Step |
Hyp |
Ref |
Expression |
1 |
|
meaiuninclem.m |
⊢ ( 𝜑 → 𝑀 ∈ Meas ) |
2 |
|
meaiuninclem.n |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
3 |
|
meaiuninclem.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑁 ) |
4 |
|
meaiuninclem.e |
⊢ ( 𝜑 → 𝐸 : 𝑍 ⟶ dom 𝑀 ) |
5 |
|
meaiuninclem.i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐸 ‘ 𝑛 ) ⊆ ( 𝐸 ‘ ( 𝑛 + 1 ) ) ) |
6 |
|
meaiuninclem.b |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) |
7 |
|
meaiuninclem.s |
⊢ 𝑆 = ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) |
8 |
|
meaiuninclem.f |
⊢ 𝐹 = ( 𝑛 ∈ 𝑍 ↦ ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ) |
9 |
|
0xr |
⊢ 0 ∈ ℝ* |
10 |
9
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 0 ∈ ℝ* ) |
11 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
12 |
11
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → +∞ ∈ ℝ* ) |
13 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑀 ∈ Meas ) |
14 |
|
eqid |
⊢ dom 𝑀 = dom 𝑀 |
15 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐸 ‘ 𝑛 ) ∈ dom 𝑀 ) |
16 |
13 14 15
|
meaxrcl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ∈ ℝ* ) |
17 |
13 15
|
meage0 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 0 ≤ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) |
18 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) |
19 |
|
simp1 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ∧ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ) |
20 |
|
simp2 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ∧ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → 𝑥 ∈ ℝ ) |
21 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ∧ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) |
22 |
19
|
simprd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ∧ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → 𝑛 ∈ 𝑍 ) |
23 |
|
rspa |
⊢ ( ( ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) |
24 |
21 22 23
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ∧ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) |
25 |
16
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ∧ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ∈ ℝ* ) |
26 |
|
rexr |
⊢ ( 𝑥 ∈ ℝ → 𝑥 ∈ ℝ* ) |
27 |
26
|
3ad2ant2 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ∧ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → 𝑥 ∈ ℝ* ) |
28 |
11
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ∧ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → +∞ ∈ ℝ* ) |
29 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ∧ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) |
30 |
|
ltpnf |
⊢ ( 𝑥 ∈ ℝ → 𝑥 < +∞ ) |
31 |
30
|
3ad2ant2 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ∧ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → 𝑥 < +∞ ) |
32 |
25 27 28 29 31
|
xrlelttrd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ∧ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) < +∞ ) |
33 |
19 20 24 32
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ∧ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) < +∞ ) |
34 |
33
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑥 ∈ ℝ → ( ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) < +∞ ) ) ) |
35 |
34
|
rexlimdv |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ∃ 𝑥 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) < +∞ ) ) |
36 |
18 35
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) < +∞ ) |
37 |
10 12 16 17 36
|
elicod |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ∈ ( 0 [,) +∞ ) ) |
38 |
37 7
|
fmptd |
⊢ ( 𝜑 → 𝑆 : 𝑍 ⟶ ( 0 [,) +∞ ) ) |
39 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
40 |
39
|
a1i |
⊢ ( 𝜑 → ( 0 [,) +∞ ) ⊆ ℝ ) |
41 |
38 40
|
fssd |
⊢ ( 𝜑 → 𝑆 : 𝑍 ⟶ ℝ ) |
42 |
3
|
peano2uzs |
⊢ ( 𝑛 ∈ 𝑍 → ( 𝑛 + 1 ) ∈ 𝑍 ) |
43 |
42
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑛 + 1 ) ∈ 𝑍 ) |
