Step |
Hyp |
Ref |
Expression |
1 |
|
esum2d.0 |
⊢ Ⅎ 𝑘 𝐹 |
2 |
|
esum2d.1 |
⊢ ( 𝑧 = 〈 𝑗 , 𝑘 〉 → 𝐹 = 𝐶 ) |
3 |
|
esum2d.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
esum2d.3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝐵 ∈ 𝑊 ) |
5 |
|
esum2d.4 |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ) ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
6 |
|
xrltso |
⊢ < Or ℝ* |
7 |
6
|
a1i |
⊢ ( 𝜑 → < Or ℝ* ) |
8 |
|
nfv |
⊢ Ⅎ 𝑐 𝜑 |
9 |
|
nfcv |
⊢ Ⅎ 𝑐 𝑠 |
10 |
|
nfmpt1 |
⊢ Ⅎ 𝑐 ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
11 |
10
|
nfrn |
⊢ Ⅎ 𝑐 ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
12 |
9 11
|
nfel |
⊢ Ⅎ 𝑐 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
13 |
8 12
|
nfan |
⊢ Ⅎ 𝑐 ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) |
14 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
15 |
|
xrge0base |
⊢ ( 0 [,] +∞ ) = ( Base ‘ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ) |
16 |
|
xrge0cmn |
⊢ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd |
17 |
16
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd ) |
18 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) |
19 |
18
|
elin2d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → 𝑐 ∈ Fin ) |
20 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑧 ∈ 𝑐 ) → 𝜑 ) |
21 |
18
|
elin1d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → 𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
22 |
21
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑧 ∈ 𝑐 ) → 𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
23 |
|
vex |
⊢ 𝑐 ∈ V |
24 |
23
|
elpw |
⊢ ( 𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ↔ 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
25 |
22 24
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑧 ∈ 𝑐 ) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
26 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑧 ∈ 𝑐 ) → 𝑧 ∈ 𝑐 ) |
27 |
25 26
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑧 ∈ 𝑐 ) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
28 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
29 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑧 |
30 |
|
nfiu1 |
⊢ Ⅎ 𝑗 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) |
31 |
29 30
|
nfel |
⊢ Ⅎ 𝑗 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) |
32 |
28 31
|
nfan |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
33 |
|
nfv |
⊢ Ⅎ 𝑘 ( ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ) |
34 |
|
nfcv |
⊢ Ⅎ 𝑘 ( 0 [,] +∞ ) |
35 |
1 34
|
nfel |
⊢ Ⅎ 𝑘 𝐹 ∈ ( 0 [,] +∞ ) |
36 |
2
|
adantl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ) ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑧 = 〈 𝑗 , 𝑘 〉 ) → 𝐹 = 𝐶 ) |
37 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ) ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑧 = 〈 𝑗 , 𝑘 〉 ) → 𝜑 ) |
38 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ) ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑧 = 〈 𝑗 , 𝑘 〉 ) → 𝑗 ∈ 𝐴 ) |
39 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ) ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑧 = 〈 𝑗 , 𝑘 〉 ) → 𝑘 ∈ 𝐵 ) |
40 |
37 38 39 5
|
syl12anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ) ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑧 = 〈 𝑗 , 𝑘 〉 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
41 |
36 40
|
eqeltrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ) ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑧 = 〈 𝑗 , 𝑘 〉 ) → 𝐹 ∈ ( 0 [,] +∞ ) ) |
42 |
|
elsnxp |
⊢ ( 𝑗 ∈ 𝐴 → ( 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ↔ ∃ 𝑘 ∈ 𝐵 𝑧 = 〈 𝑗 , 𝑘 〉 ) ) |
43 |
42
|
biimpa |
⊢ ( ( 𝑗 ∈ 𝐴 ∧ 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ) → ∃ 𝑘 ∈ 𝐵 𝑧 = 〈 𝑗 , 𝑘 〉 ) |
44 |
43
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ) → ∃ 𝑘 ∈ 𝐵 𝑧 = 〈 𝑗 , 𝑘 〉 ) |
45 |
33 35 41 44
|
r19.29af2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ) → 𝐹 ∈ ( 0 [,] +∞ ) ) |
46 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
47 |
|
eliun |
⊢ ( 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ↔ ∃ 𝑗 ∈ 𝐴 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ) |
48 |
46 47
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) → ∃ 𝑗 ∈ 𝐴 𝑧 ∈ ( { 𝑗 } × 𝐵 ) ) |
49 |
32 45 48
|
r19.29af |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) → 𝐹 ∈ ( 0 [,] +∞ ) ) |
50 |
20 27 49
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑧 ∈ 𝑐 ) → 𝐹 ∈ ( 0 [,] +∞ ) ) |
51 |
50
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ∀ 𝑧 ∈ 𝑐 𝐹 ∈ ( 0 [,] +∞ ) ) |
52 |
15 17 19 51
|
gsummptcl |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ∈ ( 0 [,] +∞ ) ) |
53 |
14 52
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ∈ ℝ* ) |
54 |
53
