii |
⊢ { 𝑥 ∣ ( ♯ ‘ 𝑥 ) = 0 } = { 𝑥 ∣ 𝑥 = ∅ } |
95 |
|
df-sn |
⊢ { ∅ } = { 𝑥 ∣ 𝑥 = ∅ } |
96 |
94 95
|
eqtr4i |
⊢ { 𝑥 ∣ ( ♯ ‘ 𝑥 ) = 0 } = { ∅ } |
97 |
96
|
ineq2i |
⊢ ( 𝐴 ∩ { 𝑥 ∣ ( ♯ ‘ 𝑥 ) = 0 } ) = ( 𝐴 ∩ { ∅ } ) |
98 |
91 97
|
eqtri |
⊢ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } = ( 𝐴 ∩ { ∅ } ) |
99 |
|
snfi |
⊢ { ∅ } ∈ Fin |
100 |
|
inss2 |
⊢ ( 𝐴 ∩ { ∅ } ) ⊆ { ∅ } |
101 |
|
ssfi |
⊢ ( ( { ∅ } ∈ Fin ∧ ( 𝐴 ∩ { ∅ } ) ⊆ { ∅ } ) → ( 𝐴 ∩ { ∅ } ) ∈ Fin ) |
102 |
99 100 101
|
mp2an |
⊢ ( 𝐴 ∩ { ∅ } ) ∈ Fin |
103 |
98 102
|
eqeltri |
⊢ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∈ Fin |
104 |
103
|
a1i |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∈ Fin ) |
105 |
|
difinf |
⊢ ( ( ¬ 𝐴 ∈ Fin ∧ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∈ Fin ) → ¬ ( 𝐴 ∖ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ) ∈ Fin ) |
106 |
90 104 105
|
syl2anc |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ¬ ( 𝐴 ∖ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ) ∈ Fin ) |
107 |
|
notrab |
⊢ ( 𝐴 ∖ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ) = { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } |
108 |
107
|
eleq1i |
⊢ ( ( 𝐴 ∖ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ) ∈ Fin ↔ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ∈ Fin ) |
109 |
106 108
|
sylnib |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ¬ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ∈ Fin ) |
110 |
50
|
a1i |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) ∧ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) → ( ♯ ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
111 |
39
|
a1i |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) ∧ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) → 𝑥 ∈ V ) |
112 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) ∧ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) → 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) |
113 |
|
rabid |
⊢ ( 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ↔ ( 𝑥 ∈ 𝐴 ∧ ¬ ( ♯ ‘ 𝑥 ) = 0 ) ) |
114 |
112 113
|
sylib |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) ∧ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) → ( 𝑥 ∈ 𝐴 ∧ ¬ ( ♯ ‘ 𝑥 ) = 0 ) ) |
115 |
114
|
simprd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) ∧ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) → ¬ ( ♯ ‘ 𝑥 ) = 0 ) |
116 |
93
|
biimpri |
⊢ ( 𝑥 = ∅ → ( ♯ ‘ 𝑥 ) = 0 ) |
117 |
116
|
necon3bi |
⊢ ( ¬ ( ♯ ‘ 𝑥 ) = 0 → 𝑥 ≠ ∅ ) |
118 |
115 117
|
syl |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) ∧ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) → 𝑥 ≠ ∅ ) |
119 |
|
hashge1 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑥 ≠ ∅ ) → 1 ≤ ( ♯ ‘ 𝑥 ) ) |
120 |
111 118 119
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) ∧ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ) → 1 ≤ ( ♯ ‘ 𝑥 ) ) |
121 |
|
1xr |
⊢ 1 ∈ ℝ* |
122 |
121
|
a1i |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → 1 ∈ ℝ* ) |
123 |
|
0lt1 |
⊢ 0 < 1 |
124 |
123
|
a1i |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → 0 < 1 ) |
125 |
88 78 89 109 110 120 122 124
|
esumpinfsum |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) = +∞ ) |
126 |
125
|
oveq2d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ( Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) +𝑒 Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ) = ( Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) +𝑒 +∞ ) ) |
127 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
128 |
79
|
adantr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∈ V ) |
129 |
50
|
a1i |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) ∧ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ) → ( ♯ ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
130 |
129
|
ralrimiva |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ∀ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
131 |
77
|
esumcl |
⊢ ( ( { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ∈ V ∧ ∀ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) → Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
132 |
128 130 131
|
syl2anc |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
133 |
127 132
|
sselid |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ∈ ℝ* ) |
134 |
|
xrge0neqmnf |
⊢ ( Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) → Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ≠ -∞ ) |
135 |
132 134
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ≠ -∞ ) |
136 |
|
xaddpnf1 |
⊢ ( ( Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ∈ ℝ* ∧ Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) ≠ -∞ ) → ( Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) +𝑒 +∞ ) = +∞ ) |
137 |
133 135 136
|
syl2anc |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ( Σ* 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ( ♯ ‘ 𝑥 ) = 0 } ( ♯ ‘ 𝑥 ) +𝑒 +∞ ) = +∞ ) |
138 |
87 126 137
|
3eqtrd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) = +∞ ) |
139 |
66 138
|
eqtr4d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
140 |
139
|
adantlr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥 ) ∧ ¬ 𝐴 ∈ Fin ) → ( ♯ ‘ ∪ 𝐴 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |
141 |
59 140
|
pm2.61dan |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥 ) → ( ♯ ‘ ∪ 𝐴 ) = Σ* 𝑥 ∈ 𝐴 ( ♯ ‘ 𝑥 ) ) |