44 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ ( 𝑛 + 1 ) ∈ 𝑍 ) → ( 𝐸 ‘ ( 𝑛 + 1 ) ) ∈ dom 𝑀 ) |
45 |
43 44
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐸 ‘ ( 𝑛 + 1 ) ) ∈ dom 𝑀 ) |
46 |
13 14 15 45 5
|
meassle |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ ( 𝑀 ‘ ( 𝐸 ‘ ( 𝑛 + 1 ) ) ) ) |
47 |
7
|
a1i |
⊢ ( 𝜑 → 𝑆 = ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) |
48 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ∈ V ) |
49 |
47 48
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑆 ‘ 𝑛 ) = ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) |
50 |
|
2fveq3 |
⊢ ( 𝑛 = 𝑚 → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) = ( 𝑀 ‘ ( 𝐸 ‘ 𝑚 ) ) ) |
51 |
50
|
cbvmptv |
⊢ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) = ( 𝑚 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑚 ) ) ) |
52 |
7 51
|
eqtri |
⊢ 𝑆 = ( 𝑚 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑚 ) ) ) |
53 |
|
2fveq3 |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑚 ) ) = ( 𝑀 ‘ ( 𝐸 ‘ ( 𝑛 + 1 ) ) ) ) |
54 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ ( 𝑛 + 1 ) ) ) ∈ V ) |
55 |
52 53 43 54
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑆 ‘ ( 𝑛 + 1 ) ) = ( 𝑀 ‘ ( 𝐸 ‘ ( 𝑛 + 1 ) ) ) ) |
56 |
49 55
|
breq12d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑆 ‘ 𝑛 ) ≤ ( 𝑆 ‘ ( 𝑛 + 1 ) ) ↔ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ ( 𝑀 ‘ ( 𝐸 ‘ ( 𝑛 + 1 ) ) ) ) ) |
57 |
46 56
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑆 ‘ 𝑛 ) ≤ ( 𝑆 ‘ ( 𝑛 + 1 ) ) ) |
58 |
49
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) = ( 𝑆 ‘ 𝑛 ) ) |
59 |
58
|
breq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ↔ ( 𝑆 ‘ 𝑛 ) ≤ 𝑥 ) ) |
60 |
59
|
ralbidva |
⊢ ( 𝜑 → ( ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ↔ ∀ 𝑛 ∈ 𝑍 ( 𝑆 ‘ 𝑛 ) ≤ 𝑥 ) ) |
61 |
60
|
biimpd |
⊢ ( 𝜑 → ( ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 → ∀ 𝑛 ∈ 𝑍 ( 𝑆 ‘ 𝑛 ) ≤ 𝑥 ) ) |
62 |
61
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 → ∀ 𝑛 ∈ 𝑍 ( 𝑆 ‘ 𝑛 ) ≤ 𝑥 ) ) |
63 |
62
|
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 → ∃ 𝑥 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( 𝑆 ‘ 𝑛 ) ≤ 𝑥 ) ) |
64 |
6 63
|
mpd |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( 𝑆 ‘ 𝑛 ) ≤ 𝑥 ) |
65 |
3 2 41 57 64
|
climsup |
⊢ ( 𝜑 → 𝑆 ⇝ sup ( ran 𝑆 , ℝ , < ) ) |
66 |
|
nfv |
⊢ Ⅎ 𝑛 𝜑 |
67 |
|
nfv |
⊢ Ⅎ 𝑥 𝜑 |
68 |
|
id |
⊢ ( 𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍 ) |
69 |
|
fvex |
⊢ ( 𝐸 ‘ 𝑛 ) ∈ V |
70 |
69
|
difexi |
⊢ ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ V |
71 |
70
|
a1i |
⊢ ( 𝑛 ∈ 𝑍 → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ V ) |
72 |
8
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ V ) → ( 𝐹 ‘ 𝑛 ) = ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ) |
73 |
68 71 72
|
syl2anc |
⊢ ( 𝑛 ∈ 𝑍 → ( 𝐹 ‘ 𝑛 ) = ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ) |
74 |
73
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑛 ) = ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ) |
75 |
1 14
|
dmmeasal |
⊢ ( 𝜑 → dom 𝑀 ∈ SAlg ) |
76 |
75
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom 𝑀 ∈ SAlg ) |
77 |
|
fzoct |
⊢ ( 𝑁 ..^ 𝑛 ) ≼ ω |
78 |
77
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑁 ..^ 𝑛 ) ≼ ω ) |
79 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ) → 𝐸 : 𝑍 ⟶ dom 𝑀 ) |
80 |
|
fzossuz |
⊢ ( 𝑁 ..^ 𝑛 ) ⊆ ( ℤ≥ ‘ 𝑁 ) |
81 |
3
|
eqcomi |
⊢ ( ℤ≥ ‘ 𝑁 ) = 𝑍 |
82 |
80 81
|
sseqtri |
⊢ ( 𝑁 ..^ 𝑛 ) ⊆ 𝑍 |
83 |
82
|
sseli |
⊢ ( 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) → 𝑖 ∈ 𝑍 ) |
84 |
83
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ) → 𝑖 ∈ 𝑍 ) |
85 |
79 84
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ) → ( 𝐸 ‘ 𝑖 ) ∈ dom 𝑀 ) |
86 |
85
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ) → ( 𝐸 ‘ 𝑖 ) ∈ dom 𝑀 ) |
87 |
76 78 86
|
saliuncl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ∈ dom 𝑀 ) |
