| Step |
Hyp |
Ref |
Expression |
| 1 |
|
oms.m |
⊢ 𝑀 = ( toOMeas ‘ 𝑅 ) |
| 2 |
|
oms.o |
⊢ ( 𝜑 → 𝑄 ∈ 𝑉 ) |
| 3 |
|
oms.r |
⊢ ( 𝜑 → 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ) |
| 4 |
|
omssubadd.a |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → 𝐴 ⊆ ∪ 𝑄 ) |
| 5 |
|
omssubadd.b |
⊢ ( 𝜑 → 𝑋 ≼ ω ) |
| 6 |
|
nnenom |
⊢ ℕ ≈ ω |
| 7 |
6
|
ensymi |
⊢ ω ≈ ℕ |
| 8 |
|
domentr |
⊢ ( ( 𝑋 ≼ ω ∧ ω ≈ ℕ ) → 𝑋 ≼ ℕ ) |
| 9 |
5 7 8
|
sylancl |
⊢ ( 𝜑 → 𝑋 ≼ ℕ ) |
| 10 |
|
brdomi |
⊢ ( 𝑋 ≼ ℕ → ∃ 𝑓 𝑓 : 𝑋 –1-1→ ℕ ) |
| 11 |
9 10
|
syl |
⊢ ( 𝜑 → ∃ 𝑓 𝑓 : 𝑋 –1-1→ ℕ ) |
| 12 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → ∃ 𝑓 𝑓 : 𝑋 –1-1→ ℕ ) |
| 13 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → 𝜑 ) |
| 14 |
|
ctex |
⊢ ( 𝑋 ≼ ω → 𝑋 ∈ V ) |
| 15 |
5 14
|
syl |
⊢ ( 𝜑 → 𝑋 ∈ V ) |
| 16 |
13 15
|
syl |
⊢ ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → 𝑋 ∈ V ) |
| 17 |
|
nfv |
⊢ Ⅎ 𝑦 𝜑 |
| 18 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑋 |
| 19 |
18
|
nfesum1 |
⊢ Ⅎ 𝑦 Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) |
| 20 |
|
nfcv |
⊢ Ⅎ 𝑦 ℝ |
| 21 |
19 20
|
nfel |
⊢ Ⅎ 𝑦 Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ |
| 22 |
17 21
|
nfan |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) |
| 23 |
|
nfv |
⊢ Ⅎ 𝑦 𝑓 : 𝑋 –1-1→ ℕ |
| 24 |
22 23
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) |
| 25 |
|
nfv |
⊢ Ⅎ 𝑦 𝑒 ∈ ℝ+ |
| 26 |
24 25
|
nfan |
⊢ Ⅎ 𝑦 ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) |
| 27 |
13
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → 𝜑 ) |
| 28 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → 𝑦 ∈ 𝑋 ) |
| 29 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → 𝑋 ∈ V ) |
| 30 |
|
omsf |
⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ) → ( toOMeas ‘ 𝑅 ) : 𝒫 ∪ dom 𝑅 ⟶ ( 0 [,] +∞ ) ) |
| 31 |
1
|
feq1i |
⊢ ( 𝑀 : 𝒫 ∪ dom 𝑅 ⟶ ( 0 [,] +∞ ) ↔ ( toOMeas ‘ 𝑅 ) : 𝒫 ∪ dom 𝑅 ⟶ ( 0 [,] +∞ ) ) |
| 32 |
30 31
|
sylibr |
⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ) → 𝑀 : 𝒫 ∪ dom 𝑅 ⟶ ( 0 [,] +∞ ) ) |
| 33 |
2 3 32
|
syl2anc |
⊢ ( 𝜑 → 𝑀 : 𝒫 ∪ dom 𝑅 ⟶ ( 0 [,] +∞ ) ) |
| 34 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → 𝑀 : 𝒫 ∪ dom 𝑅 ⟶ ( 0 [,] +∞ ) ) |
| 35 |
3
|
fdmd |
⊢ ( 𝜑 → dom 𝑅 = 𝑄 ) |
| 36 |
35
|
unieqd |
⊢ ( 𝜑 → ∪ dom 𝑅 = ∪ 𝑄 ) |
| 37 |
36
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → ∪ dom 𝑅 = ∪ 𝑄 ) |
| 38 |
4 37
|
sseqtrrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → 𝐴 ⊆ ∪ dom 𝑅 ) |
| 39 |
2
|
uniexd |
⊢ ( 𝜑 → ∪ 𝑄 ∈ V ) |
| 40 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → ∪ 𝑄 ∈ V ) |
| 41 |
|
ssexg |
⊢ ( ( 𝐴 ⊆ ∪ 𝑄 ∧ ∪ 𝑄 ∈ V ) → 𝐴 ∈ V ) |
| 42 |
4 40 41
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → 𝐴 ∈ V ) |
| 43 |
|
elpwg |
⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 ∪ dom 𝑅 ↔ 𝐴 ⊆ ∪ dom 𝑅 ) ) |
| 44 |
42 43
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → ( 𝐴 ∈ 𝒫 ∪ dom 𝑅 ↔ 𝐴 ⊆ ∪ dom 𝑅 ) ) |
| 45 |
38 44
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) |
| 46 |
34 45
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
| 47 |
46
|
adantlr |
⊢ ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
| 48 |
|
simpr |
⊢ ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) |
| 49 |
22 29 47 48
|
esumcvgre |
⊢ ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) |
| 50 |
49
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) |
| 51 |
50
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) |
| 52 |
|
rpssre |
⊢ ℝ+ ⊆ ℝ |
| 53 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → 𝑒 ∈ ℝ+ ) |
| 54 |
|
2rp |
⊢ 2 ∈ ℝ+ |
| 55 |
54
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → 2 ∈ ℝ+ ) |
| 56 |
|
df-f1 |
⊢ ( 𝑓 : 𝑋 –1-1→ ℕ ↔ ( 𝑓 : 𝑋 ⟶ ℕ ∧ Fun ◡ 𝑓 ) ) |
| 57 |
56
|
simplbi |
⊢ ( 𝑓 : 𝑋 –1-1→ ℕ → 𝑓 : 𝑋 ⟶ ℕ ) |
| 58 |
57
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → 𝑓 : 𝑋 ⟶ ℕ ) |
| 59 |
58
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑓 ‘ 𝑦 ) ∈ ℕ ) |
| 60 |
59
|
nnzd |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑓 ‘ 𝑦 ) ∈ ℤ ) |
| 61 |
55 60
|
rpexpcld |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ∈ ℝ+ ) |
| 62 |
61
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ∈ ℝ+ ) |
| 63 |
53 62
|
rpdivcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ+ ) |
| 64 |
52 63
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ ) |
| 65 |
64
|
adantl3r |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ ) |
| 66 |
|
rexadd |
⊢ ( ( ( 𝑀 ‘ 𝐴 ) ∈ ℝ ∧ ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ ) → ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) = ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 67 |
51 65 66
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) = ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 68 |
13 46
|
sylan |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
| 69 |
|
dfrp2 |
⊢ ℝ+ = ( 0 (,) +∞ ) |
| 70 |
|
ioossicc |
⊢ ( 0 (,) +∞ ) ⊆ ( 0 [,] +∞ ) |
| 71 |
69 70
|
eqsstri |
⊢ ℝ+ ⊆ ( 0 [,] +∞ ) |
| 72 |
71 63
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 73 |
72
|
adantl3r |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 74 |
68 73
|
xrge0addcld |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 75 |
67 74
|
eqeltrrd |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 76 |
52 53
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → 𝑒 ∈ ℝ ) |
| 77 |
76
|
adantl3r |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → 𝑒 ∈ ℝ ) |
| 78 |
52 61
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ∈ ℝ ) |
| 79 |
78
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ∈ ℝ ) |
| 80 |
79
|
adantl3r |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ∈ ℝ ) |
| 81 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → 𝑒 ∈ ℝ+ ) |
| 82 |
81
|
rpgt0d |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → 0 < 𝑒 ) |
| 83 |
|
2re |
⊢ 2 ∈ ℝ |
| 84 |
83
|
a1i |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → 2 ∈ ℝ ) |
| 85 |
60
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑓 ‘ 𝑦 ) ∈ ℤ ) |
| 86 |
85
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑓 ‘ 𝑦 ) ∈ ℤ ) |
| 87 |
|
2pos |
⊢ 0 < 2 |
| 88 |
87
|
a1i |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → 0 < 2 ) |
| 89 |
|
expgt0 |
⊢ ( ( 2 ∈ ℝ ∧ ( 𝑓 ‘ 𝑦 ) ∈ ℤ ∧ 0 < 2 ) → 0 < ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) |
| 90 |
84 86 88 89
|
syl3anc |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → 0 < ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) |
| 91 |
77 80 82 90
|
divgt0d |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → 0 < ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) |
| 92 |
65 51
|
ltaddposd |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 0 < ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ↔ ( 𝑀 ‘ 𝐴 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 93 |
91 92
|
mpbid |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 94 |
1
|
fveq1i |
⊢ ( 𝑀 ‘ 𝐴 ) = ( ( toOMeas ‘ 𝑅 ) ‘ 𝐴 ) |
| 95 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → 𝑄 ∈ 𝑉 ) |
