Step |
Hyp |
Ref |
Expression |
1 |
|
esumlub.f |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
esumlub.0 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
esumlub.1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
4 |
|
esumlub.2 |
⊢ ( 𝜑 → 𝑋 ∈ ℝ* ) |
5 |
|
esumlub.3 |
⊢ ( 𝜑 → 𝑋 < Σ* 𝑘 ∈ 𝐴 𝐵 ) |
6 |
|
nfcv |
⊢ Ⅎ 𝑘 𝐴 |
7 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) |
8 |
1 6 2 3 7
|
esumval |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 = sup ( ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) , ℝ* , < ) ) |
9 |
8
|
breq2d |
⊢ ( 𝜑 → ( 𝑋 < Σ* 𝑘 ∈ 𝐴 𝐵 ↔ 𝑋 < sup ( ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) , ℝ* , < ) ) ) |
10 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
11 |
|
xrge0base |
⊢ ( 0 [,] +∞ ) = ( Base ‘ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ) |
12 |
|
xrge0cmn |
⊢ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd |
13 |
12
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd ) |
14 |
|
inss2 |
⊢ ( 𝒫 𝐴 ∩ Fin ) ⊆ Fin |
15 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
16 |
14 15
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑥 ∈ Fin ) |
17 |
|
nfv |
⊢ Ⅎ 𝑘 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) |
18 |
1 17
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
19 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑥 ) → 𝜑 ) |
20 |
|
inss1 |
⊢ ( 𝒫 𝐴 ∩ Fin ) ⊆ 𝒫 𝐴 |
21 |
20
|
sseli |
⊢ ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) → 𝑥 ∈ 𝒫 𝐴 ) |
22 |
21
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑥 ) → 𝑥 ∈ 𝒫 𝐴 ) |
23 |
22
|
elpwid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑥 ) → 𝑥 ⊆ 𝐴 ) |
24 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑥 ) → 𝑘 ∈ 𝑥 ) |
25 |
23 24
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑥 ) → 𝑘 ∈ 𝐴 ) |
26 |
19 25 3
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑥 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
27 |
26
|
ex |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝑘 ∈ 𝑥 → 𝐵 ∈ ( 0 [,] +∞ ) ) ) |
28 |
18 27
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ∀ 𝑘 ∈ 𝑥 𝐵 ∈ ( 0 [,] +∞ ) ) |
29 |
11 13 16 28
|
gsummptcl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ∈ ( 0 [,] +∞ ) ) |
30 |
10 29
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ∈ ℝ* ) |
31 |
30
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ∈ ℝ* ) |
32 |
|
eqid |
⊢ ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) = ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) |
33 |
32
|
rnmptss |
⊢ ( ∀ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ∈ ℝ* → ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ⊆ ℝ* ) |
34 |
31 33
|
syl |
⊢ ( 𝜑 → ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ⊆ ℝ* ) |
35 |
|
supxrlub |
⊢ ( ( ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ⊆ ℝ* ∧ 𝑋 ∈ ℝ* ) → ( 𝑋 < sup ( ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) , ℝ* , < ) ↔ ∃ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) 𝑋 < 𝑦 ) ) |
36 |
34 4 35
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 < sup ( ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) , ℝ* , < ) ↔ ∃ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) 𝑋 < 𝑦 ) ) |
37 |
9 36
|
bitrd |
⊢ ( 𝜑 → ( 𝑋 < Σ* 𝑘 ∈ 𝐴 𝐵 ↔ ∃ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) 𝑋 < 𝑦 ) ) |
38 |
5 37
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) 𝑋 < 𝑦 ) |
39 |
|
ovex |
⊢ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ∈ V |
40 |
39
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ∈ V ) |
41 |
|
mpteq1 |
⊢ ( 𝑥 = 𝑎 → ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) = ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) |
42 |
41
|
oveq2d |
⊢ ( 𝑥 = 𝑎 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) |
43 |
42
|
cbvmptv |
⊢ ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) = ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) |
44 |
43 39
|
elrnmpti |
⊢ ( 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ↔ ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) 𝑦 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) |
45 |
44
|
a1i |
⊢ ( 𝜑 → ( 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ↔ ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) 𝑦 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) ) |
46 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) → 𝑦 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) |
47 |
46
|
breq2d |
⊢ ( ( 𝜑 ∧ 𝑦 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) → ( 𝑋 < 𝑦 ↔ 𝑋 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) ) |
48 |
40 45 47
|
rexxfr2d |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) 𝑋 < 𝑦 ↔ ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) 𝑋 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) ) |
49 |
38 48
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) 𝑋 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) |
50 |
|
nfv |
⊢ Ⅎ 𝑘 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) |
51 |
1 50
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
52 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
53 |
14 52
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑎 ∈ Fin ) |
54 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑎 ) → 𝜑 ) |
55 |
20
|
sseli |
⊢ ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) → 𝑎 ∈ 𝒫 𝐴 ) |
56 |
55
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑎 ) → 𝑎 ∈ 𝒫 𝐴 ) |
57 |
56
|
elpwid |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑎 ) → 𝑎 ⊆ 𝐴 ) |
58 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑎 ) → 𝑘 ∈ 𝑎 ) |
59 |
57 58
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑎 ) → 𝑘 ∈ 𝐴 ) |
60 |
54 59 3
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑎 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
61 |
51 53 60
|
gsumesum |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) = Σ* 𝑘 ∈ 𝑎 𝐵 ) |
62 |
61
|
breq2d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝑋 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ↔ 𝑋 < Σ* 𝑘 ∈ 𝑎 𝐵 ) ) |
63 |
62
|
biimpd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝑋 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) → 𝑋 < Σ* 𝑘 ∈ 𝑎 𝐵 ) ) |
64 |
63
|
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) 𝑋 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) → ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) 𝑋 < Σ* 𝑘 ∈ 𝑎 𝐵 ) ) |
65 |
49 64
|
mpd |
⊢ ( 𝜑 → ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) 𝑋 < Σ* 𝑘 ∈ 𝑎 𝐵 ) |