Step |
Hyp |
Ref |
Expression |
1 |
|
smflimsuplem5.a |
⊢ Ⅎ 𝑛 𝜑 |
2 |
|
smflimsuplem5.b |
⊢ Ⅎ 𝑚 𝜑 |
3 |
|
smflimsuplem5.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
4 |
|
smflimsuplem5.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
5 |
|
smflimsuplem5.s |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |
6 |
|
smflimsuplem5.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( SMblFn ‘ 𝑆 ) ) |
7 |
|
smflimsuplem5.e |
⊢ 𝐸 = ( 𝑛 ∈ 𝑍 ↦ { 𝑥 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } ) |
8 |
|
smflimsuplem5.h |
⊢ 𝐻 = ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ ( 𝐸 ‘ 𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ) |
9 |
|
smflimsuplem5.r |
⊢ ( 𝜑 → ( lim sup ‘ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ ) |
10 |
|
smflimsuplem5.n |
⊢ ( 𝜑 → 𝑁 ∈ 𝑍 ) |
11 |
|
smflimsuplem5.x |
⊢ ( 𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) dom ( 𝐹 ‘ 𝑚 ) ) |
12 |
4
|
eleq2i |
⊢ ( 𝑁 ∈ 𝑍 ↔ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
13 |
12
|
biimpi |
⊢ ( 𝑁 ∈ 𝑍 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
14 |
|
uzss |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
15 |
13 14
|
syl |
⊢ ( 𝑁 ∈ 𝑍 → ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
16 |
15 4
|
sseqtrrdi |
⊢ ( 𝑁 ∈ 𝑍 → ( ℤ≥ ‘ 𝑁 ) ⊆ 𝑍 ) |
17 |
10 16
|
syl |
⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) ⊆ 𝑍 ) |
18 |
17
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑛 ∈ 𝑍 ) |
19 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑍 |
20 |
|
nfrab1 |
⊢ Ⅎ 𝑥 { 𝑥 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } |
21 |
19 20
|
nfmpt |
⊢ Ⅎ 𝑥 ( 𝑛 ∈ 𝑍 ↦ { 𝑥 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } ) |
22 |
7 21
|
nfcxfr |
⊢ Ⅎ 𝑥 𝐸 |
23 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑛 |
24 |
22 23
|
nffv |
⊢ Ⅎ 𝑥 ( 𝐸 ‘ 𝑛 ) |
25 |
|
fvex |
⊢ ( 𝐸 ‘ 𝑛 ) ∈ V |
26 |
24 25
|
mptexf |
⊢ ( 𝑥 ∈ ( 𝐸 ‘ 𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ∈ V |
27 |
26
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( 𝑥 ∈ ( 𝐸 ‘ 𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ∈ V ) |
28 |
8
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ ( 𝑥 ∈ ( 𝐸 ‘ 𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ∈ V ) → ( 𝐻 ‘ 𝑛 ) = ( 𝑥 ∈ ( 𝐸 ‘ 𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ) |
29 |
18 27 28
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( 𝐻 ‘ 𝑛 ) = ( 𝑥 ∈ ( 𝐸 ‘ 𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ) |
30 |
29
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( ( 𝐻 ‘ 𝑛 ) ‘ 𝑋 ) = ( ( 𝑥 ∈ ( 𝐸 ‘ 𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ‘ 𝑋 ) ) |
31 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 𝐸 ‘ 𝑛 ) |
32 |
|
nfcv |
⊢ Ⅎ 𝑦 sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) |
33 |
|
nfcv |
⊢ Ⅎ 𝑥 sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) |
34 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) = ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) |
35 |
34
|
mpteq2dv |
⊢ ( 𝑥 = 𝑦 → ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) = ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) ) |
36 |
35
|
rneqd |
⊢ ( 𝑥 = 𝑦 → ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) = ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) ) |
37 |
36
|
supeq1d |
⊢ ( 𝑥 = 𝑦 → sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) = sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) ) |
38 |
24 31 32 33 37
|
cbvmptf |
⊢ ( 𝑥 ∈ ( 𝐸 ‘ 𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) = ( 𝑦 ∈ ( 𝐸 ‘ 𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) ) |
39 |
|
simpl |
⊢ ( ( 𝑦 = 𝑋 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑦 = 𝑋 ) |
40 |
39
|
fveq2d |
⊢ ( ( 𝑦 = 𝑋 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) = ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) |
41 |
40
|
mpteq2dva |
⊢ ( 𝑦 = 𝑋 → ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) = ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) |
42 |
41
|
rneqd |
⊢ ( 𝑦 = 𝑋 → ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) = ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) |
43 |
42
|
supeq1d |
⊢ ( 𝑦 = 𝑋 → sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) = sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ) |
44 |
43
|
eleq1d |