| 96 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ) |
| 97 |
|
omsfval |
⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → ( ( toOMeas ‘ 𝑅 ) ‘ 𝐴 ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) ) |
| 98 |
95 96 4 97
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → ( ( toOMeas ‘ 𝑅 ) ‘ 𝐴 ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) ) |
| 99 |
94 98
|
eqtrid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) ) |
| 100 |
13 99
|
sylan |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) ) |
| 101 |
100
|
eqcomd |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) = ( 𝑀 ‘ 𝐴 ) ) |
| 102 |
101
|
breq1d |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ↔ ( 𝑀 ‘ 𝐴 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 103 |
93 102
|
mpbird |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 104 |
75 103
|
jca |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ( 0 [,] +∞ ) ∧ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 105 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
| 106 |
|
xrltso |
⊢ < Or ℝ* |
| 107 |
|
soss |
⊢ ( ( 0 [,] +∞ ) ⊆ ℝ* → ( < Or ℝ* → < Or ( 0 [,] +∞ ) ) ) |
| 108 |
105 106 107
|
mp2 |
⊢ < Or ( 0 [,] +∞ ) |
| 109 |
|
biid |
⊢ ( < Or ( 0 [,] +∞ ) ↔ < Or ( 0 [,] +∞ ) ) |
| 110 |
108 109
|
mpbi |
⊢ < Or ( 0 [,] +∞ ) |
| 111 |
110
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → < Or ( 0 [,] +∞ ) ) |
| 112 |
|
omscl |
⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) → ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ⊆ ( 0 [,] +∞ ) ) |
| 113 |
95 96 45 112
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ⊆ ( 0 [,] +∞ ) ) |
| 114 |
|
xrge0infss |
⊢ ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ⊆ ( 0 [,] +∞ ) → ∃ 𝑣 ∈ ( 0 [,] +∞ ) ( ∀ ℎ ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ¬ ℎ < 𝑣 ∧ ∀ ℎ ∈ ( 0 [,] +∞ ) ( 𝑣 < ℎ → ∃ 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) 𝑢 < ℎ ) ) ) |
| 115 |
113 114
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → ∃ 𝑣 ∈ ( 0 [,] +∞ ) ( ∀ ℎ ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ¬ ℎ < 𝑣 ∧ ∀ ℎ ∈ ( 0 [,] +∞ ) ( 𝑣 < ℎ → ∃ 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) 𝑢 < ℎ ) ) ) |
| 116 |
111 115
|
infglb |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → ( ( ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ( 0 [,] +∞ ) ∧ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → ∃ 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 117 |
116
|
imp |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) ∧ ( ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ( 0 [,] +∞ ) ∧ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∃ 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 118 |
27 28 104 117
|
syl21anc |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ∃ 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 119 |
|
eqid |
⊢ ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) = ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) |
| 120 |
|
esumex |
⊢ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ∈ V |
| 121 |
119 120
|
elrnmpti |
⊢ ( 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ↔ ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) |
| 122 |
121
|
anbi1i |
⊢ ( ( 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ↔ ( ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 123 |
|
r19.41v |
⊢ ( ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ( 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ↔ ( ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 124 |
122 123
|
bitr4i |
⊢ ( ( 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ↔ ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ( 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 125 |
124
|
exbii |
⊢ ( ∃ 𝑢 ( 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ↔ ∃ 𝑢 ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ( 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 126 |
|
df-rex |
⊢ ( ∃ 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ↔ ∃ 𝑢 ( 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 127 |
|
rexcom4 |
⊢ ( ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ∃ 𝑢 ( 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ↔ ∃ 𝑢 ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ( 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 128 |
125 126 127
|
3bitr4i |
⊢ ( ∃ 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ↔ ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ∃ 𝑢 ( 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 129 |
|
breq1 |
⊢ ( 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) → ( 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ↔ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 130 |
|
idd |
⊢ ( 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) → ( Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) → Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 131 |
129 130
|
sylbid |
⊢ ( 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) → ( 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) → Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 132 |
131
|
imp |
⊢ ( ( 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 133 |
132
|
exlimiv |
⊢ ( ∃ 𝑢 ( 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 134 |
133
|
reximi |
⊢ ( ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ∃ 𝑢 ( 𝑢 = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ∧ 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 135 |
128 134
|
sylbi |
⊢ ( ∃ 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) 𝑢 < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 136 |
118 135
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 137 |
|
simpr |
⊢ ( ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) → 𝑧 ≼ ω ) |
| 138 |
137
|
a1i |
⊢ ( 𝑧 ∈ 𝒫 dom 𝑅 → ( ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) → 𝑧 ≼ ω ) ) |
| 139 |
138
|
ss2rabi |
⊢ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ⊆ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } |
| 140 |
|
rexss |
⊢ ( { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ⊆ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } → ( ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ↔ ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ) |
| 141 |
139 140
|
ax-mp |
⊢ ( ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ↔ ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 142 |
|
unieq |
⊢ ( 𝑧 = 𝑥 → ∪ 𝑧 = ∪ 𝑥 ) |
| 143 |
142
|
sseq2d |
⊢ ( 𝑧 = 𝑥 → ( 𝐴 ⊆ ∪ 𝑧 ↔ 𝐴 ⊆ ∪ 𝑥 ) ) |
| 144 |
|
breq1 |
⊢ ( 𝑧 = 𝑥 → ( 𝑧 ≼ ω ↔ 𝑥 ≼ ω ) ) |
| 145 |
143 144
|
anbi12d |
⊢ ( 𝑧 = 𝑥 → ( ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) ↔ ( 𝐴 ⊆ ∪ 𝑥 ∧ 𝑥 ≼ ω ) ) ) |
| 146 |
145
|
elrab |
⊢ ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↔ ( 𝑥 ∈ 𝒫 dom 𝑅 ∧ ( 𝐴 ⊆ ∪ 𝑥 ∧ 𝑥 ≼ ω ) ) ) |
| 147 |
146
|
simprbi |
⊢ ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } → ( 𝐴 ⊆ ∪ 𝑥 ∧ 𝑥 ≼ ω ) ) |
| 148 |
147
|
simpld |
⊢ ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } → 𝐴 ⊆ ∪ 𝑥 ) |