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ∈ ℝ* ) |
55 |
|
eqid |
⊢ ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) = ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
56 |
55
|
rnmptss |
⊢ ( ∀ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ∈ ℝ* → ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ⊆ ℝ* ) |
57 |
54 56
|
syl |
⊢ ( 𝜑 → ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ⊆ ℝ* ) |
58 |
57
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ⊆ ℝ* ) |
59 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) |
60 |
58 59
|
sseldd |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → 𝑠 ∈ ℝ* ) |
61 |
|
snex |
⊢ { 𝑗 } ∈ V |
62 |
|
xpexg |
⊢ ( ( { 𝑗 } ∈ V ∧ 𝐵 ∈ 𝑊 ) → ( { 𝑗 } × 𝐵 ) ∈ V ) |
63 |
61 4 62
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ( { 𝑗 } × 𝐵 ) ∈ V ) |
64 |
63
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∈ V ) |
65 |
|
iunexg |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∈ V ) → ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∈ V ) |
66 |
3 64 65
|
syl2anc |
⊢ ( 𝜑 → ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∈ V ) |
67 |
49
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ∈ ( 0 [,] +∞ ) ) |
68 |
|
nfcv |
⊢ Ⅎ 𝑧 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) |
69 |
68
|
esumcl |
⊢ ( ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∈ V ∧ ∀ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ∈ ( 0 [,] +∞ ) ) → Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ∈ ( 0 [,] +∞ ) ) |
70 |
66 67 69
|
syl2anc |
⊢ ( 𝜑 → Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ∈ ( 0 [,] +∞ ) ) |
71 |
14 70
|
sselid |
⊢ ( 𝜑 → Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ∈ ℝ* ) |
72 |
71
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ∈ ℝ* ) |
73 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
74 |
|
nfv |
⊢ Ⅎ 𝑧 ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) |
75 |
|
nfcv |
⊢ Ⅎ 𝑧 𝑐 |
76 |
74 75 19 50
|
esumgsum |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → Σ* 𝑧 ∈ 𝑐 𝐹 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
77 |
66
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∈ V ) |
78 |
49
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) → 𝐹 ∈ ( 0 [,] +∞ ) ) |
79 |
21 24
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
80 |
74 77 78 79
|
esummono |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → Σ* 𝑧 ∈ 𝑐 𝐹 ≤ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) |
81 |
76 80
|
eqbrtrrd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ≤ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) |
82 |
81
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ≤ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) |
83 |
82
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ≤ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) |
84 |
73 83
|
eqbrtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → 𝑠 ≤ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) |
85 |
|
xrlenlt |
⊢ ( ( 𝑠 ∈ ℝ* ∧ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ∈ ℝ* ) → ( 𝑠 ≤ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ↔ ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < 𝑠 ) ) |
86 |
85
|
biimpa |
⊢ ( ( ( 𝑠 ∈ ℝ* ∧ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ∈ ℝ* ) ∧ 𝑠 ≤ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < 𝑠 ) |
87 |
60 72 84 86
|
syl21anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < 𝑠 ) |
88 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) → 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) |
89 |
|
ovex |
⊢ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ∈ V |
90 |
55 89
|
elrnmpti |
⊢ ( 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ↔ ∃ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
91 |
88 90
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) → ∃ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
92 |
13 87 91
|
r19.29af |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) → ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < 𝑠 ) |
93 |
92
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < 𝑠 ) |
94 |
|
nfv |
⊢ Ⅎ 𝑐 ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) |
95 |
|
nfv |
⊢ Ⅎ 𝑐 𝑠 < 𝑡 |
96 |
11 95
|
nfrex |
⊢ Ⅎ 𝑐 ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 |
97 |
76
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → Σ* 𝑧 ∈ 𝑐 𝐹 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
98 |
97
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → Σ* 𝑧 ∈ 𝑐 𝐹 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
99 |
98
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 < Σ* 𝑧 ∈ 𝑐 𝐹 ) → Σ* 𝑧 ∈ 𝑐 𝐹 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
100 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 < Σ* 𝑧 ∈ 𝑐 𝐹 ) → 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) |