⊢ ( 𝑦 = 𝑋 → ( sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) ∈ ℝ ↔ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ∈ ℝ ) ) |
45 |
|
uzss |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) → ( ℤ≥ ‘ 𝑛 ) ⊆ ( ℤ≥ ‘ 𝑁 ) ) |
46 |
|
iinss1 |
⊢ ( ( ℤ≥ ‘ 𝑛 ) ⊆ ( ℤ≥ ‘ 𝑁 ) → ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) dom ( 𝐹 ‘ 𝑚 ) ⊆ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) |
47 |
45 46
|
syl |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) → ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) dom ( 𝐹 ‘ 𝑚 ) ⊆ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) |
48 |
47
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) dom ( 𝐹 ‘ 𝑚 ) ⊆ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) |
49 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) dom ( 𝐹 ‘ 𝑚 ) ) |
50 |
48 49
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) |
51 |
|
nfv |
⊢ Ⅎ 𝑚 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) |
52 |
2 51
|
nfan |
⊢ Ⅎ 𝑚 ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
53 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑛 ) = ( ℤ≥ ‘ 𝑛 ) |
54 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝜑 ) |
55 |
45
|
sselda |
⊢ ( ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
56 |
55
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
57 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑆 ∈ SAlg ) |
58 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝜑 ) |
59 |
17
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑚 ∈ 𝑍 ) |
60 |
6
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑚 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
61 |
58 59 60
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( 𝐹 ‘ 𝑚 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
62 |
|
eqid |
⊢ dom ( 𝐹 ‘ 𝑚 ) = dom ( 𝐹 ‘ 𝑚 ) |
63 |
57 61 62
|
smff |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( 𝐹 ‘ 𝑚 ) : dom ( 𝐹 ‘ 𝑚 ) ⟶ ℝ ) |
64 |
|
eliin |
⊢ ( 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) dom ( 𝐹 ‘ 𝑚 ) → ( 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) dom ( 𝐹 ‘ 𝑚 ) ↔ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) 𝑋 ∈ dom ( 𝐹 ‘ 𝑚 ) ) ) |
65 |
11 64
|
syl |
⊢ ( 𝜑 → ( 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) dom ( 𝐹 ‘ 𝑚 ) ↔ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) 𝑋 ∈ dom ( 𝐹 ‘ 𝑚 ) ) ) |
66 |
11 65
|
mpbid |
⊢ ( 𝜑 → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) 𝑋 ∈ dom ( 𝐹 ‘ 𝑚 ) ) |
67 |
66
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) 𝑋 ∈ dom ( 𝐹 ‘ 𝑚 ) ) |
68 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
69 |
|
rspa |
⊢ ( ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) 𝑋 ∈ dom ( 𝐹 ‘ 𝑚 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑋 ∈ dom ( 𝐹 ‘ 𝑚 ) ) |
70 |
67 68 69
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑋 ∈ dom ( 𝐹 ‘ 𝑚 ) ) |
71 |
63 70
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ∈ ℝ ) |
72 |
54 56 71
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ∈ ℝ ) |
73 |
|
eluzelz |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) → 𝑛 ∈ ℤ ) |
74 |
73
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑛 ∈ ℤ ) |
75 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑀 ∈ ℤ ) |
76 |
|
fvex |
⊢ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ∈ V |
77 |
76
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) ∧ 𝑚 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ∈ V ) |
78 |
52 74 75 53 4 72 77
|
limsupequzmpt |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( lim sup ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) = ( lim sup ‘ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ) |
79 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( lim sup ‘ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ ) |
80 |
78 79
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( lim sup ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ ) |
81 |
80
|
renepnfd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( lim sup ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ≠ +∞ ) |
82 |
52 53 72 81
|
limsupubuzmpt |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ≤ 𝑦 ) |
83 |
|
uzid2 |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
84 |
83
|
ne0d |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) → ( ℤ≥ ‘ 𝑛 ) ≠ ∅ ) |
85 |
84