| 149 |
148
|
a1i |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } → 𝐴 ⊆ ∪ 𝑥 ) ) |
| 150 |
149
|
anim1d |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → ( 𝐴 ⊆ ∪ 𝑥 ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ) |
| 151 |
150
|
reximdv |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ( 𝐴 ⊆ ∪ 𝑥 ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ) |
| 152 |
141 151
|
biimtrid |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ( 𝐴 ⊆ ∪ 𝑥 ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ) |
| 153 |
136 152
|
mpd |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ( 𝐴 ⊆ ∪ 𝑥 ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 154 |
153
|
ex |
⊢ ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ( 𝑦 ∈ 𝑋 → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ( 𝐴 ⊆ ∪ 𝑥 ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ) |
| 155 |
26 154
|
ralrimi |
⊢ ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ∀ 𝑦 ∈ 𝑋 ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ( 𝐴 ⊆ ∪ 𝑥 ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 156 |
|
unieq |
⊢ ( 𝑥 = ( 𝑔 ‘ 𝑦 ) → ∪ 𝑥 = ∪ ( 𝑔 ‘ 𝑦 ) ) |
| 157 |
156
|
sseq2d |
⊢ ( 𝑥 = ( 𝑔 ‘ 𝑦 ) → ( 𝐴 ⊆ ∪ 𝑥 ↔ 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ) ) |
| 158 |
|
esumeq1 |
⊢ ( 𝑥 = ( 𝑔 ‘ 𝑦 ) → Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) = Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ) |
| 159 |
158
|
breq1d |
⊢ ( 𝑥 = ( 𝑔 ‘ 𝑦 ) → ( Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ↔ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 160 |
157 159
|
anbi12d |
⊢ ( 𝑥 = ( 𝑔 ‘ 𝑦 ) → ( ( 𝐴 ⊆ ∪ 𝑥 ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ↔ ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ) |
| 161 |
160
|
ac6sg |
⊢ ( 𝑋 ∈ V → ( ∀ 𝑦 ∈ 𝑋 ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ( 𝐴 ⊆ ∪ 𝑥 ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → ∃ 𝑔 ( 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ) ) |
| 162 |
161
|
imp |
⊢ ( ( 𝑋 ∈ V ∧ ∀ 𝑦 ∈ 𝑋 ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ( 𝐴 ⊆ ∪ 𝑥 ∧ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∃ 𝑔 ( 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ) |
| 163 |
16 155 162
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ∃ 𝑔 ( 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ) |
| 164 |
13
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → 𝜑 ) |
| 165 |
38
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ dom 𝑅 ) |
| 166 |
|
iunss |
⊢ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ dom 𝑅 ↔ ∀ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ dom 𝑅 ) |
| 167 |
165 166
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ dom 𝑅 ) |
| 168 |
42
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝑋 𝐴 ∈ V ) |
| 169 |
|
iunexg |
⊢ ( ( 𝑋 ∈ V ∧ ∀ 𝑦 ∈ 𝑋 𝐴 ∈ V ) → ∪ 𝑦 ∈ 𝑋 𝐴 ∈ V ) |
| 170 |
15 168 169
|
syl2anc |
⊢ ( 𝜑 → ∪ 𝑦 ∈ 𝑋 𝐴 ∈ V ) |
| 171 |
|
elpwg |
⊢ ( ∪ 𝑦 ∈ 𝑋 𝐴 ∈ V → ( ∪ 𝑦 ∈ 𝑋 𝐴 ∈ 𝒫 ∪ dom 𝑅 ↔ ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ dom 𝑅 ) ) |
| 172 |
170 171
|
syl |
⊢ ( 𝜑 → ( ∪ 𝑦 ∈ 𝑋 𝐴 ∈ 𝒫 ∪ dom 𝑅 ↔ ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ dom 𝑅 ) ) |
| 173 |
167 172
|
mpbird |
⊢ ( 𝜑 → ∪ 𝑦 ∈ 𝑋 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) |
| 174 |
33 173
|
ffvelcdmd |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
| 175 |
105 174
|
sselid |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ∈ ℝ* ) |
| 176 |
164 175
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ∈ ℝ* ) |
| 177 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) |
| 178 |
29
|
ad4antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → 𝑋 ∈ V ) |
| 179 |
177 178
|
fexd |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → 𝑔 ∈ V ) |
| 180 |
|
rnexg |
⊢ ( 𝑔 ∈ V → ran 𝑔 ∈ V ) |
| 181 |
|
uniexg |
⊢ ( ran 𝑔 ∈ V → ∪ ran 𝑔 ∈ V ) |
| 182 |
179 180 181
|
3syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∪ ran 𝑔 ∈ V ) |
| 183 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → 𝜑 ) |
| 184 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑐 ∈ ∪ ran 𝑔 ) → 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ) |
| 185 |
|
frn |
⊢ ( 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } → ran 𝑔 ⊆ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) |
| 186 |
|
ssrab2 |
⊢ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ⊆ 𝒫 dom 𝑅 |
| 187 |
185 186
|
sstrdi |
⊢ ( 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } → ran 𝑔 ⊆ 𝒫 dom 𝑅 ) |
| 188 |
187
|
unissd |
⊢ ( 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } → ∪ ran 𝑔 ⊆ ∪ 𝒫 dom 𝑅 ) |
| 189 |
|
unipw |
⊢ ∪ 𝒫 dom 𝑅 = dom 𝑅 |
| 190 |
188 189
|
sseqtrdi |
⊢ ( 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } → ∪ ran 𝑔 ⊆ dom 𝑅 ) |
| 191 |
190
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → ∪ ran 𝑔 ⊆ dom 𝑅 ) |
| 192 |
35
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → dom 𝑅 = 𝑄 ) |
| 193 |
191 192
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → ∪ ran 𝑔 ⊆ 𝑄 ) |
| 194 |
193
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑐 ∈ ∪ ran 𝑔 ) → 𝑐 ∈ 𝑄 ) |
| 195 |
184 194
|
ffvelcdmd |
⊢ ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑐 ∈ ∪ ran 𝑔 ) → ( 𝑅 ‘ 𝑐 ) ∈ ( 0 [,] +∞ ) ) |
| 196 |
195
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → ∀ 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ ( 0 [,] +∞ ) ) |
| 197 |
183 177 196
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∀ 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ ( 0 [,] +∞ ) ) |
| 198 |
|
nfcv |
⊢ Ⅎ 𝑐 ∪ ran 𝑔 |
| 199 |
198
|
esumcl |
⊢ ( ( ∪ ran 𝑔 ∈ V ∧ ∀ 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ ( 0 [,] +∞ ) ) → Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ ( 0 [,] +∞ ) ) |
| 200 |
182 197 199
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ ( 0 [,] +∞ ) ) |
| 201 |
105 200
|
sselid |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ ℝ* ) |
| 202 |
|
simp-5r |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) |
| 203 |
202
|
rexrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ) |
| 204 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → 𝑒 ∈ ℝ+ ) |
| 205 |
204
|
rpxrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → 𝑒 ∈ ℝ* ) |
| 206 |
203 205
|
xaddcld |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) +𝑒 𝑒 ) ∈ ℝ* ) |
| 207 |
185
|
ad2antlr |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ran 𝑔 ⊆ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) |
| 208 |
|
sstr |
⊢ ( ( ran 𝑔 ⊆ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ∧ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ⊆ 𝒫 dom 𝑅 ) → ran 𝑔 ⊆ 𝒫 dom 𝑅 ) |
| 209 |
186 208
|
mpan2 |
⊢ ( ran 𝑔 ⊆ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } → ran 𝑔 ⊆ 𝒫 dom 𝑅 ) |
| 210 |
|
sspwuni |
⊢ ( ran 𝑔 ⊆ 𝒫 dom 𝑅 ↔ ∪ ran 𝑔 ⊆ dom 𝑅 ) |
| 211 |
209 210
|
sylib |
⊢ ( ran 𝑔 ⊆ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } → ∪ ran 𝑔 ⊆ dom 𝑅 ) |
| 212 |
207 211