101 |
89
|
a1i |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 < Σ* 𝑧 ∈ 𝑐 𝐹 ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ∈ V ) |
102 |
55
|
elrnmpt1 |
⊢ ( ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ∧ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ∈ V ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) |
103 |
100 101 102
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 < Σ* 𝑧 ∈ 𝑐 𝐹 ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) |
104 |
99 103
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 < Σ* 𝑧 ∈ 𝑐 𝐹 ) → Σ* 𝑧 ∈ 𝑐 𝐹 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) |
105 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 < Σ* 𝑧 ∈ 𝑐 𝐹 ) ∧ 𝑡 = Σ* 𝑧 ∈ 𝑐 𝐹 ) → 𝑡 = Σ* 𝑧 ∈ 𝑐 𝐹 ) |
106 |
105
|
breq2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 < Σ* 𝑧 ∈ 𝑐 𝐹 ) ∧ 𝑡 = Σ* 𝑧 ∈ 𝑐 𝐹 ) → ( 𝑠 < 𝑡 ↔ 𝑠 < Σ* 𝑧 ∈ 𝑐 𝐹 ) ) |
107 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 < Σ* 𝑧 ∈ 𝑐 𝐹 ) → 𝑠 < Σ* 𝑧 ∈ 𝑐 𝐹 ) |
108 |
104 106 107
|
rspcedvd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑠 < Σ* 𝑧 ∈ 𝑐 𝐹 ) → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) |
109 |
|
nfv |
⊢ Ⅎ 𝑧 ( 𝜑 ∧ 𝑠 ∈ ℝ* ) |
110 |
|
nfcv |
⊢ Ⅎ 𝑧 𝑠 |
111 |
|
nfcv |
⊢ Ⅎ 𝑧 < |
112 |
68
|
nfesum1 |
⊢ Ⅎ 𝑧 Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 |
113 |
110 111 112
|
nfbr |
⊢ Ⅎ 𝑧 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 |
114 |
109 113
|
nfan |
⊢ Ⅎ 𝑧 ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) |
115 |
66
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∈ V ) |
116 |
49
|
3ad2antr3 |
⊢ ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) ) → 𝐹 ∈ ( 0 [,] +∞ ) ) |
117 |
116
|
3anassrs |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) → 𝐹 ∈ ( 0 [,] +∞ ) ) |
118 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → 𝑠 ∈ ℝ* ) |
119 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) |
120 |
114 115 117 118 119
|
esumlub |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → ∃ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) 𝑠 < Σ* 𝑧 ∈ 𝑐 𝐹 ) |
121 |
94 96 108 120
|
r19.29af2 |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) |
122 |
121
|
ex |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) → ( 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ) |
123 |
122
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑠 ∈ ℝ* ( 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ) |
124 |
93 123
|
jca |
⊢ ( 𝜑 → ( ∀ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < 𝑠 ∧ ∀ 𝑠 ∈ ℝ* ( 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ) ) |
125 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑟 = Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → 𝑟 = Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) |
126 |
125
|
breq1d |
⊢ ( ( 𝜑 ∧ 𝑟 = Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → ( 𝑟 < 𝑠 ↔ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < 𝑠 ) ) |
127 |
126
|
notbid |
⊢ ( ( 𝜑 ∧ 𝑟 = Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → ( ¬ 𝑟 < 𝑠 ↔ ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < 𝑠 ) ) |
128 |
127
|
ralbidv |
⊢ ( ( 𝜑 ∧ 𝑟 = Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → ( ∀ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ¬ 𝑟 < 𝑠 ↔ ∀ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < 𝑠 ) ) |
129 |
125
|
breq2d |
⊢ ( ( 𝜑 ∧ 𝑟 = Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → ( 𝑠 < 𝑟 ↔ 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) ) |
130 |
129
|
imbi1d |
⊢ ( ( 𝜑 ∧ 𝑟 = Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → ( ( 𝑠 < 𝑟 → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ↔ ( 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ) ) |
131 |
130
|
ralbidv |
⊢ ( ( 𝜑 ∧ 𝑟 = Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → ( ∀ 𝑠 ∈ ℝ* ( 𝑠 < 𝑟 → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ↔ ∀ 𝑠 ∈ ℝ* ( 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ) ) |
132 |
128 131
|
anbi12d |
⊢ ( ( 𝜑 ∧ 𝑟 = Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) → ( ( ∀ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ¬ 𝑟 < 𝑠 ∧ ∀ 𝑠 ∈ ℝ* ( 𝑠 < 𝑟 → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ) ↔ ( ∀ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < 𝑠 ∧ ∀ 𝑠 ∈ ℝ* ( 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ) ) ) |
133 |
71 132
|
rspcedv |
⊢ ( 𝜑 → ( ( ∀ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < 𝑠 ∧ ∀ 𝑠 ∈ ℝ* ( 𝑠 < Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ) → ∃ 𝑟 ∈ ℝ* ( ∀ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ¬ 𝑟 < 𝑠 ∧ ∀ 𝑠 ∈ ℝ* ( 𝑠 < 𝑟 → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ) ) ) |
134 |
124 133
|
mpd |