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( ℤ≥ ‘ 𝑛 ) ≠ ∅ ) |
86 |
52 85 72
|
supxrre3rnmpt |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ∈ ℝ ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ≤ 𝑦 ) ) |
87 |
82 86
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ∈ ℝ ) |
88 |
44 50 87
|
elrabd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑋 ∈ { 𝑦 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) ∈ ℝ } ) |
89 |
|
simpl |
⊢ ( ( 𝑦 = 𝑥 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑦 = 𝑥 ) |
90 |
89
|
fveq2d |
⊢ ( ( 𝑦 = 𝑥 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) = ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) |
91 |
90
|
mpteq2dva |
⊢ ( 𝑦 = 𝑥 → ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) = ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ) |
92 |
91
|
rneqd |
⊢ ( 𝑦 = 𝑥 → ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) = ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ) |
93 |
92
|
supeq1d |
⊢ ( 𝑦 = 𝑥 → sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) = sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) |
94 |
93
|
eleq1d |
⊢ ( 𝑦 = 𝑥 → ( sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) ∈ ℝ ↔ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ ) ) |
95 |
94
|
cbvrabv |
⊢ { 𝑦 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) ∈ ℝ } = { 𝑥 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } |
96 |
88 95
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑋 ∈ { 𝑥 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } ) |
97 |
|
eqid |
⊢ { 𝑥 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } = { 𝑥 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } |
98 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑚 ) ∈ V |
99 |
98
|
dmex |
⊢ dom ( 𝐹 ‘ 𝑚 ) ∈ V |
100 |
99
|
rgenw |
⊢ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V |
101 |
100
|
a1i |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V ) |
102 |
84 101
|
iinexd |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) → ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V ) |
103 |
102
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V ) |
104 |
97 103
|
rabexd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → { 𝑥 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } ∈ V ) |
105 |
7
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ { 𝑥 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } ∈ V ) → ( 𝐸 ‘ 𝑛 ) = { 𝑥 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } ) |
106 |
18 104 105
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( 𝐸 ‘ 𝑛 ) = { 𝑥 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } ) |
107 |
96 106
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑋 ∈ ( 𝐸 ‘ 𝑛 ) ) |
108 |
38 43 107 87
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( ( 𝑥 ∈ ( 𝐸 ‘ 𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ‘ 𝑋 ) = sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ) |
109 |
30 108
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( ( 𝐻 ‘ 𝑛 ) ‘ 𝑋 ) = sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ) |
110 |
1 109
|
mpteq2da |
⊢ ( 𝜑 → ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ↦ ( ( 𝐻 ‘ 𝑛 ) ‘ 𝑋 ) ) = ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ) ) |
111 |
4
|
eluzelz2 |
⊢ ( 𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ ) |
112 |
10 111
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
113 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑁 ) = ( ℤ≥ ‘ 𝑁 ) |
114 |
76
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ∈ V ) |
115 |
76
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ∈ V ) |
116 |
2 112 3 113 4 114 115
|
limsupequzmpt |
⊢ ( 𝜑 → ( lim sup ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) = ( lim sup ‘ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ) |
117 |
116 9
|
eqeltrd |
⊢ ( 𝜑 → ( lim sup ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ ) |
118 |
2 112 113 71 117
|
supcnvlimsupmpt |
⊢ ( 𝜑 → ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ) ⇝ ( lim sup ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ) |
119 |
110 118
|
eqbrtrd |
⊢ ( 𝜑 → ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ↦ ( ( 𝐻 ‘ 𝑛 ) ‘ 𝑋 ) ) ⇝ ( lim sup ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑁 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ) |