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∪ ran 𝑔 ⊆ dom 𝑅 ) |
| 213 |
|
ffn |
⊢ ( 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } → 𝑔 Fn 𝑋 ) |
| 214 |
213
|
ad2antlr |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → 𝑔 Fn 𝑋 ) |
| 215 |
164 5
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → 𝑋 ≼ ω ) |
| 216 |
|
fnct |
⊢ ( ( 𝑔 Fn 𝑋 ∧ 𝑋 ≼ ω ) → 𝑔 ≼ ω ) |
| 217 |
|
rnct |
⊢ ( 𝑔 ≼ ω → ran 𝑔 ≼ ω ) |
| 218 |
216 217
|
syl |
⊢ ( ( 𝑔 Fn 𝑋 ∧ 𝑋 ≼ ω ) → ran 𝑔 ≼ ω ) |
| 219 |
|
dfss3 |
⊢ ( ran 𝑔 ⊆ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ↔ ∀ 𝑤 ∈ ran 𝑔 𝑤 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) |
| 220 |
219
|
biimpi |
⊢ ( ran 𝑔 ⊆ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } → ∀ 𝑤 ∈ ran 𝑔 𝑤 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) |
| 221 |
|
breq1 |
⊢ ( 𝑧 = 𝑤 → ( 𝑧 ≼ ω ↔ 𝑤 ≼ ω ) ) |
| 222 |
221
|
elrab |
⊢ ( 𝑤 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ↔ ( 𝑤 ∈ 𝒫 dom 𝑅 ∧ 𝑤 ≼ ω ) ) |
| 223 |
222
|
simprbi |
⊢ ( 𝑤 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } → 𝑤 ≼ ω ) |
| 224 |
223
|
ralimi |
⊢ ( ∀ 𝑤 ∈ ran 𝑔 𝑤 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } → ∀ 𝑤 ∈ ran 𝑔 𝑤 ≼ ω ) |
| 225 |
220 224
|
syl |
⊢ ( ran 𝑔 ⊆ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } → ∀ 𝑤 ∈ ran 𝑔 𝑤 ≼ ω ) |
| 226 |
|
unictb |
⊢ ( ( ran 𝑔 ≼ ω ∧ ∀ 𝑤 ∈ ran 𝑔 𝑤 ≼ ω ) → ∪ ran 𝑔 ≼ ω ) |
| 227 |
218 225 226
|
syl2an |
⊢ ( ( ( 𝑔 Fn 𝑋 ∧ 𝑋 ≼ ω ) ∧ ran 𝑔 ⊆ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → ∪ ran 𝑔 ≼ ω ) |
| 228 |
214 215 207 227
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∪ ran 𝑔 ≼ ω ) |
| 229 |
|
ctex |
⊢ ( ∪ ran 𝑔 ≼ ω → ∪ ran 𝑔 ∈ V ) |
| 230 |
|
elpwg |
⊢ ( ∪ ran 𝑔 ∈ V → ( ∪ ran 𝑔 ∈ 𝒫 dom 𝑅 ↔ ∪ ran 𝑔 ⊆ dom 𝑅 ) ) |
| 231 |
228 229 230
|
3syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( ∪ ran 𝑔 ∈ 𝒫 dom 𝑅 ↔ ∪ ran 𝑔 ⊆ dom 𝑅 ) ) |
| 232 |
212 231
|
mpbird |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∪ ran 𝑔 ∈ 𝒫 dom 𝑅 ) |
| 233 |
|
simpl |
⊢ ( ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ) |
| 234 |
233
|
ralimi |
⊢ ( ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → ∀ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ) |
| 235 |
|
fvssunirn |
⊢ ( 𝑔 ‘ 𝑦 ) ⊆ ∪ ran 𝑔 |
| 236 |
235
|
unissi |
⊢ ∪ ( 𝑔 ‘ 𝑦 ) ⊆ ∪ ∪ ran 𝑔 |
| 237 |
|
sstr |
⊢ ( ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ ∪ ( 𝑔 ‘ 𝑦 ) ⊆ ∪ ∪ ran 𝑔 ) → 𝐴 ⊆ ∪ ∪ ran 𝑔 ) |
| 238 |
236 237
|
mpan2 |
⊢ ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) → 𝐴 ⊆ ∪ ∪ ran 𝑔 ) |
| 239 |
238
|
ralimi |
⊢ ( ∀ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) → ∀ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ∪ ran 𝑔 ) |
| 240 |
|
iunss |
⊢ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ∪ ran 𝑔 ↔ ∀ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ∪ ran 𝑔 ) |
| 241 |
239 240
|
sylibr |
⊢ ( ∀ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) → ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ∪ ran 𝑔 ) |
| 242 |
234 241
|
syl |
⊢ ( ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ∪ ran 𝑔 ) |
| 243 |
242
|
adantl |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ∪ ran 𝑔 ) |
| 244 |
232 243 228
|
jca32 |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( ∪ ran 𝑔 ∈ 𝒫 dom 𝑅 ∧ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ∪ ran 𝑔 ∧ ∪ ran 𝑔 ≼ ω ) ) ) |
| 245 |
|
unieq |
⊢ ( 𝑧 = ∪ ran 𝑔 → ∪ 𝑧 = ∪ ∪ ran 𝑔 ) |
| 246 |
245
|
sseq2d |
⊢ ( 𝑧 = ∪ ran 𝑔 → ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ↔ ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ∪ ran 𝑔 ) ) |
| 247 |
|
breq1 |
⊢ ( 𝑧 = ∪ ran 𝑔 → ( 𝑧 ≼ ω ↔ ∪ ran 𝑔 ≼ ω ) ) |
| 248 |
246 247
|
anbi12d |
⊢ ( 𝑧 = ∪ ran 𝑔 → ( ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) ↔ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ∪ ran 𝑔 ∧ ∪ ran 𝑔 ≼ ω ) ) ) |
| 249 |
248
|
elrab |
⊢ ( ∪ ran 𝑔 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↔ ( ∪ ran 𝑔 ∈ 𝒫 dom 𝑅 ∧ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ ∪ ran 𝑔 ∧ ∪ ran 𝑔 ≼ ω ) ) ) |
| 250 |
244 249
|
sylibr |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∪ ran 𝑔 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) |
| 251 |
|
fveq2 |
⊢ ( 𝑐 = 𝑤 → ( 𝑅 ‘ 𝑐 ) = ( 𝑅 ‘ 𝑤 ) ) |
| 252 |
251
|
cbvesumv |
⊢ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) = Σ* 𝑤 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑤 ) |
| 253 |
|
esumeq1 |
⊢ ( 𝑥 = ∪ ran 𝑔 → Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) = Σ* 𝑤 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑤 ) ) |
| 254 |
253
|
rspceeqv |
⊢ ( ( ∪ ran 𝑔 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ∧ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) = Σ* 𝑤 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑤 ) ) → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) |
| 255 |
250 252 254
|
sylancl |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) |
| 256 |
|
esumex |
⊢ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ V |
| 257 |
|
eqid |
⊢ ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) = ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) |
| 258 |
257
|
elrnmpt |
⊢ ( Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ V → ( Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ↔ ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ) |
| 259 |
256 258
|
ax-mp |
⊢ ( Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ↔ ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) = Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) |
| 260 |
255 259
|
sylibr |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ) |
| 261 |
110
|
a1i |
⊢ ( 𝜑 → < Or ( 0 [,] +∞ ) ) |
| 262 |
|
omscl |
⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ ∪ 𝑦 ∈ 𝑋 𝐴 ∈ 𝒫 ∪ dom 𝑅 ) → ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ⊆ ( 0 [,] +∞ ) ) |
| 263 |
2 3 173 262
|
syl3anc |
⊢ ( 𝜑 → ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ⊆ ( 0 [,] +∞ ) ) |
| 264 |
|
xrge0infss |
⊢ ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ⊆ ( 0 [,] +∞ ) → ∃ 𝑒 ∈ ( 0 [,] +∞ ) ( ∀ 𝑡 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ¬ 𝑡 < 𝑒 ∧ ∀ 𝑡 ∈ ( 0 [,] +∞ ) ( 𝑒 < 𝑡 → ∃ 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) 𝑢 < 𝑡 ) ) ) |
| 265 |
263 264
|
syl |
⊢ ( 𝜑 → ∃ 𝑒 ∈ ( 0 [,] +∞ ) ( ∀ 𝑡 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) ¬ 𝑡 < 𝑒 ∧ ∀ 𝑡 ∈ ( 0 [,] +∞ ) ( 𝑒 < 𝑡 → ∃ 𝑢 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) 𝑢 < 𝑡 ) ) ) |
| 266 |
261 265
|
inflb |
⊢ ( 𝜑 → ( Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) → ¬ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) < inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) ) ) |
| 267 |
1
|
fveq1i |
⊢ ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) = ( ( toOMeas ‘ 𝑅 ) ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) |
| 268 |
167 36
|
sseqtrd |
⊢ ( 𝜑 → ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑄 ) |
| 269 |
|
omsfval |
⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑄 ) → ( ( toOMeas ‘ 𝑅 ) ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) ) |