⊢ ( 𝜑 → ∃ 𝑟 ∈ ℝ* ( ∀ 𝑠 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ¬ 𝑟 < 𝑠 ∧ ∀ 𝑠 ∈ ℝ* ( 𝑠 < 𝑟 → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ) ) |
135 |
7 134
|
supcl |
⊢ ( 𝜑 → sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ∈ ℝ* ) |
136 |
|
nfv |
⊢ Ⅎ 𝑎 𝜑 |
137 |
|
nfcv |
⊢ Ⅎ 𝑎 𝑠 |
138 |
|
nfmpt1 |
⊢ Ⅎ 𝑎 ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
139 |
138
|
nfrn |
⊢ Ⅎ 𝑎 ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
140 |
137 139
|
nfel |
⊢ Ⅎ 𝑎 𝑠 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
141 |
136 140
|
nfan |
⊢ Ⅎ 𝑎 ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) |
142 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
143 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑗 ∈ 𝑎 ) → 𝜑 ) |
144 |
142
|
elin1d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑎 ∈ 𝒫 𝐴 ) |
145 |
|
elpwi |
⊢ ( 𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴 ) |
146 |
144 145
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑎 ⊆ 𝐴 ) |
147 |
146
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑗 ∈ 𝑎 ) → 𝑗 ∈ 𝐴 ) |
148 |
143 147 4
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑗 ∈ 𝑎 ) → 𝐵 ∈ 𝑊 ) |
149 |
143
|
adantrr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵 ) ) → 𝜑 ) |
150 |
147
|
adantrr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵 ) ) → 𝑗 ∈ 𝐴 ) |
151 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵 ) ) → 𝑘 ∈ 𝐵 ) |
152 |
149 150 151 5
|
syl12anc |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑗 ∈ 𝑎 ∧ 𝑘 ∈ 𝐵 ) ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
153 |
142
|
elin2d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑎 ∈ Fin ) |
154 |
1 2 142 148 152 153
|
esum2dlem |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → Σ* 𝑗 ∈ 𝑎 Σ* 𝑘 ∈ 𝐵 𝐶 = Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ( { 𝑗 } × 𝐵 ) 𝐹 ) |
155 |
|
nfv |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
156 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑎 |
157 |
5
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑘 ∈ 𝐵 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
158 |
157
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ∀ 𝑘 ∈ 𝐵 𝐶 ∈ ( 0 [,] +∞ ) ) |
159 |
|
nfcv |
⊢ Ⅎ 𝑘 𝐵 |
160 |
159
|
esumcl |
⊢ ( ( 𝐵 ∈ 𝑊 ∧ ∀ 𝑘 ∈ 𝐵 𝐶 ∈ ( 0 [,] +∞ ) ) → Σ* 𝑘 ∈ 𝐵 𝐶 ∈ ( 0 [,] +∞ ) ) |
161 |
4 158 160
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → Σ* 𝑘 ∈ 𝐵 𝐶 ∈ ( 0 [,] +∞ ) ) |
162 |
143 147 161
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑗 ∈ 𝑎 ) → Σ* 𝑘 ∈ 𝐵 𝐶 ∈ ( 0 [,] +∞ ) ) |
163 |
155 156 153 162
|
esumgsum |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → Σ* 𝑗 ∈ 𝑎 Σ* 𝑘 ∈ 𝐵 𝐶 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
164 |
154 163
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ( { 𝑗 } × 𝐵 ) 𝐹 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
165 |
|
nfv |
⊢ Ⅎ 𝑧 ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
166 |
66
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∈ V ) |
167 |
49
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) → 𝐹 ∈ ( 0 [,] +∞ ) ) |
168 |
|
iunss1 |
⊢ ( 𝑎 ⊆ 𝐴 → ∪ 𝑗 ∈ 𝑎 ( { 𝑗 } × 𝐵 ) ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
169 |
146 168
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ∪ 𝑗 ∈ 𝑎 ( { 𝑗 } × 𝐵 ) ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
170 |
165 166 167 169
|
esummono |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ( { 𝑗 } × 𝐵 ) 𝐹 ≤ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) |
171 |
164 170
|
eqbrtrrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ≤ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) |
172 |
16
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd ) |
173 |
162
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ∀ 𝑗 ∈ 𝑎 Σ* 𝑘 ∈ 𝐵 𝐶 ∈ ( 0 [,] +∞ ) ) |
174 |
15 172 153 173
|
gsummptcl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ∈ ( 0 [,] +∞ ) ) |
175 |
14 174
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ∈ ℝ* ) |
176 |
71
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ∈ ℝ* ) |
177 |
|
xrlenlt |
⊢ ( ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ∈ ℝ* ∧ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ∈ ℝ* ) → ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ≤ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ↔ ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) |
178 |
175 176 177
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ≤ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ↔ ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) |
179 |
171 178
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ¬ Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
180 |
|
nfv |
⊢ Ⅎ 𝑧 𝜑 |
181 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
182 |
180 68 66 49 181
|
