Step |
Hyp |
Ref |
Expression |
1 |
|
gsumesum.0 |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
gsumesum.1 |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
3 |
|
gsumesum.2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
4 |
|
nfcv |
⊢ Ⅎ 𝑘 𝐴 |
5 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) |
6 |
1 4 2 3 5
|
esumval |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 = sup ( ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) , ℝ* , < ) ) |
7 |
|
xrltso |
⊢ < Or ℝ* |
8 |
7
|
a1i |
⊢ ( 𝜑 → < Or ℝ* ) |
9 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
10 |
|
xrge0base |
⊢ ( 0 [,] +∞ ) = ( Base ‘ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ) |
11 |
|
xrge0cmn |
⊢ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd |
12 |
11
|
a1i |
⊢ ( 𝜑 → ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd ) |
13 |
3
|
ex |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 → 𝐵 ∈ ( 0 [,] +∞ ) ) ) |
14 |
1 13
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) ) |
15 |
10 12 2 14
|
gsummptcl |
⊢ ( 𝜑 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ∈ ( 0 [,] +∞ ) ) |
16 |
9 15
|
sselid |
⊢ ( 𝜑 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ∈ ℝ* ) |
17 |
|
pwidg |
⊢ ( 𝐴 ∈ Fin → 𝐴 ∈ 𝒫 𝐴 ) |
18 |
2 17
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ 𝒫 𝐴 ) |
19 |
18 2
|
elind |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
20 |
|
eqidd |
⊢ ( 𝜑 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |
21 |
|
mpteq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) |
22 |
21
|
oveq2d |
⊢ ( 𝑥 = 𝐴 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |
23 |
22
|
rspceeqv |
⊢ ( ( 𝐴 ∈ ( 𝒫 𝐴 ∩ Fin ) ∧ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) → ∃ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) |
24 |
19 20 23
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) |
25 |
|
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 ) ) ) |
26 |
|
ovex |
⊢ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ∈ V |
27 |
25 26
|
elrnmpti |
⊢ ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ↔ ∃ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) |
28 |
24 27
|
sylibr |
⊢ ( 𝜑 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) |
29 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) → 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) |
30 |
|
mpteq1 |
⊢ ( 𝑥 = 𝑎 → ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) = ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) |
31 |
30
|
oveq2d |
⊢ ( 𝑥 = 𝑎 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) |
32 |
31
|
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 ) ) ) |
33 |
|
ovex |
⊢ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ∈ V |
34 |
32 33
|
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 ) ) ) |
35 |
29 34
|
sylib |
⊢ ( ( 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 ) ) ) |
36 |
11
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd ) |
37 |
|
inss2 |
⊢ ( 𝒫 𝐴 ∩ Fin ) ⊆ Fin |
38 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
39 |
37 38
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑎 ∈ Fin ) |
40 |
|
nfv |
⊢ Ⅎ 𝑘 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) |
41 |
1 40
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
42 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑎 ) → 𝜑 ) |
43 |
|
inss1 |
⊢ ( 𝒫 𝐴 ∩ Fin ) ⊆ 𝒫 𝐴 |
44 |
43
|
sseli |
⊢ ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) → 𝑎 ∈ 𝒫 𝐴 ) |
45 |
44
|
elpwid |
⊢ ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) → 𝑎 ⊆ 𝐴 ) |
46 |
45
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑎 ) → 𝑎 ⊆ 𝐴 ) |
47 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑎 ) → 𝑘 ∈ 𝑎 ) |
48 |
46 47
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑎 ) → 𝑘 ∈ 𝐴 ) |
49 |
42 48 3
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑎 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
50 |
49
|
ex |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝑘 ∈ 𝑎 → 𝐵 ∈ ( 0 [,] +∞ ) ) ) |
51 |
41 50
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ∀ 𝑘 ∈ 𝑎 𝐵 ∈ ( 0 [,] +∞ ) ) |
52 |
10 36 39 51
|
gsummptcl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ∈ ( 0 [,] +∞ ) ) |
53 |
9 52
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ∈ ℝ* ) |
54 |
|
diffi |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∖ 𝑎 ) ∈ Fin ) |
55 |
2 54
|
syl |
⊢ ( 𝜑 → ( 𝐴 ∖ 𝑎 ) ∈ Fin ) |
56 |
55
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝐴 ∖ 𝑎 ) ∈ Fin ) |
57 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ) → 𝜑 ) |
58 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ) → 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ) |
59 |
58
|
eldifad |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ) → 𝑘 ∈ 𝐴 ) |
60 |
57 59 3
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
61 |
60
|
ex |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) ) |
62 |
41 61