| 270 |
2 3 268 269
|
syl3anc |
⊢ ( 𝜑 → ( ( toOMeas ‘ 𝑅 ) ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) ) |
| 271 |
267 270
|
eqtrid |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) ) |
| 272 |
271
|
breq2d |
⊢ ( 𝜑 → ( Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) < ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ↔ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) < inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) ) ) |
| 273 |
272
|
notbid |
⊢ ( 𝜑 → ( ¬ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) < ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ↔ ¬ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) < inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) , ( 0 [,] +∞ ) , < ) ) ) |
| 274 |
266 273
|
sylibrd |
⊢ ( 𝜑 → ( Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∪ 𝑦 ∈ 𝑋 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑤 ∈ 𝑥 ( 𝑅 ‘ 𝑤 ) ) → ¬ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) < ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ) ) |
| 275 |
164 260 274
|
sylc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ¬ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) < ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ) |
| 276 |
|
biid |
⊢ ( ¬ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) < ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ↔ ¬ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) < ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ) |
| 277 |
275 276
|
sylib |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ¬ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) < ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ) |
| 278 |
|
xrlenlt |
⊢ ( ( ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ∈ ℝ* ∧ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ∈ ℝ* ) → ( ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ↔ ¬ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) < ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ) ) |
| 279 |
176 201 278
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ↔ ¬ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) < ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ) ) |
| 280 |
277 279
|
mpbird |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ) |
| 281 |
|
nfv |
⊢ Ⅎ 𝑦 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } |
| 282 |
26 281
|
nfan |
⊢ Ⅎ 𝑦 ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) |
| 283 |
|
nfra1 |
⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 284 |
282 283
|
nfan |
⊢ Ⅎ 𝑦 ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 285 |
|
simp-6l |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ∧ 𝑦 ∈ 𝑋 ) → 𝜑 ) |
| 286 |
|
simpllr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ∧ 𝑦 ∈ 𝑋 ) → 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) |
| 287 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ∧ 𝑦 ∈ 𝑋 ) → 𝑦 ∈ 𝑋 ) |
| 288 |
3
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ) → 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ) |
| 289 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ) → 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) |
| 290 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ) → 𝑦 ∈ 𝑋 ) |
| 291 |
289 290
|
ffvelcdmd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ) → ( 𝑔 ‘ 𝑦 ) ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) |
| 292 |
186 291
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ) → ( 𝑔 ‘ 𝑦 ) ∈ 𝒫 dom 𝑅 ) |
| 293 |
292
|
elpwid |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ) → ( 𝑔 ‘ 𝑦 ) ⊆ dom 𝑅 ) |
| 294 |
288 293
|
fssdmd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ) → ( 𝑔 ‘ 𝑦 ) ⊆ 𝑄 ) |
| 295 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ) → 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ) |
| 296 |
294 295
|
sseldd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ) → 𝑤 ∈ 𝑄 ) |
| 297 |
288 296
|
ffvelcdmd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ) → ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) |
| 298 |
297
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → ∀ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) |
| 299 |
|
fvex |
⊢ ( 𝑔 ‘ 𝑦 ) ∈ V |
| 300 |
|
nfcv |
⊢ Ⅎ 𝑤 ( 𝑔 ‘ 𝑦 ) |
| 301 |
300
|
esumcl |
⊢ ( ( ( 𝑔 ‘ 𝑦 ) ∈ V ∧ ∀ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) |
| 302 |
299 301
|
mpan |
⊢ ( ∀ 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) |
| 303 |
298 302
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) |
| 304 |
285 286 287 303
|
syl21anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ∧ 𝑦 ∈ 𝑋 ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) |
| 305 |
304
|
ex |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( 𝑦 ∈ 𝑋 → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) ) |
| 306 |
284 305
|
ralrimi |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∀ 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) |
| 307 |
18
|
esumcl |
⊢ ( ( 𝑋 ∈ V ∧ ∀ 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) → Σ* 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) |
| 308 |
178 306 307
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) |
| 309 |
105 308
|
sselid |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ℝ* ) |
| 310 |
|
nfv |
⊢ Ⅎ 𝑤 ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) |
| 311 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) |
| 312 |
|
fniunfv |
⊢ ( 𝑔 Fn 𝑋 → ∪ 𝑦 ∈ 𝑋 ( 𝑔 ‘ 𝑦 ) = ∪ ran 𝑔 ) |
| 313 |
311 213 312
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → ∪ 𝑦 ∈ 𝑋 ( 𝑔 ‘ 𝑦 ) = ∪ ran 𝑔 ) |
| 314 |
310 313
|
esumeq1d |
⊢ ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → Σ* 𝑤 ∈ ∪ 𝑦 ∈ 𝑋 ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) = Σ* 𝑤 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑤 ) ) |
| 315 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → 𝑋 ∈ V ) |
| 316 |
299
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑔 ‘ 𝑦 ) ∈ V ) |
| 317 |
315 316 297
|
esumiun |
⊢ ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → Σ* 𝑤 ∈ ∪ 𝑦 ∈ 𝑋 ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ Σ* 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ) |
| 318 |
314 317
|
eqbrtrrd |
⊢ ( ( 𝜑 ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → Σ* 𝑤 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑤 ) ≤ Σ* 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ) |
| 319 |
13 318
|
sylan |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → Σ* 𝑤 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑤 ) ≤ Σ* 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ) |
| 320 |
319
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑤 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑤 ) ≤ Σ* 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ) |
| 321 |
252 320
|
eqbrtrid |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ≤ Σ* 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ) |
| 322 |