88 |
|
saldifcl2 |
⊢ ( ( dom 𝑀 ∈ SAlg ∧ ( 𝐸 ‘ 𝑛 ) ∈ dom 𝑀 ∧ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ∈ dom 𝑀 ) → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ dom 𝑀 ) |
89 |
76 15 87 88
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ dom 𝑀 ) |
90 |
74 89
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑛 ) ∈ dom 𝑀 ) |
91 |
13 14 90
|
meaxrcl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ* ) |
92 |
13 90
|
meage0 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 0 ≤ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
93 |
|
difssd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ⊆ ( 𝐸 ‘ 𝑛 ) ) |
94 |
74 93
|
eqsstrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐸 ‘ 𝑛 ) ) |
95 |
13 14 90 15 94
|
meassle |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) |
96 |
91 16 12 95 36
|
xrlelttrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) < +∞ ) |
97 |
10 12 91 92 96
|
elicod |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ( 0 [,) +∞ ) ) |
98 |
|
2fveq3 |
⊢ ( 𝑛 = 𝑖 → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) = ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ) |
99 |
98
|
breq1d |
⊢ ( 𝑛 = 𝑖 → ( ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ↔ ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ≤ 𝑥 ) ) |
100 |
99
|
cbvralvw |
⊢ ( ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ↔ ∀ 𝑖 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ≤ 𝑥 ) |
101 |
100
|
biimpi |
⊢ ( ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 → ∀ 𝑖 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ≤ 𝑥 ) |
102 |
101
|
adantl |
⊢ ( ( 𝜑 ∧ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → ∀ 𝑖 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ≤ 𝑥 ) |
103 |
|
eleq1w |
⊢ ( 𝑛 = 𝑖 → ( 𝑛 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍 ) ) |
104 |
103
|
anbi2d |
⊢ ( 𝑛 = 𝑖 → ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ↔ ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) ) ) |
105 |
|
oveq2 |
⊢ ( 𝑛 = 𝑖 → ( 𝑁 ... 𝑛 ) = ( 𝑁 ... 𝑖 ) ) |
106 |
105
|
sumeq1d |
⊢ ( 𝑛 = 𝑖 → Σ 𝑚 ∈ ( 𝑁 ... 𝑛 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) = Σ 𝑚 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) ) |
107 |
98 106
|
eqeq12d |
⊢ ( 𝑛 = 𝑖 → ( ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) = Σ 𝑚 ∈ ( 𝑁 ... 𝑛 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) ↔ ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) = Σ 𝑚 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) |
108 |
104 107
|
imbi12d |
⊢ ( 𝑛 = 𝑖 → ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) = Σ 𝑚 ∈ ( 𝑁 ... 𝑛 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) = Σ 𝑚 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) |
109 |
|
eleq1w |
⊢ ( 𝑚 = 𝑛 → ( 𝑚 ∈ 𝑍 ↔ 𝑛 ∈ 𝑍 ) ) |
110 |
109
|
anbi2d |
⊢ ( 𝑚 = 𝑛 → ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) ↔ ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ) ) |
111 |
|
oveq2 |
⊢ ( 𝑚 = 𝑛 → ( 𝑁 ... 𝑚 ) = ( 𝑁 ... 𝑛 ) ) |
112 |
111
|
iuneq1d |
⊢ ( 𝑚 = 𝑛 → ∪ 𝑖 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑖 ) = ∪ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐹 ‘ 𝑖 ) ) |
113 |
111
|
iuneq1d |
⊢ ( 𝑚 = 𝑛 → ∪ 𝑖 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑖 ) = ∪ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) |
114 |
112 113
|
eqeq12d |
⊢ ( 𝑚 = 𝑛 → ( ∪ 𝑖 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑖 ) = ∪ 𝑖 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑖 ) ↔ ∪ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐹 ‘ 𝑖 ) = ∪ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ) |
115 |
110 114
|
imbi12d |