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ∀ 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) 𝐵 ∈ ( 0 [,] +∞ ) ) |
63 |
10 36 56 62
|
gsummptcl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ∈ ( 0 [,] +∞ ) ) |
64 |
9 63
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ∈ ℝ* ) |
65 |
|
elxrge0 |
⊢ ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ∈ ( 0 [,] +∞ ) ↔ ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ∈ ℝ* ∧ 0 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ) ) |
66 |
65
|
simprbi |
⊢ ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ∈ ( 0 [,] +∞ ) → 0 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ) |
67 |
63 66
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 0 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ) |
68 |
|
xraddge02 |
⊢ ( ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ∈ ℝ* ∧ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ∈ ℝ* ) → ( 0 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ≤ ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) +𝑒 ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ) ) ) |
69 |
68
|
imp |
⊢ ( ( ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ∈ ℝ* ∧ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ∈ ℝ* ) ∧ 0 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ≤ ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) +𝑒 ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ) ) |
70 |
53 64 67 69
|
syl21anc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ≤ ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) +𝑒 ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ) ) |
71 |
70
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ≤ ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) +𝑒 ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ) ) |
72 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝜑 ) |
73 |
45
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑎 ⊆ 𝐴 ) |
74 |
|
xrge00 |
⊢ 0 = ( 0g ‘ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ) |
75 |
|
xrge0plusg |
⊢ +𝑒 = ( +g ‘ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ) |
76 |
11
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐴 ) → ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd ) |
77 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐴 ) → 𝐴 ∈ Fin ) |
78 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) |
79 |
1 3 78
|
fmptdf |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ( 0 [,] +∞ ) ) |
80 |
79
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐴 ) → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ( 0 [,] +∞ ) ) |
81 |
78
|
fnmpt |
⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) Fn 𝐴 ) |
82 |
14 81
|
syl |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) Fn 𝐴 ) |
83 |
|
c0ex |
⊢ 0 ∈ V |
84 |
83
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
85 |
82 2 84
|
fndmfifsupp |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) finSupp 0 ) |
86 |
85
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐴 ) → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) finSupp 0 ) |
87 |
|
disjdif |
⊢ ( 𝑎 ∩ ( 𝐴 ∖ 𝑎 ) ) = ∅ |
88 |
87
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐴 ) → ( 𝑎 ∩ ( 𝐴 ∖ 𝑎 ) ) = ∅ ) |
89 |
|
undif |
⊢ ( 𝑎 ⊆ 𝐴 ↔ ( 𝑎 ∪ ( 𝐴 ∖ 𝑎 ) ) = 𝐴 ) |
90 |
89
|
biimpi |
⊢ ( 𝑎 ⊆ 𝐴 → ( 𝑎 ∪ ( 𝐴 ∖ 𝑎 ) ) = 𝐴 ) |
91 |
90
|
eqcomd |
⊢ ( 𝑎 ⊆ 𝐴 → 𝐴 = ( 𝑎 ∪ ( 𝐴 ∖ 𝑎 ) ) ) |
92 |
91
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐴 ) → 𝐴 = ( 𝑎 ∪ ( 𝐴 ∖ 𝑎 ) ) ) |
93 |
10 74 75 76 77 80 86 88 92
|
gsumsplit |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐴 ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝑎 ) ) +𝑒 ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ ( 𝐴 ∖ 𝑎 ) ) ) ) ) |
94 |
|
resmpt |
⊢ ( 𝑎 ⊆ 𝐴 → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝑎 ) = ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) |
95 |
94
|
oveq2d |
⊢ ( 𝑎 ⊆ 𝐴 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝑎 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) |
96 |
95
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐴 ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝑎 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ) |
97 |
|
difss |
⊢ ( 𝐴 ∖ 𝑎 ) ⊆ 𝐴 |
98 |
|
resmpt |
⊢ ( ( 𝐴 ∖ 𝑎 ) ⊆ 𝐴 → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ ( 𝐴 ∖ 𝑎 ) ) = ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) |
99 |
97 98
|
ax-mp |
⊢ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ ( 𝐴 ∖ 𝑎 ) ) = ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) |
100 |
99
|
oveq2i |
⊢ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ ( 𝐴 ∖ 𝑎 ) ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) |
101 |
100
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐴 ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ ( 𝐴 ∖ 𝑎 ) ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ) |
102 |
96 101
