Step |
Hyp |
Ref |
Expression |
1 |
|
sge0ltfirp.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
2 |
|
sge0ltfirp.f |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) |
3 |
|
sge0ltfirp.y |
⊢ ( 𝜑 → 𝑌 ∈ ℝ+ ) |
4 |
|
sge0ltfirp.re |
⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) ∈ ℝ ) |
5 |
1 2 4
|
sge0rern |
⊢ ( 𝜑 → ¬ +∞ ∈ ran 𝐹 ) |
6 |
2 5
|
fge0iccico |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ( 0 [,) +∞ ) ) |
7 |
6
|
sge0rnre |
⊢ ( 𝜑 → ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ⊆ ℝ ) |
8 |
|
sge0rnn0 |
⊢ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ≠ ∅ |
9 |
8
|
a1i |
⊢ ( 𝜑 → ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ≠ ∅ ) |
10 |
1 2 4
|
sge0rnbnd |
⊢ ( 𝜑 → ∃ 𝑧 ∈ ℝ ∀ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) 𝑤 ≤ 𝑧 ) |
11 |
7 9 10 3
|
suprltrp |
⊢ ( 𝜑 → ∃ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 ) |
12 |
|
nfv |
⊢ Ⅎ 𝑤 𝜑 |
13 |
|
nfv |
⊢ Ⅎ 𝑤 ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( Σ^ ‘ 𝐹 ) < ( ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) + 𝑌 ) |
14 |
|
simp1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ∧ ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 ) → 𝜑 ) |
15 |
|
vex |
⊢ 𝑤 ∈ V |
16 |
|
eqid |
⊢ ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) = ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
17 |
16
|
elrnmpt |
⊢ ( 𝑤 ∈ V → ( 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ) |
18 |
15 17
|
ax-mp |
⊢ ( 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
19 |
18
|
biimpi |
⊢ ( 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ∧ ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 ) → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
21 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
22 |
21
|
nfrn |
⊢ Ⅎ 𝑥 ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
23 |
|
nfcv |
⊢ Ⅎ 𝑥 ℝ |
24 |
|
nfcv |
⊢ Ⅎ 𝑥 < |
25 |
22 23 24
|
nfsup |
⊢ Ⅎ 𝑥 sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) |
26 |
|
nfcv |
⊢ Ⅎ 𝑥 − |
27 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑌 |
28 |
25 26 27
|
nfov |
⊢ Ⅎ 𝑥 ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) |
29 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑤 |
30 |
28 24 29
|
nfbr |
⊢ Ⅎ 𝑥 ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 |
31 |
|
simpl |
⊢ ( ( ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 ∧ 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 ) |
32 |
|
simpr |
⊢ ( ( ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 ∧ 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
33 |
31 32
|
breqtrd |
⊢ ( ( ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 ∧ 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
34 |
33
|
ex |
⊢ ( ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 → ( 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) → ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ) |
35 |
34
|
a1d |
⊢ ( ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 → ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) → ( 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) → ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ) ) |
36 |
30 35
|
reximdai |
⊢ ( ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 → ( ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ) |
37 |
36
|
adantl |
⊢ ( ( 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ∧ ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 ) → ( ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ) |
38 |
20 37
|
mpd |
⊢ ( ( 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ∧ ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 ) → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
39 |
38
|
3adant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ∧ ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 ) → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
40 |
|
simpl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ) |
41 |
1 2 4
|
sge0supre |
⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) = sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) ) |
42 |
41
|
oveq1d |
⊢ ( 𝜑 → ( ( Σ^ ‘ 𝐹 ) − 𝑌 ) = ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) ) |
43 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( ( Σ^ ‘ 𝐹 ) − 𝑌 ) = ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) ) |
44 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
45 |
43 44
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( ( Σ^ ‘ 𝐹 ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
46 |
45
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( ( Σ^ ‘ 𝐹 ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