285 287 46
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
| 323 |
|
simplll |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ∧ 𝑦 ∈ 𝑋 ) → ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ) |
| 324 |
323 287 73
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 325 |
322 324
|
xrge0addcld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 326 |
325
|
ex |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( 𝑦 ∈ 𝑋 → ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ( 0 [,] +∞ ) ) ) |
| 327 |
284 326
|
ralrimi |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∀ 𝑦 ∈ 𝑋 ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 328 |
18
|
esumcl |
⊢ ( ( 𝑋 ∈ V ∧ ∀ 𝑦 ∈ 𝑋 ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ( 0 [,] +∞ ) ) → Σ* 𝑦 ∈ 𝑋 ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 329 |
178 327 328
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 330 |
105 329
|
sselid |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ℝ* ) |
| 331 |
215 14
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → 𝑋 ∈ V ) |
| 332 |
|
simp-4l |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ) |
| 333 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → 𝑦 ∈ 𝑋 ) |
| 334 |
332 333 49
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) |
| 335 |
334
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) |
| 336 |
65
|
adantlr |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ ) |
| 337 |
336
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ ) |
| 338 |
|
id |
⊢ ( Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 339 |
338
|
adantl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 340 |
66
|
breq2d |
⊢ ( ( ( 𝑀 ‘ 𝐴 ) ∈ ℝ ∧ ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ ) → ( Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ↔ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 341 |
340
|
biimpar |
⊢ ( ( ( ( 𝑀 ‘ 𝐴 ) ∈ ℝ ∧ ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 342 |
335 337 339 341
|
syl21anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 343 |
342
|
ex |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → ( Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 344 |
332
|
simpld |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → 𝜑 ) |
| 345 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) |
| 346 |
344 345 333 303
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) |
| 347 |
105 346
|
sselid |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ℝ* ) |
| 348 |
334
|
rexrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ) |
| 349 |
336
|
rexrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ* ) |
| 350 |
348 349
|
xaddcld |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ℝ* ) |
| 351 |
|
xrltle |
⊢ ( ( Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ∈ ℝ* ∧ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ∈ ℝ* ) → ( Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 352 |
347 350 351
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → ( Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 353 |
343 352
|
syld |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → ( Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 354 |
353
|
adantld |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 355 |
354
|
ex |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → ( 𝑦 ∈ 𝑋 → ( ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ) |
| 356 |
282 355
|
ralrimi |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → ∀ 𝑦 ∈ 𝑋 ( ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 357 |
|
ralim |
⊢ ( ∀ 𝑦 ∈ 𝑋 ( ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → ( ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → ∀ 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 358 |
356 357
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) → ( ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) → ∀ 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) |
| 359 |
358
|
imp |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∀ 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 360 |
359
|
r19.21bi |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ∧ 𝑦 ∈ 𝑋 ) → Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 361 |
284 18 331 304 325 360
|
esumlef |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ Σ* 𝑦 ∈ 𝑋 ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 362 |
164 46
|
sylan |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
| 363 |
284 18 331 362 324
|
esumaddf |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) = ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) +𝑒 Σ* 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 364 |
324
|
ex |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( 𝑦 ∈ 𝑋 → ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ( 0 [,] +∞ ) ) ) |
| 365 |
284 364
|
ralrimi |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ∀ 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 366 |
18
|
esumcl |
⊢ ( ( 𝑋 ∈ V ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ( 0 [,] +∞ ) ) → Σ* 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 367 |
178 365 366
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 368 |
105 367
|
sselid |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ* ) |
| 369 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → 𝑓 : 𝑋 –1-1→ ℕ ) |
| 370 |
|
vex |
⊢ 𝑓 ∈ V |
| 371 |
370
|
rnex |
⊢ ran 𝑓 ∈ V |
| 372 |
371
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ran 𝑓 ∈ V ) |
| 373 |
58
|
frnd |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → ran 𝑓 ⊆ ℕ ) |
| 374 |
373
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ran 𝑓 ⊆ ℕ ) |
| 375 |
374
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑧 ∈ ran 𝑓 ) → 𝑧 ∈ ℕ ) |
| 376 |
54
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑧 ∈ ℕ ) → 2 ∈ ℝ+ ) |
| 377 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑧 ∈ ℕ ) → 𝑧 ∈ ℕ ) |
| 378 |
377
|
nnzd |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑧 ∈ ℕ ) → 𝑧 ∈ ℤ ) |
| 379 |
376 378
|
rpexpcld |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑧 ∈ ℕ ) → ( 2 ↑ 𝑧 ) ∈ ℝ+ ) |
| 380 |
379
|
rpreccld |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑧 ∈ ℕ ) → ( 1 / ( 2 ↑ 𝑧 ) ) ∈ ℝ+ ) |
| 381 |
71 380
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑧 ∈ ℕ ) → ( 1 / ( 2 ↑ 𝑧 ) ) ∈ ( 0 [,] +∞ ) ) |
| 382 |
381
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑧 ∈ ℕ ) → ( 1 / ( 2 ↑ 𝑧 ) ) ∈ ( 0 [,] +∞ ) ) |
| 383 |
375 382
|
syldan |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑧 ∈ ran 𝑓 ) → ( 1 / ( 2 ↑ 𝑧 ) ) ∈ ( 0 [,] +∞ ) ) |
| 384 |
383
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ∀ 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ∈ ( 0 [,] +∞ ) ) |
| 385 |
|
nfcv |
⊢ Ⅎ 𝑧 ran 𝑓 |
| 386 |
385
|
esumcl |
⊢ ( ( ran 𝑓 ∈ V ∧ ∀ 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ∈ ( 0 [,] +∞ ) ) → Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ∈ ( 0 [,] +∞ ) ) |
| 387 |
372 384 386