esumval |
⊢ ( 𝜑 → Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 = sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) |
183 |
182
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 = sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) |
184 |
183
|
breq1d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ↔ sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) |
185 |
179 184
|
mtbid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ¬ sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
186 |
185
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ¬ sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
187 |
186
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) → ¬ sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
188 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) → 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
189 |
188
|
breq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) → ( sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) < 𝑠 ↔ sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) |
190 |
189
|
notbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) → ( ¬ sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) < 𝑠 ↔ ¬ sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) |
191 |
187 190
|
mpbird |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) → ¬ sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) < 𝑠 ) |
192 |
|
eqid |
⊢ ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) = ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
193 |
|
ovex |
⊢ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ∈ V |
194 |
192 193
|
elrnmpti |
⊢ ( 𝑠 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ↔ ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
195 |
194
|
biimpi |
⊢ ( 𝑠 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) → ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
196 |
195
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) → ∃ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) 𝑠 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
197 |
141 191 196
|
r19.29af |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) → ¬ sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) < 𝑠 ) |
198 |
9
|
nfel1 |
⊢ Ⅎ 𝑐 𝑠 ∈ ℝ* |
199 |
|
nfcv |
⊢ Ⅎ 𝑐 < |
200 |
|
nfcv |
⊢ Ⅎ 𝑐 ℝ* |
201 |
11 200 199
|
nfsup |
⊢ Ⅎ 𝑐 sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) |
202 |
9 199 201
|
nfbr |
⊢ Ⅎ 𝑐 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) |
203 |
198 202
|
nfan |
⊢ Ⅎ 𝑐 ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) |
204 |
8 203
|
nfan |
⊢ Ⅎ 𝑐 ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) |
205 |
|
nfcv |
⊢ Ⅎ 𝑐 𝑢 |
206 |
205 11
|
nfel |
⊢ Ⅎ 𝑐 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
207 |
204 206
|
nfan |
⊢ Ⅎ 𝑐 ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) |
208 |
|
nfv |
⊢ Ⅎ 𝑐 𝑠 < 𝑢 |
209 |
207 208
|
nfan |
⊢ Ⅎ 𝑐 ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) |
210 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → 𝜑 ) |
211 |
|
simpr1l |
⊢ ( ( 𝜑 ∧ ( ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ ( 𝑠 < 𝑢 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ) → 𝑠 ∈ ℝ* ) |
212 |
211
|
3anassrs |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ ( 𝑠 < 𝑢 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) → 𝑠 ∈ ℝ* ) |
213 |
212
|
3anassrs |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → 𝑠 ∈ ℝ* ) |
214 |
210 213
|
jca |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ) |
215 |
|
simpr1r |
⊢ ( ( 𝜑 ∧ ( ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ ( 𝑠 < 𝑢 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ) → 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) |
216 |
215
|
3anassrs |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ ( 𝑠 < 𝑢 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) → 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) |
217 |
216
|
3anassrs |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) |
218 |
214 217
|
jca |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) |
219 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → 𝑠 < 𝑢 ) |
220 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
221 |
219 220
|
breqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
222 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) |
223 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) |
224 |
223
|
elin1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → 𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
225 |
|
elpwi |
⊢ ( 𝑐 ∈ 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
226 |
|
dmss |
⊢ ( 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) → dom 𝑐 ⊆ dom ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