⊢ ( 𝑚 = 𝑛 → ( ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) → ∪ 𝑖 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑖 ) = ∪ 𝑖 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑖 ) ) ↔ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ∪ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐹 ‘ 𝑖 ) = ∪ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ) ) |
116 |
|
fveq2 |
⊢ ( 𝑖 = 𝑛 → ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑛 ) ) |
117 |
116
|
cbviunv |
⊢ ∪ 𝑖 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑖 ) = ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑛 ) |
118 |
117
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) → ∪ 𝑖 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑖 ) = ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑛 ) ) |
119 |
66 3 4 8
|
iundjiun |
⊢ ( 𝜑 → ( ( ∀ 𝑚 ∈ 𝑍 ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑛 ) ∧ ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ∧ Disj 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) ) |
120 |
119
|
simplld |
⊢ ( 𝜑 → ∀ 𝑚 ∈ 𝑍 ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑛 ) ) |
121 |
120
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) → ∀ 𝑚 ∈ 𝑍 ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑛 ) ) |
122 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) → 𝑚 ∈ 𝑍 ) |
123 |
|
rspa |
⊢ ( ( ∀ 𝑚 ∈ 𝑍 ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑛 ) ∧ 𝑚 ∈ 𝑍 ) → ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑛 ) ) |
124 |
121 122 123
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) → ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑛 ) ) |
125 |
|
fveq2 |
⊢ ( 𝑛 = 𝑖 → ( 𝐸 ‘ 𝑛 ) = ( 𝐸 ‘ 𝑖 ) ) |
126 |
125
|
cbviunv |
⊢ ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑛 ) = ∪ 𝑖 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑖 ) |
127 |
126
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) → ∪ 𝑛 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑛 ) = ∪ 𝑖 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑖 ) ) |
128 |
118 124 127
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) → ∪ 𝑖 ∈ ( 𝑁 ... 𝑚 ) ( 𝐹 ‘ 𝑖 ) = ∪ 𝑖 ∈ ( 𝑁 ... 𝑚 ) ( 𝐸 ‘ 𝑖 ) ) |
129 |
115 128
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ∪ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐹 ‘ 𝑖 ) = ∪ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) |
130 |
68 3
|
eleqtrdi |
⊢ ( 𝑛 ∈ 𝑍 → 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
131 |
130
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
132 |
|
fvoveq1 |
⊢ ( 𝑛 = 𝑖 → ( 𝐸 ‘ ( 𝑛 + 1 ) ) = ( 𝐸 ‘ ( 𝑖 + 1 ) ) ) |
133 |
125 132
|
sseq12d |
⊢ ( 𝑛 = 𝑖 → ( ( 𝐸 ‘ 𝑛 ) ⊆ ( 𝐸 ‘ ( 𝑛 + 1 ) ) ↔ ( 𝐸 ‘ 𝑖 ) ⊆ ( 𝐸 ‘ ( 𝑖 + 1 ) ) ) ) |
134 |
104 133
|
imbi12d |
⊢ ( 𝑛 = 𝑖 → ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐸 ‘ 𝑛 ) ⊆ ( 𝐸 ‘ ( 𝑛 + 1 ) ) ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ( 𝐸 ‘ 𝑖 ) ⊆ ( 𝐸 ‘ ( 𝑖 + 1 ) ) ) ) ) |
135 |
134 5
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ( 𝐸 ‘ 𝑖 ) ⊆ ( 𝐸 ‘ ( 𝑖 + 1 ) ) ) |
136 |
84 135
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ) → ( 𝐸 ‘ 𝑖 ) ⊆ ( 𝐸 ‘ ( 𝑖 + 1 ) ) ) |
137 |
136
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ) → ( 𝐸 ‘ 𝑖 ) ⊆ ( 𝐸 ‘ ( 𝑖 + 1 ) ) ) |
138 |
131 137
|
iunincfi |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ∪ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐸 ‘ 𝑖 ) = ( 𝐸 ‘ 𝑛 ) ) |
139 |
129 138
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐸 ‘ 𝑛 ) = ∪ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐹 ‘ 𝑖 ) ) |
140 |
139
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) = ( 𝑀 ‘ ∪ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐹 ‘ 𝑖 ) ) ) |
141 |
|
nfv |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) |
142 |
|
elfzuz |
⊢ ( 𝑖 ∈ ( 𝑁 ... 𝑛 ) → 𝑖 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
143 |
142 81
|
eleqtrdi |
⊢ ( 𝑖 ∈ ( 𝑁 ... 𝑛 ) → 𝑖 ∈ 𝑍 ) |
144 |
143
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ) → 𝑖 ∈ 𝑍 ) |
145 |
|
fveq2 |
