Step |
Hyp |
Ref |
Expression |
1 |
|
sge0fsum.x |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
2 |
|
sge0fsum.f |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ( 0 [,) +∞ ) ) |
3 |
2
|
fge0icoicc |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) |
4 |
1 3
|
sge0xrcl |
⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) ∈ ℝ* ) |
5 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
6 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) |
7 |
5 6
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
8 |
1 7
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
9 |
8
|
rexrd |
⊢ ( 𝜑 → Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ∈ ℝ* ) |
10 |
1 2
|
sge0reval |
⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) = sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) , ℝ* , < ) ) |
11 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ) → 𝑤 ∈ ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ) |
12 |
|
vex |
⊢ 𝑤 ∈ V |
13 |
12
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ) → 𝑤 ∈ V ) |
14 |
|
eqid |
⊢ ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) = ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) |
15 |
14
|
elrnmpt |
⊢ ( 𝑤 ∈ V → ( 𝑤 ∈ ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ↔ ∃ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ) |
16 |
13 15
|
syl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ) → ( 𝑤 ∈ ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ↔ ∃ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ) |
17 |
11 16
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ) → ∃ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) |
18 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ∧ 𝑤 = Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) → 𝑤 = Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) |
19 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝑋 ∈ Fin ) |
20 |
2
|
fge0npnf |
⊢ ( 𝜑 → ¬ +∞ ∈ ran 𝐹 ) |
21 |
3 20
|
fge0iccre |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ℝ ) |
22 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝐹 : 𝑋 ⟶ ℝ ) |
23 |
22
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑥 ∈ 𝑋 ) → 𝐹 : 𝑋 ⟶ ℝ ) |
24 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑥 ∈ 𝑋 ) → 𝑥 ∈ 𝑋 ) |
25 |
23 24
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
26 |
|
0xr |
⊢ 0 ∈ ℝ* |
27 |
26
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑥 ∈ 𝑋 ) → 0 ∈ ℝ* ) |
28 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
29 |
28
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑥 ∈ 𝑋 ) → +∞ ∈ ℝ* ) |
30 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) |
31 |
30
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
32 |
|
iccgelb |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ( 𝐹 ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) → 0 ≤ ( 𝐹 ‘ 𝑥 ) ) |
33 |
27 29 31 32
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑥 ∈ 𝑋 ) → 0 ≤ ( 𝐹 ‘ 𝑥 ) ) |
34 |
|
elinel1 |
⊢ ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) → 𝑦 ∈ 𝒫 𝑋 ) |
35 |
|
elpwi |
⊢ ( 𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋 ) |
36 |
34 35
|
syl |
⊢ ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) → 𝑦 ⊆ 𝑋 ) |
37 |
36
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝑦 ⊆ 𝑋 ) |
38 |
19 25 33 37
|
fsumless |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) |
39 |
38
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ∧ 𝑤 = Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) → Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) |
40 |
18 39
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ∧ 𝑤 = Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) → 𝑤 ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) |
41 |
40
|
3exp |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) → ( 𝑤 = Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) → 𝑤 ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) ) ) |
42 |
41
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) → 𝑤 ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) ) |
43 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ) → ( ∃ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) → 𝑤 ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) ) |
44 |
17 43
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ) → 𝑤 ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) |
45 |
44
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑤 ∈ ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) 𝑤 ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) |
46 |
|
elinel2 |
⊢ ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) → 𝑦 ∈ Fin ) |
47 |
46
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝑦 ∈ Fin ) |
48 |
22
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑥 ∈ 𝑦 ) → 𝐹 : 𝑋 ⟶ ℝ ) |
49 |
37
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑥 ∈ 𝑦 ) → 𝑥 ∈ 𝑋 ) |
50 |
48 49
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) ∧ 𝑥 ∈ 𝑦 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
51 |
47 50
|
fsumrecl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
52 |
51
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ∈ ℝ* ) |
53 |
52
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ∈ ℝ* ) |
54 |
14
|
rnmptss |
⊢ ( ∀ 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ∈ ℝ* → ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ⊆ ℝ* ) |
55 |
53 54
|
syl |
⊢ ( 𝜑 → ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ⊆ ℝ* ) |
56 |
|
supxrleub |
⊢ ( ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) ⊆ ℝ* ∧ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ∈ ℝ* ) → ( sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) , ℝ* , < ) ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ↔ ∀ 𝑤 ∈ ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) 𝑤 ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) ) |
57 |
55 9 56
|
syl2anc |
⊢ ( 𝜑 → ( sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) , ℝ* , < ) ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ↔ ∀ 𝑤 ∈ ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) 𝑤 ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) ) |
58 |
45 57
|
mpbird |
⊢ ( 𝜑 → sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑥 ∈ 𝑦 ( 𝐹 ‘ 𝑥 ) ) , ℝ* , < ) ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) |
59 |
10 58
|
eqbrtrd |
⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) ≤ Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) |
60 |
|
ssid |
⊢ 𝑋 ⊆ 𝑋 |
61 |
60
|
a1i |
⊢ ( 𝜑 → 𝑋 ⊆ 𝑋 ) |
62 |
1 2 61 1
|
fsumlesge0 |
⊢ ( 𝜑 → Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ≤ ( Σ^ ‘ 𝐹 ) ) |
63 |
4 9 59 62
|
xrletrid |
⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) = Σ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) |