227 |
|
dmiun |
⊢ dom ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) = ∪ 𝑗 ∈ 𝐴 dom ( { 𝑗 } × 𝐵 ) |
228 |
226 227
|
sseqtrdi |
⊢ ( 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) → dom 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 dom ( { 𝑗 } × 𝐵 ) ) |
229 |
|
dmxpss |
⊢ dom ( { 𝑗 } × 𝐵 ) ⊆ { 𝑗 } |
230 |
229
|
a1i |
⊢ ( 𝑗 ∈ 𝐴 → dom ( { 𝑗 } × 𝐵 ) ⊆ { 𝑗 } ) |
231 |
|
snssi |
⊢ ( 𝑗 ∈ 𝐴 → { 𝑗 } ⊆ 𝐴 ) |
232 |
230 231
|
sstrd |
⊢ ( 𝑗 ∈ 𝐴 → dom ( { 𝑗 } × 𝐵 ) ⊆ 𝐴 ) |
233 |
232
|
rgen |
⊢ ∀ 𝑗 ∈ 𝐴 dom ( { 𝑗 } × 𝐵 ) ⊆ 𝐴 |
234 |
|
iunss |
⊢ ( ∪ 𝑗 ∈ 𝐴 dom ( { 𝑗 } × 𝐵 ) ⊆ 𝐴 ↔ ∀ 𝑗 ∈ 𝐴 dom ( { 𝑗 } × 𝐵 ) ⊆ 𝐴 ) |
235 |
233 234
|
mpbir |
⊢ ∪ 𝑗 ∈ 𝐴 dom ( { 𝑗 } × 𝐵 ) ⊆ 𝐴 |
236 |
228 235
|
sstrdi |
⊢ ( 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) → dom 𝑐 ⊆ 𝐴 ) |
237 |
23
|
dmex |
⊢ dom 𝑐 ∈ V |
238 |
237
|
elpw |
⊢ ( dom 𝑐 ∈ 𝒫 𝐴 ↔ dom 𝑐 ⊆ 𝐴 ) |
239 |
236 238
|
sylibr |
⊢ ( 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) → dom 𝑐 ∈ 𝒫 𝐴 ) |
240 |
224 225 239
|
3syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → dom 𝑐 ∈ 𝒫 𝐴 ) |
241 |
223
|
elin2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → 𝑐 ∈ Fin ) |
242 |
|
dmfi |
⊢ ( 𝑐 ∈ Fin → dom 𝑐 ∈ Fin ) |
243 |
241 242
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → dom 𝑐 ∈ Fin ) |
244 |
240 243
|
elind |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → dom 𝑐 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
245 |
|
ovex |
⊢ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ∈ V |
246 |
245
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ∈ V ) |
247 |
|
mpteq1 |
⊢ ( 𝑎 = dom 𝑐 → ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) = ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) |
248 |
247
|
oveq2d |
⊢ ( 𝑎 = dom 𝑐 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
249 |
192 248
|
elrnmpt1s |
⊢ ( ( dom 𝑐 ∈ ( 𝒫 𝐴 ∩ Fin ) ∧ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ∈ V ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) |
250 |
244 246 249
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) |
251 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑡 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) → 𝑡 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
252 |
251
|
breq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑡 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) → ( 𝑠 < 𝑡 ↔ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) ) |
253 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → 𝑠 ∈ ℝ* ) |
254 |
16
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd ) |
255 |
|
nfcv |
⊢ Ⅎ 𝑧 ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) |
256 |
|
nfcv |
⊢ Ⅎ 𝑧 Σg |
257 |
|
nfmpt1 |
⊢ Ⅎ 𝑧 ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) |
258 |
255 256 257
|
nfov |
⊢ Ⅎ 𝑧 ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) |
259 |
110 111 258
|
nfbr |
⊢ Ⅎ 𝑧 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) |
260 |
109 259
|
nfan |
⊢ Ⅎ 𝑧 ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
261 |
|
nfv |
⊢ Ⅎ 𝑧 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) |
262 |
260 261
|
nfan |
⊢ Ⅎ 𝑧 ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) |
263 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑧 ∈ 𝑐 ) → 𝜑 ) |
264 |
224 225
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
265 |
264
|
sselda |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑧 ∈ 𝑐 ) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
266 |
263 265 49
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑧 ∈ 𝑐 ) → 𝐹 ∈ ( 0 [,] +∞ ) ) |
267 |
266
|
ex |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( 𝑧 ∈ 𝑐 → 𝐹 ∈ ( 0 [,] +∞ ) ) ) |
268 |
262 267
|
ralrimi |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ∀ 𝑧 ∈ 𝑐 𝐹 ∈ ( 0 [,] +∞ ) ) |
269 |
15 254 241 268
|
gsummptcl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ∈ ( 0 [,] +∞ ) ) |
270 |
14 269
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ∈ ℝ* ) |
271 |
|
nfv |
⊢ Ⅎ 𝑗 ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
272 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑐 |
273 |
30
|
nfpw |
⊢ Ⅎ 𝑗 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) |
274 |
|
nfcv |
⊢ Ⅎ 𝑗 Fin |
275 |
273 274
|
nfin |
⊢ Ⅎ 𝑗 ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) |
276 |
272 275
|
nfel |
⊢ Ⅎ 𝑗 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) |
277 |
271 276
|
nfan |
⊢ Ⅎ 𝑗 ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) |
278 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → 𝜑 ) |
279 |
79 236
|
syl |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → dom 𝑐 ⊆ 𝐴 ) |
280 |
279
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → 𝑗 ∈ 𝐴 ) |
281 |
278 280 161