⊢ ( 𝑛 = 𝑖 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑖 ) ) |
146 |
145
|
eleq1d |
⊢ ( 𝑛 = 𝑖 → ( ( 𝐹 ‘ 𝑛 ) ∈ dom 𝑀 ↔ ( 𝐹 ‘ 𝑖 ) ∈ dom 𝑀 ) ) |
147 |
104 146
|
imbi12d |
⊢ ( 𝑛 = 𝑖 → ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑛 ) ∈ dom 𝑀 ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑖 ) ∈ dom 𝑀 ) ) ) |
148 |
147 90
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑖 ) ∈ dom 𝑀 ) |
149 |
144 148
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ) → ( 𝐹 ‘ 𝑖 ) ∈ dom 𝑀 ) |
150 |
149
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ) → ( 𝐹 ‘ 𝑖 ) ∈ dom 𝑀 ) |
151 |
|
fzct |
⊢ ( 𝑁 ... 𝑛 ) ≼ ω |
152 |
151
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑁 ... 𝑛 ) ≼ ω ) |
153 |
144
|
ssd |
⊢ ( 𝜑 → ( 𝑁 ... 𝑛 ) ⊆ 𝑍 ) |
154 |
119
|
simprd |
⊢ ( 𝜑 → Disj 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) |
155 |
145
|
cbvdisjv |
⊢ ( Disj 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ↔ Disj 𝑖 ∈ 𝑍 ( 𝐹 ‘ 𝑖 ) ) |
156 |
154 155
|
sylib |
⊢ ( 𝜑 → Disj 𝑖 ∈ 𝑍 ( 𝐹 ‘ 𝑖 ) ) |
157 |
|
disjss1 |
⊢ ( ( 𝑁 ... 𝑛 ) ⊆ 𝑍 → ( Disj 𝑖 ∈ 𝑍 ( 𝐹 ‘ 𝑖 ) → Disj 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐹 ‘ 𝑖 ) ) ) |
158 |
153 156 157
|
sylc |
⊢ ( 𝜑 → Disj 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐹 ‘ 𝑖 ) ) |
159 |
158
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → Disj 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐹 ‘ 𝑖 ) ) |
160 |
141 13 14 150 152 159
|
meadjiun |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ∪ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝐹 ‘ 𝑖 ) ) = ( Σ^ ‘ ( 𝑖 ∈ ( 𝑁 ... 𝑛 ) ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) |
161 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑁 ... 𝑛 ) ∈ Fin ) |
162 |
|
2fveq3 |
⊢ ( 𝑛 = 𝑖 → ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) = ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) ) |
163 |
162
|
eleq1d |
⊢ ( 𝑛 = 𝑖 → ( ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ( 0 [,) +∞ ) ↔ ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ ( 0 [,) +∞ ) ) ) |
164 |
104 163
|
imbi12d |
⊢ ( 𝑛 = 𝑖 → ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ( 0 [,) +∞ ) ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ ( 0 [,) +∞ ) ) ) ) |
165 |
164 97
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ ( 0 [,) +∞ ) ) |
166 |
144 165
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ ( 0 [,) +∞ ) ) |
167 |
166
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ ( 0 [,) +∞ ) ) |
168 |
161 167
|
sge0fsummpt |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( Σ^ ‘ ( 𝑖 ∈ ( 𝑁 ... 𝑛 ) ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) = Σ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) ) |
169 |
|
2fveq3 |
⊢ ( 𝑖 = 𝑚 → ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) = ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) ) |
170 |
169
|
cbvsumv |
⊢ Σ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) = Σ 𝑚 ∈ ( 𝑁 ... 𝑛 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) |
171 |
170
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → Σ 𝑖 ∈ ( 𝑁 ... 𝑛 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) = Σ 𝑚 ∈ ( 𝑁 ... 𝑛 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) ) |
172 |
168 171
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( Σ^ ‘ ( 𝑖 ∈ ( 𝑁 ... 𝑛 ) ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) = Σ 𝑚 ∈ ( 𝑁 ... 𝑛 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) ) |
173 |
140 160 172
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) = Σ 𝑚 ∈ ( 𝑁 ... 𝑛 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) ) |
174 |
108 173
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) = Σ 𝑚 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) ) |
175 |
|
2fveq3 |
⊢ ( 𝑚 = 𝑛 → ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) = ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
176 |
175
|
cbvsumv |