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ∈ ( 0 [,] +∞ ) ) |
| 388 |
105 387
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ∈ ℝ* ) |
| 389 |
|
1xr |
⊢ 1 ∈ ℝ* |
| 390 |
389
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → 1 ∈ ℝ* ) |
| 391 |
71
|
sseli |
⊢ ( 𝑒 ∈ ℝ+ → 𝑒 ∈ ( 0 [,] +∞ ) ) |
| 392 |
391
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → 𝑒 ∈ ( 0 [,] +∞ ) ) |
| 393 |
|
elxrge0 |
⊢ ( 𝑒 ∈ ( 0 [,] +∞ ) ↔ ( 𝑒 ∈ ℝ* ∧ 0 ≤ 𝑒 ) ) |
| 394 |
392 393
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ( 𝑒 ∈ ℝ* ∧ 0 ≤ 𝑒 ) ) |
| 395 |
|
nfv |
⊢ Ⅎ 𝑧 ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) |
| 396 |
|
nnex |
⊢ ℕ ∈ V |
| 397 |
396
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → ℕ ∈ V ) |
| 398 |
395 397 381 373
|
esummono |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ≤ Σ* 𝑧 ∈ ℕ ( 1 / ( 2 ↑ 𝑧 ) ) ) |
| 399 |
|
oveq2 |
⊢ ( 𝑧 = 𝑤 → ( 2 ↑ 𝑧 ) = ( 2 ↑ 𝑤 ) ) |
| 400 |
399
|
oveq2d |
⊢ ( 𝑧 = 𝑤 → ( 1 / ( 2 ↑ 𝑧 ) ) = ( 1 / ( 2 ↑ 𝑤 ) ) ) |
| 401 |
|
ioossico |
⊢ ( 0 (,) +∞ ) ⊆ ( 0 [,) +∞ ) |
| 402 |
69 401
|
eqsstri |
⊢ ℝ+ ⊆ ( 0 [,) +∞ ) |
| 403 |
402 380
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑧 ∈ ℕ ) → ( 1 / ( 2 ↑ 𝑧 ) ) ∈ ( 0 [,) +∞ ) ) |
| 404 |
|
eqidd |
⊢ ( 𝑧 ∈ ℕ → ( 𝑤 ∈ ℕ ↦ ( 1 / ( 2 ↑ 𝑤 ) ) ) = ( 𝑤 ∈ ℕ ↦ ( 1 / ( 2 ↑ 𝑤 ) ) ) ) |
| 405 |
|
simpr |
⊢ ( ( 𝑧 ∈ ℕ ∧ 𝑤 = 𝑧 ) → 𝑤 = 𝑧 ) |
| 406 |
405
|
oveq2d |
⊢ ( ( 𝑧 ∈ ℕ ∧ 𝑤 = 𝑧 ) → ( 2 ↑ 𝑤 ) = ( 2 ↑ 𝑧 ) ) |
| 407 |
406
|
oveq2d |
⊢ ( ( 𝑧 ∈ ℕ ∧ 𝑤 = 𝑧 ) → ( 1 / ( 2 ↑ 𝑤 ) ) = ( 1 / ( 2 ↑ 𝑧 ) ) ) |
| 408 |
|
id |
⊢ ( 𝑧 ∈ ℕ → 𝑧 ∈ ℕ ) |
| 409 |
|
ovexd |
⊢ ( 𝑧 ∈ ℕ → ( 1 / ( 2 ↑ 𝑧 ) ) ∈ V ) |
| 410 |
404 407 408 409
|
fvmptd |
⊢ ( 𝑧 ∈ ℕ → ( ( 𝑤 ∈ ℕ ↦ ( 1 / ( 2 ↑ 𝑤 ) ) ) ‘ 𝑧 ) = ( 1 / ( 2 ↑ 𝑧 ) ) ) |
| 411 |
410
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑧 ∈ ℕ ) → ( ( 𝑤 ∈ ℕ ↦ ( 1 / ( 2 ↑ 𝑤 ) ) ) ‘ 𝑧 ) = ( 1 / ( 2 ↑ 𝑧 ) ) ) |
| 412 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 413 |
|
eqid |
⊢ ( 𝑤 ∈ ℕ ↦ ( 1 / ( 2 ↑ 𝑤 ) ) ) = ( 𝑤 ∈ ℕ ↦ ( 1 / ( 2 ↑ 𝑤 ) ) ) |
| 414 |
413
|
geo2lim |
⊢ ( 1 ∈ ℂ → seq 1 ( + , ( 𝑤 ∈ ℕ ↦ ( 1 / ( 2 ↑ 𝑤 ) ) ) ) ⇝ 1 ) |
| 415 |
412 414
|
ax-mp |
⊢ seq 1 ( + , ( 𝑤 ∈ ℕ ↦ ( 1 / ( 2 ↑ 𝑤 ) ) ) ) ⇝ 1 |
| 416 |
415
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → seq 1 ( + , ( 𝑤 ∈ ℕ ↦ ( 1 / ( 2 ↑ 𝑤 ) ) ) ) ⇝ 1 ) |
| 417 |
|
1re |
⊢ 1 ∈ ℝ |
| 418 |
417
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → 1 ∈ ℝ ) |
| 419 |
400 403 411 416 418
|
esumcvgsum |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → Σ* 𝑧 ∈ ℕ ( 1 / ( 2 ↑ 𝑧 ) ) = Σ 𝑧 ∈ ℕ ( 1 / ( 2 ↑ 𝑧 ) ) ) |
| 420 |
|
geoihalfsum |
⊢ Σ 𝑧 ∈ ℕ ( 1 / ( 2 ↑ 𝑧 ) ) = 1 |
| 421 |
419 420
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → Σ* 𝑧 ∈ ℕ ( 1 / ( 2 ↑ 𝑧 ) ) = 1 ) |
| 422 |
398 421
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ≤ 1 ) |
| 423 |
422
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ≤ 1 ) |
| 424 |
|
xlemul2a |
⊢ ( ( ( Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ∈ ℝ* ∧ 1 ∈ ℝ* ∧ ( 𝑒 ∈ ℝ* ∧ 0 ≤ 𝑒 ) ) ∧ Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ≤ 1 ) → ( 𝑒 ·e Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ) ≤ ( 𝑒 ·e 1 ) ) |
| 425 |
388 390 394 423 424
|
syl31anc |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ( 𝑒 ·e Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ) ≤ ( 𝑒 ·e 1 ) ) |
| 426 |
17 23
|
nfan |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) |
| 427 |
426 25
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) |
| 428 |
76
|
recnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → 𝑒 ∈ ℂ ) |
| 429 |
78
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ∈ ℂ ) |
| 430 |
429
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ∈ ℂ ) |
| 431 |
|
2cn |
⊢ 2 ∈ ℂ |
| 432 |
431
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → 2 ∈ ℂ ) |
| 433 |
|
2ne0 |
⊢ 2 ≠ 0 |
| 434 |
433
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → 2 ≠ 0 ) |
| 435 |
432 434 60
|
expne0d |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ≠ 0 ) |
| 436 |
435
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ≠ 0 ) |
| 437 |
428 430 436
|
divrecd |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) = ( 𝑒 · ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 438 |
|
1rp |
⊢ 1 ∈ ℝ+ |
| 439 |
438
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → 1 ∈ ℝ+ ) |
| 440 |
439 61
|
rpdivcld |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ+ ) |
| 441 |
52 440
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ ) |
| 442 |
441
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ ) |
| 443 |
|
rexmul |
⊢ ( ( 𝑒 ∈ ℝ ∧ ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ ) → ( 𝑒 ·e ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) = ( 𝑒 · ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 444 |
76 442 443
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑒 ·e ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) = ( 𝑒 · ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 445 |
437 444
|
eqtr4d |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) = ( 𝑒 ·e ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 446 |
445
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ∀ 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) = ( 𝑒 ·e ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 447 |
427 446
|
esumeq2d |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → Σ* 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) = Σ* 𝑦 ∈ 𝑋 ( 𝑒 ·e ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 448 |
15
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → 𝑋 ∈ V ) |
| 449 |
71 440
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑦 ∈ 𝑋 ) → ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 450 |
449
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝑋 ) → ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 451 |
402
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → ℝ+ ⊆ ( 0 [,) +∞ ) ) |
| 452 |
451
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → 𝑒 ∈ ( 0 [,) +∞ ) ) |
| 453 |
448 450 452
|
esummulc2 |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ( 𝑒 ·e Σ* 𝑦 ∈ 𝑋 ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) = Σ* 𝑦 ∈ 𝑋 ( 𝑒 ·e ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
| 454 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 1 / ( 2 ↑ 𝑧 ) ) |
| 455 |
|
oveq2 |
⊢ ( 𝑧 = ( 𝑓 ‘ 𝑦 ) → ( 2 ↑ 𝑧 ) = ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) |
| 456 |
455
|
oveq2d |
⊢ ( 𝑧 = ( 𝑓 ‘ 𝑦 ) → ( 1 / ( 2 ↑ 𝑧 ) ) = ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) |
| 457 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → 𝑋 ∈ V ) |
| 458 |
56
|
simprbi |
⊢ ( 𝑓 : 𝑋 –1-1→ ℕ → Fun ◡ 𝑓 ) |
| 459 |
57
|
feqmptd |
⊢ ( 𝑓 : 𝑋 –1-1→ ℕ → 𝑓 = ( 𝑦 ∈ 𝑋 ↦ ( 𝑓 ‘ 𝑦 ) ) ) |
| 460 |
459
|
cnveqd |
⊢ ( 𝑓 : 𝑋 –1-1→ ℕ → ◡ 𝑓 = ◡ ( 𝑦 ∈ 𝑋 ↦ ( 𝑓 ‘ 𝑦 ) ) ) |
| 461 |
460
|
funeqd |
⊢ ( 𝑓 : 𝑋 –1-1→ ℕ → ( Fun ◡ 𝑓 ↔ Fun ◡ ( 𝑦 ∈ 𝑋 ↦ ( 𝑓 ‘ 𝑦 ) ) ) ) |
| 462 |
458 461
|
mpbid |
⊢ ( 𝑓 : 𝑋 –1-1→ ℕ → Fun ◡ ( 𝑦 ∈ 𝑋 ↦ ( 𝑓 ‘ 𝑦 ) ) ) |