47 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ ( ( Σ^ ‘ 𝐹 ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( ( Σ^ ‘ 𝐹 ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
48 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → ( Σ^ ‘ 𝐹 ) ∈ ℝ ) |
49 |
3
|
rpred |
⊢ ( 𝜑 → 𝑌 ∈ ℝ ) |
50 |
49
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝑌 ∈ ℝ ) |
51 |
|
elinel2 |
⊢ ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) → 𝑥 ∈ Fin ) |
52 |
51
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝑥 ∈ Fin ) |
53 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
54 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝐹 : 𝑋 ⟶ ( 0 [,) +∞ ) ) |
55 |
54
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝐹 : 𝑋 ⟶ ( 0 [,) +∞ ) ) |
56 |
|
elpwinss |
⊢ ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) → 𝑥 ⊆ 𝑋 ) |
57 |
56
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝑥 ⊆ 𝑋 ) |
58 |
57
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ 𝑋 ) |
59 |
55 58
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ∈ ( 0 [,) +∞ ) ) |
60 |
53 59
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ∈ ℝ ) |
61 |
52 60
|
fsumrecl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ∈ ℝ ) |
62 |
48 50 61
|
ltsubaddd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → ( ( ( Σ^ ‘ 𝐹 ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ↔ ( Σ^ ‘ 𝐹 ) < ( Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) + 𝑌 ) ) ) |
63 |
62
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ ( ( Σ^ ‘ 𝐹 ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( ( ( Σ^ ‘ 𝐹 ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ↔ ( Σ^ ‘ 𝐹 ) < ( Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) + 𝑌 ) ) ) |
64 |
47 63
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ ( ( Σ^ ‘ 𝐹 ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( Σ^ ‘ 𝐹 ) < ( Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) + 𝑌 ) ) |
65 |
54 57
|
fssresd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → ( 𝐹 ↾ 𝑥 ) : 𝑥 ⟶ ( 0 [,) +∞ ) ) |
66 |
52 65
|
sge0fsum |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) = Σ 𝑦 ∈ 𝑥 ( ( 𝐹 ↾ 𝑥 ) ‘ 𝑦 ) ) |
67 |
|
fvres |
⊢ ( 𝑦 ∈ 𝑥 → ( ( 𝐹 ↾ 𝑥 ) ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
68 |
67
|
sumeq2i |
⊢ Σ 𝑦 ∈ 𝑥 ( ( 𝐹 ↾ 𝑥 ) ‘ 𝑦 ) = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) |
69 |
68
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → Σ 𝑦 ∈ 𝑥 ( ( 𝐹 ↾ 𝑥 ) ‘ 𝑦 ) = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) |
70 |
66 69
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) = ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) ) |
71 |
70
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → ( Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) + 𝑌 ) = ( ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) + 𝑌 ) ) |
72 |
71
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ ( ( Σ^ ‘ 𝐹 ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) + 𝑌 ) = ( ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) + 𝑌 ) ) |
73 |
64 72
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ ( ( Σ^ ‘ 𝐹 ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( Σ^ ‘ 𝐹 ) < ( ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) + 𝑌 ) ) |
74 |
40 46 73
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( Σ^ ‘ 𝐹 ) < ( ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) + 𝑌 ) ) |
75 |
74
|
ex |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → ( ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) → ( Σ^ ‘ 𝐹 ) < ( ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) + 𝑌 ) ) ) |
76 |
75
|
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( Σ^ ‘ 𝐹 ) < ( ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) + 𝑌 ) ) ) |
77 |
76
|
imp |
⊢ ( ( 𝜑 ∧ ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( Σ^ ‘ 𝐹 ) < ( ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) + 𝑌 ) ) |
78 |
14 39 77
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ∧ ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 ) → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( Σ^ ‘ 𝐹 ) < ( ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) + 𝑌 ) ) |
79 |
78
|
3exp |
⊢ ( 𝜑 → ( 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ( ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( Σ^ ‘ 𝐹 ) < ( ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) + 𝑌 ) ) ) ) |
80 |
12 13 79
|
rexlimd |
⊢ ( 𝜑 → ( ∃ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ( sup ( ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) , ℝ , < ) − 𝑌 ) < 𝑤 → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( Σ^ ‘ 𝐹 ) < ( ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) + 𝑌 ) ) ) |
81 |
11 80
|
mpd |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ( Σ^ ‘ 𝐹 ) < ( ( Σ^ ‘ ( 𝐹 ↾ 𝑥 ) ) + 𝑌 ) ) |