⊢ Σ 𝑚 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) = Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) |
177 |
176
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → Σ 𝑚 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑚 ) ) = Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
178 |
174 177
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) = Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
179 |
178
|
breq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑍 ) → ( ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ≤ 𝑥 ↔ Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ 𝑥 ) ) |
180 |
179
|
ralbidva |
⊢ ( 𝜑 → ( ∀ 𝑖 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ≤ 𝑥 ↔ ∀ 𝑖 ∈ 𝑍 Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ 𝑥 ) ) |
181 |
180
|
biimpd |
⊢ ( 𝜑 → ( ∀ 𝑖 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ≤ 𝑥 → ∀ 𝑖 ∈ 𝑍 Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ 𝑥 ) ) |
182 |
181
|
imp |
⊢ ( ( 𝜑 ∧ ∀ 𝑖 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ≤ 𝑥 ) → ∀ 𝑖 ∈ 𝑍 Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ 𝑥 ) |
183 |
102 182
|
syldan |
⊢ ( ( 𝜑 ∧ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 ) → ∀ 𝑖 ∈ 𝑍 Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ 𝑥 ) |
184 |
183
|
ex |
⊢ ( 𝜑 → ( ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 → ∀ 𝑖 ∈ 𝑍 Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ 𝑥 ) ) |
185 |
184
|
reximdv |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∀ 𝑛 ∈ 𝑍 ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ≤ 𝑥 → ∃ 𝑥 ∈ ℝ ∀ 𝑖 ∈ 𝑍 Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ 𝑥 ) ) |
186 |
6 185
|
mpd |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑖 ∈ 𝑍 Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ 𝑥 ) |
187 |
66 67 2 3 97 186
|
sge0reuzb |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) = sup ( ran ( 𝑖 ∈ 𝑍 ↦ Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) , ℝ , < ) ) |
188 |
98
|
cbvmptv |
⊢ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) = ( 𝑖 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ) |
189 |
7 188
|
eqtri |
⊢ 𝑆 = ( 𝑖 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ) |
190 |
189
|
a1i |
⊢ ( 𝜑 → 𝑆 = ( 𝑖 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ) ) |
191 |
178
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑖 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑖 ) ) ) = ( 𝑖 ∈ 𝑍 ↦ Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
192 |
190 191
|
eqtrd |
⊢ ( 𝜑 → 𝑆 = ( 𝑖 ∈ 𝑍 ↦ Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
193 |
192
|
rneqd |
⊢ ( 𝜑 → ran 𝑆 = ran ( 𝑖 ∈ 𝑍 ↦ Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
194 |
193
|
supeq1d |
⊢ ( 𝜑 → sup ( ran 𝑆 , ℝ , < ) = sup ( ran ( 𝑖 ∈ 𝑍 ↦ Σ 𝑛 ∈ ( 𝑁 ... 𝑖 ) ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) , ℝ , < ) ) |
195 |
187 194
|
eqtr4d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) = sup ( ran 𝑆 , ℝ , < ) ) |
196 |
195
|
eqcomd |
⊢ ( 𝜑 → sup ( ran 𝑆 , ℝ , < ) = ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
197 |
3
|
uzct |
⊢ 𝑍 ≼ ω |
198 |
197
|
a1i |
⊢ ( 𝜑 → 𝑍 ≼ ω ) |
199 |
66 1 14 90 198 154
|
meadjiun |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) = ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
200 |
199
|
eqcomd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) = ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) ) |
201 |
119
|
simplrd |
⊢ ( 𝜑 → ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) |
202 |
201
|
fveq2d |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) = ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ) |
203 |
196 200 202
|
3eqtrd |
⊢ ( 𝜑 → sup ( ran 𝑆 , ℝ , < ) = ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ) |
204 |
65 203
|
breqtrd |
⊢ ( 𝜑 → 𝑆 ⇝ ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ) |