| 463 |
462
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → Fun ◡ ( 𝑦 ∈ 𝑋 ↦ ( 𝑓 ‘ 𝑦 ) ) ) |
| 464 |
454 426 18 456 457 463 449 59
|
esumc |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → Σ* 𝑦 ∈ 𝑋 ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) = Σ* 𝑧 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ 𝑋 𝑥 = ( 𝑓 ‘ 𝑦 ) } ( 1 / ( 2 ↑ 𝑧 ) ) ) |
| 465 |
|
ffn |
⊢ ( 𝑓 : 𝑋 ⟶ ℕ → 𝑓 Fn 𝑋 ) |
| 466 |
|
fnrnfv |
⊢ ( 𝑓 Fn 𝑋 → ran 𝑓 = { 𝑥 ∣ ∃ 𝑦 ∈ 𝑋 𝑥 = ( 𝑓 ‘ 𝑦 ) } ) |
| 467 |
58 465 466
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → ran 𝑓 = { 𝑥 ∣ ∃ 𝑦 ∈ 𝑋 𝑥 = ( 𝑓 ‘ 𝑦 ) } ) |
| 468 |
395 467
|
esumeq1d |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) = Σ* 𝑧 ∈ { 𝑥 ∣ ∃ 𝑦 ∈ 𝑋 𝑥 = ( 𝑓 ‘ 𝑦 ) } ( 1 / ( 2 ↑ 𝑧 ) ) ) |
| 469 |
464 468
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → Σ* 𝑦 ∈ 𝑋 ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) = Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ) |
| 470 |
469
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → Σ* 𝑦 ∈ 𝑋 ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) = Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ) |
| 471 |
470
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ( 𝑒 ·e Σ* 𝑦 ∈ 𝑋 ( 1 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) = ( 𝑒 ·e Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ) ) |
| 472 |
447 453 471
|
3eqtr2rd |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ( 𝑒 ·e Σ* 𝑧 ∈ ran 𝑓 ( 1 / ( 2 ↑ 𝑧 ) ) ) = Σ* 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) |
| 473 |
394
|
simpld |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → 𝑒 ∈ ℝ* ) |
| 474 |
|
xmulrid |
⊢ ( 𝑒 ∈ ℝ* → ( 𝑒 ·e 1 ) = 𝑒 ) |
| 475 |
473 474
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ( 𝑒 ·e 1 ) = 𝑒 ) |
| 476 |
425 472 475
|
3brtr3d |
⊢ ( ( ( 𝜑 ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → Σ* 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ≤ 𝑒 ) |
| 477 |
164 369 204 476
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ≤ 𝑒 ) |
| 478 |
|
xleadd2a |
⊢ ( ( ( Σ* 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ∈ ℝ* ∧ 𝑒 ∈ ℝ* ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ) ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ≤ 𝑒 ) → ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) +𝑒 Σ* 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) +𝑒 𝑒 ) ) |
| 479 |
368 205 203 477 478
|
syl31anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) +𝑒 Σ* 𝑦 ∈ 𝑋 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) +𝑒 𝑒 ) ) |
| 480 |
363 479
|
eqbrtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) +𝑒 𝑒 ) ) |
| 481 |
309 330 206 361 480
|
xrletrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑦 ∈ 𝑋 Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) +𝑒 𝑒 ) ) |
| 482 |
201 309 206 321 481
|
xrletrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → Σ* 𝑐 ∈ ∪ ran 𝑔 ( 𝑅 ‘ 𝑐 ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) +𝑒 𝑒 ) ) |
| 483 |
176 201 206 280 482
|
xrletrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) +𝑒 𝑒 ) ) |
| 484 |
204
|
rpred |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → 𝑒 ∈ ℝ ) |
| 485 |
|
rexadd |
⊢ ( ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ∧ 𝑒 ∈ ℝ ) → ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) +𝑒 𝑒 ) = ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) + 𝑒 ) ) |
| 486 |
202 484 485
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) +𝑒 𝑒 ) = ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) + 𝑒 ) ) |
| 487 |
483 486
|
breqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) + 𝑒 ) ) |
| 488 |
487
|
anasss |
⊢ ( ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) ∧ ( 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) + 𝑒 ) ) |
| 489 |
488
|
ex |
⊢ ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ( ( 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) + 𝑒 ) ) ) |
| 490 |
489
|
exlimdv |
⊢ ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ( ∃ 𝑔 ( 𝑔 : 𝑋 ⟶ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ 𝑧 ≼ ω } ∧ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ⊆ ∪ ( 𝑔 ‘ 𝑦 ) ∧ Σ* 𝑤 ∈ ( 𝑔 ‘ 𝑦 ) ( 𝑅 ‘ 𝑤 ) < ( ( 𝑀 ‘ 𝐴 ) + ( 𝑒 / ( 2 ↑ ( 𝑓 ‘ 𝑦 ) ) ) ) ) ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) + 𝑒 ) ) ) |
| 491 |
163 490
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) ∧ 𝑒 ∈ ℝ+ ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) + 𝑒 ) ) |
| 492 |
491
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → ∀ 𝑒 ∈ ℝ+ ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) + 𝑒 ) ) |
| 493 |
|
xralrple |
⊢ ( ( ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ∈ ℝ* ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → ( ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ↔ ∀ 𝑒 ∈ ℝ+ ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) + 𝑒 ) ) ) |
| 494 |
175 493
|
sylan |
⊢ ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → ( ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ↔ ∀ 𝑒 ∈ ℝ+ ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) + 𝑒 ) ) ) |
| 495 |
494
|
adantr |
⊢ ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → ( ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ↔ ∀ 𝑒 ∈ ℝ+ ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) + 𝑒 ) ) ) |
| 496 |
492 495
|
mpbird |
⊢ ( ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) ∧ 𝑓 : 𝑋 –1-1→ ℕ ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ) |
| 497 |
496
|
ex |
⊢ ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → ( 𝑓 : 𝑋 –1-1→ ℕ → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ) ) |
| 498 |
497
|
exlimdv |
⊢ ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → ( ∃ 𝑓 𝑓 : 𝑋 –1-1→ ℕ → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ) ) |
| 499 |
12 498
|
mpd |
⊢ ( ( 𝜑 ∧ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ) |
| 500 |
175
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ∈ ℝ* ) |
| 501 |
|
pnfge |
⊢ ( ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ∈ ℝ* → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ +∞ ) |
| 502 |
500 501
|
syl |
⊢ ( ( 𝜑 ∧ ¬ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ +∞ ) |
| 503 |
46
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
| 504 |
18
|
esumcl |
⊢ ( ( 𝑋 ∈ V ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) → Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
| 505 |
15 503 504
|
syl2anc |
⊢ ( 𝜑 → Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
| 506 |
|
xrge0nre |
⊢ ( ( Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ∧ ¬ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) = +∞ ) |
| 507 |
505 506
|
sylan |
⊢ ( ( 𝜑 ∧ ¬ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) = +∞ ) |
| 508 |
502 507
|
breqtrrd |
⊢ ( ( 𝜑 ∧ ¬ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ∈ ℝ ) → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ) |
| 509 |
499 508
|
pm2.61dan |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑦 ∈ 𝑋 𝐴 ) ≤ Σ* 𝑦 ∈ 𝑋 ( 𝑀 ‘ 𝐴 ) ) |