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐴 ) → ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝑎 ) ) +𝑒 ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ ( 𝐴 ∖ 𝑎 ) ) ) ) = ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) +𝑒 ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ) ) |
103 |
93 102
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐴 ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) +𝑒 ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ) ) |
104 |
72 73 103
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) +𝑒 ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝐴 ∖ 𝑎 ) ↦ 𝐵 ) ) ) ) |
105 |
71 104
|
breqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |
106 |
105
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) → ∀ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |
107 |
|
r19.29r |
⊢ ( ( ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) 𝑦 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ∧ ∀ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) → ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ( 𝑦 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ∧ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) ) |
108 |
|
breq1 |
⊢ ( 𝑦 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) → ( 𝑦 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ↔ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) ) |
109 |
108
|
biimpar |
⊢ ( ( 𝑦 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ∧ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) → 𝑦 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |
110 |
109
|
rexlimivw |
⊢ ( ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ( 𝑦 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ∧ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) → 𝑦 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |
111 |
107 110
|
syl |
⊢ ( ( ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) 𝑦 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ∧ ∀ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑎 ↦ 𝐵 ) ) ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) → 𝑦 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |
112 |
35 106 111
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) → 𝑦 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |
113 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ∈ ℝ* ) |
114 |
11
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd ) |
115 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
116 |
37 115
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑥 ∈ Fin ) |
117 |
|
nfv |
⊢ Ⅎ 𝑘 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) |
118 |
1 117
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
119 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑥 ) → 𝜑 ) |
120 |
43
|
sseli |
⊢ ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) → 𝑥 ∈ 𝒫 𝐴 ) |
121 |
120
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑥 ) → 𝑥 ∈ 𝒫 𝐴 ) |
122 |
121
|
elpwid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑥 ) → 𝑥 ⊆ 𝐴 ) |
123 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑥 ) → 𝑘 ∈ 𝑥 ) |
124 |
122 123
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑥 ) → 𝑘 ∈ 𝐴 ) |
125 |
119 124 3
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑥 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
126 |
125
|
ex |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝑘 ∈ 𝑥 → 𝐵 ∈ ( 0 [,] +∞ ) ) ) |
127 |
118 126
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ∀ 𝑘 ∈ 𝑥 𝐵 ∈ ( 0 [,] +∞ ) ) |
128 |
10 114 116 127
|
gsummptcl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ∈ ( 0 [,] +∞ ) ) |
129 |
9 128
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ∈ ℝ* ) |
130 |
129
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ∈ ℝ* ) |
131 |
25
|
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* ) |
132 |
130 131
|
syl |
⊢ ( 𝜑 → ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ⊆ ℝ* ) |
133 |
132
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) → 𝑦 ∈ ℝ* ) |
134 |
|
xrltnle |
⊢ ( ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) < 𝑦 ↔ ¬ 𝑦 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) ) |
135 |
134
|
con2bid |
⊢ ( ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( 𝑦 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ↔ ¬ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) < 𝑦 ) ) |
136 |
113 133 135
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) → ( 𝑦 ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ↔ ¬ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) < 𝑦 ) ) |
137 |
112 136
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) ) → ¬ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) < 𝑦 ) |
138 |
8 16 28 137
|
supmax |
⊢ ( 𝜑 → sup ( ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) ) , ℝ* , < ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |
139 |
6 138
|
eqtr2d |
⊢ ( 𝜑 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = Σ* 𝑘 ∈ 𝐴 𝐵 ) |