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → Σ* 𝑘 ∈ 𝐵 𝐶 ∈ ( 0 [,] +∞ ) ) |
282 |
281
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → Σ* 𝑘 ∈ 𝐵 𝐶 ∈ ( 0 [,] +∞ ) ) |
283 |
282
|
adantllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → Σ* 𝑘 ∈ 𝐵 𝐶 ∈ ( 0 [,] +∞ ) ) |
284 |
283
|
ex |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( 𝑗 ∈ dom 𝑐 → Σ* 𝑘 ∈ 𝐵 𝐶 ∈ ( 0 [,] +∞ ) ) ) |
285 |
277 284
|
ralrimi |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ∀ 𝑗 ∈ dom 𝑐 Σ* 𝑘 ∈ 𝐵 𝐶 ∈ ( 0 [,] +∞ ) ) |
286 |
15 254 243 285
|
gsummptcl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ∈ ( 0 [,] +∞ ) ) |
287 |
14 286
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ∈ ℝ* ) |
288 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
289 |
28 276
|
nfan |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) |
290 |
|
id |
⊢ ( 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
291 |
|
xpss |
⊢ ( { 𝑗 } × 𝐵 ) ⊆ ( V × V ) |
292 |
291
|
rgenw |
⊢ ∀ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ⊆ ( V × V ) |
293 |
|
iunss |
⊢ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ⊆ ( V × V ) ↔ ∀ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ⊆ ( V × V ) ) |
294 |
292 293
|
mpbir |
⊢ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ⊆ ( V × V ) |
295 |
294
|
a1i |
⊢ ( 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) → ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ⊆ ( V × V ) ) |
296 |
290 295
|
sstrd |
⊢ ( 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) → 𝑐 ⊆ ( V × V ) ) |
297 |
79 296
|
syl |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → 𝑐 ⊆ ( V × V ) ) |
298 |
|
df-rel |
⊢ ( Rel 𝑐 ↔ 𝑐 ⊆ ( V × V ) ) |
299 |
297 298
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → Rel 𝑐 ) |
300 |
1 289 15 2 299 19 17 50
|
gsummpt2d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) ↦ 𝐶 ) ) ) ) ) |
301 |
|
nfcv |
⊢ Ⅎ 𝑗 dom 𝑐 |
302 |
237
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → dom 𝑐 ∈ V ) |
303 |
278
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) ∧ 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) ) → 𝜑 ) |
304 |
280
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) ∧ 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) ) → 𝑗 ∈ 𝐴 ) |
305 |
79
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
306 |
|
imass1 |
⊢ ( 𝑐 ⊆ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) → ( 𝑐 “ { 𝑗 } ) ⊆ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) “ { 𝑗 } ) ) |
307 |
305 306
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → ( 𝑐 “ { 𝑗 } ) ⊆ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) “ { 𝑗 } ) ) |
308 |
3 4
|
iunsnima |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) “ { 𝑗 } ) = 𝐵 ) |
309 |
278 280 308
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) “ { 𝑗 } ) = 𝐵 ) |
310 |
307 309
|
sseqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → ( 𝑐 “ { 𝑗 } ) ⊆ 𝐵 ) |
311 |
310
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) ∧ 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) ) → 𝑘 ∈ 𝐵 ) |
312 |
303 304 311 5
|
syl12anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) ∧ 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
313 |
312
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → ∀ 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 ∈ ( 0 [,] +∞ ) ) |
314 |
|
imaexg |
⊢ ( 𝑐 ∈ V → ( 𝑐 “ { 𝑗 } ) ∈ V ) |
315 |
23 314
|
ax-mp |
⊢ ( 𝑐 “ { 𝑗 } ) ∈ V |
316 |
|
nfcv |
⊢ Ⅎ 𝑘 ( 𝑐 “ { 𝑗 } ) |
317 |
316
|
esumcl |
⊢ ( ( ( 𝑐 “ { 𝑗 } ) ∈ V ∧ ∀ 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 ∈ ( 0 [,] +∞ ) ) → Σ* 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 ∈ ( 0 [,] +∞ ) ) |
318 |
315 317
|
mpan |
⊢ ( ∀ 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 ∈ ( 0 [,] +∞ ) → Σ* 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 ∈ ( 0 [,] +∞ ) ) |
319 |
313 318
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → Σ* 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 ∈ ( 0 [,] +∞ ) ) |
320 |
|
nfv |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) |
321 |
278 280 4
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → 𝐵 ∈ 𝑊 ) |
322 |
278
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) ∧ 𝑘 ∈ 𝐵 ) → 𝜑 ) |
323 |
280
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) ∧ 𝑘 ∈ 𝐵 ) → 𝑗 ∈ 𝐴 ) |
324 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) ∧ 𝑘 ∈ 𝐵 ) → 𝑘 ∈ 𝐵 ) |
325 |
322 323 324 5
|
syl12anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) ∧ 𝑘 ∈ 𝐵 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
326 |
320 321 325 310
|
esummono |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → Σ* 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 ≤ Σ* 𝑘 ∈ 𝐵 𝐶 ) |
327 |
289 301 302 319 281 326
|
esumlef |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → Σ* 𝑗 ∈ dom 𝑐 Σ* 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 ≤ Σ* 𝑗 ∈ dom 𝑐 Σ* 𝑘 ∈ 𝐵 𝐶 ) |
328 |
19 242
|
syl |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → dom 𝑐 ∈ Fin ) |
329 |
289 301 328 319
|
esumgsum |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → Σ* 𝑗 ∈ dom 𝑐 Σ* 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 ) ) ) |
330 |
19
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → 𝑐 ∈ Fin ) |
331 |
|
imafi2 |
⊢ ( 𝑐 ∈ Fin → ( 𝑐 “ { 𝑗 } ) ∈ Fin ) |
332 |
330 331
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → ( 𝑐 “ { 𝑗 } ) ∈ Fin ) |
333 |
320 316 332 312
|
esumgsum |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑗 ∈ dom 𝑐 ) → Σ* 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) ↦ 𝐶 ) ) ) |
334 |
289 333
|
mpteq2da |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 ) = ( 𝑗 ∈ dom 𝑐 ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) ↦ 𝐶 ) ) ) ) |
335 |
334
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) ↦ 𝐶 ) ) ) ) ) |
336 |
329 335
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → Σ* 𝑗 ∈ dom 𝑐 Σ* 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) 𝐶 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) ↦ 𝐶 ) ) ) ) ) |
337 |
289 301 328 281
|
esumgsum |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → Σ* 𝑗 ∈ dom 𝑐 Σ* 𝑘 ∈ 𝐵 𝐶 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
338 |
327 336 337
|
3brtr3d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ ( 𝑐 “ { 𝑗 } ) ↦ 𝐶 ) ) ) ) ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
339 |
300 338
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
340 |
339
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
341 |
340
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ≤ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
342 |
253 270 287 288 341
|
xrltletrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ dom 𝑐 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
343 |
250 252 342
|
rspcedvd |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ∃ 𝑡 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) 𝑠 < 𝑡 ) |
344 |
343
|
adantllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ ℝ* ) ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ∧ 𝑠 < ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) → ∃ 𝑡 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) 𝑠 < 𝑡 ) |
345 |
218 221 222 344
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) ∧ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ) ∧ 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → ∃ 𝑡 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) 𝑠 < 𝑡 ) |
346 |
55 89
|
elrnmpti |
⊢ ( 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ↔ ∃ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
347 |
346
|
biimpi |
⊢ ( 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) → ∃ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
348 |
347
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) → ∃ 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) 𝑢 = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) |
349 |
209 345 348
|
r19.29af |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) ∧ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) ) ∧ 𝑠 < 𝑢 ) → ∃ 𝑡 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) 𝑠 < 𝑡 ) |
350 |
7 134
|
suplub |
⊢ ( 𝜑 → ( ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) ) |
351 |
350
|
imp |
⊢ ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) → ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ) |
352 |
|
breq2 |
⊢ ( 𝑡 = 𝑢 → ( 𝑠 < 𝑡 ↔ 𝑠 < 𝑢 ) ) |
353 |
352
|
cbvrexvw |
⊢ ( ∃ 𝑡 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑡 ↔ ∃ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑢 ) |
354 |
351 353
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) → ∃ 𝑢 ∈ ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) 𝑠 < 𝑢 ) |
355 |
349 354
|
r19.29a |
⊢ ( ( 𝜑 ∧ ( 𝑠 ∈ ℝ* ∧ 𝑠 < sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) ) → ∃ 𝑡 ∈ ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) 𝑠 < 𝑡 ) |
356 |
7 135 197 355
|
eqsupd |
⊢ ( 𝜑 → sup ( ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) , ℝ* , < ) = sup ( ran ( 𝑐 ∈ ( 𝒫 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑧 ∈ 𝑐 ↦ 𝐹 ) ) ) , ℝ* , < ) ) |
357 |
|
nfcv |
⊢ Ⅎ 𝑗 𝐴 |
358 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) |
359 |
28 357 3 161 358
|
esumval |
⊢ ( 𝜑 → Σ* 𝑗 ∈ 𝐴 Σ* 𝑘 ∈ 𝐵 𝐶 = sup ( ran ( 𝑎 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑗 ∈ 𝑎 ↦ Σ* 𝑘 ∈ 𝐵 𝐶 ) ) ) , ℝ* , < ) ) |
360 |
356 359 182
|
3eqtr4d |
⊢ ( 𝜑 → Σ* 𝑗 ∈ 𝐴 Σ* 𝑘 ∈ 𝐵 𝐶 = Σ* 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) 𝐹 ) |