Step |
Hyp |
Ref |
Expression |
1 |
|
smflimlem1.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
smflimlem1.2 |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |
3 |
|
smflimlem1.3 |
⊢ 𝐷 = { 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ∈ dom ⇝ } |
4 |
|
smflimlem1.4 |
⊢ 𝑃 = ( 𝑚 ∈ 𝑍 , 𝑘 ∈ ℕ ↦ { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } ) |
5 |
|
smflimlem1.5 |
⊢ 𝐻 = ( 𝑚 ∈ 𝑍 , 𝑘 ∈ ℕ ↦ ( 𝐶 ‘ ( 𝑚 𝑃 𝑘 ) ) ) |
6 |
|
smflimlem1.6 |
⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( 𝑚 𝐻 𝑘 ) |
7 |
|
smflimlem1.7 |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ran 𝑃 ) → ( 𝐶 ‘ 𝑟 ) ∈ 𝑟 ) |
8 |
|
fvex |
⊢ ( ℤ≥ ‘ 𝑀 ) ∈ V |
9 |
1 8
|
eqeltri |
⊢ 𝑍 ∈ V |
10 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
11 |
1
|
eleq2i |
⊢ ( 𝑛 ∈ 𝑍 ↔ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
12 |
11
|
biimpi |
⊢ ( 𝑛 ∈ 𝑍 → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
13 |
10 12
|
sselid |
⊢ ( 𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ ) |
14 |
|
uzid |
⊢ ( 𝑛 ∈ ℤ → 𝑛 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
15 |
13 14
|
syl |
⊢ ( 𝑛 ∈ 𝑍 → 𝑛 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
16 |
15
|
ne0d |
⊢ ( 𝑛 ∈ 𝑍 → ( ℤ≥ ‘ 𝑛 ) ≠ ∅ ) |
17 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑚 ) ∈ V |
18 |
17
|
dmex |
⊢ dom ( 𝐹 ‘ 𝑚 ) ∈ V |
19 |
18
|
rgenw |
⊢ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V |
20 |
19
|
a1i |
⊢ ( 𝑛 ∈ 𝑍 → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V ) |
21 |
|
iinexg |
⊢ ( ( ( ℤ≥ ‘ 𝑛 ) ≠ ∅ ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V ) → ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V ) |
22 |
16 20 21
|
syl2anc |
⊢ ( 𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V ) |
23 |
22
|
rgen |
⊢ ∀ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V |
24 |
|
iunexg |
⊢ ( ( 𝑍 ∈ V ∧ ∀ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V ) → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V ) |
25 |
9 23 24
|
mp2an |
⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∈ V |
26 |
25
|
rabex |
⊢ { 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ∈ dom ⇝ } ∈ V |
27 |
3 26
|
eqeltri |
⊢ 𝐷 ∈ V |
28 |
27
|
a1i |
⊢ ( 𝜑 → 𝐷 ∈ V ) |
29 |
|
nnct |
⊢ ℕ ≼ ω |
30 |
29
|
a1i |
⊢ ( 𝜑 → ℕ ≼ ω ) |
31 |
|
nnn0 |
⊢ ℕ ≠ ∅ |
32 |
31
|
a1i |
⊢ ( 𝜑 → ℕ ≠ ∅ ) |
33 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑆 ∈ SAlg ) |
34 |
1
|
uzct |
⊢ 𝑍 ≼ ω |
35 |
34
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑍 ≼ ω ) |
36 |
33
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ 𝑛 ∈ 𝑍 ) → 𝑆 ∈ SAlg ) |
37 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑛 ) = ( ℤ≥ ‘ 𝑛 ) |
38 |
37
|
uzct |
⊢ ( ℤ≥ ‘ 𝑛 ) ≼ ω |
39 |
38
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ 𝑛 ∈ 𝑍 ) → ( ℤ≥ ‘ 𝑛 ) ≼ ω ) |
40 |
16
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ 𝑛 ∈ 𝑍 ) → ( ℤ≥ ‘ 𝑛 ) ≠ ∅ ) |
41 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝜑 ) |
42 |
41
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝜑 ) |
43 |
|
simpll |
⊢ ( ( ( 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑘 ∈ ℕ ) |
44 |
43
|
adantlll |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑘 ∈ ℕ ) |
45 |
1
|
uztrn2 |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑗 ∈ 𝑍 ) |
46 |
45
|
ssd |
⊢ ( 𝑛 ∈ 𝑍 → ( ℤ≥ ‘ 𝑛 ) ⊆ 𝑍 ) |
47 |
46
|
sselda |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑚 ∈ 𝑍 ) |
48 |
47
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑚 ∈ 𝑍 ) |
49 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → 𝑚 ∈ 𝑍 ) |
50 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → 𝑘 ∈ ℕ ) |
51 |
|
fvex |
⊢ ( 𝐶 ‘ ( 𝑚 𝑃 𝑘 ) ) ∈ V |
52 |
51
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → ( 𝐶 ‘ ( 𝑚 𝑃 𝑘 ) ) ∈ V ) |
53 |
5
|
ovmpt4g |
⊢ ( ( 𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ ∧ ( 𝐶 ‘ ( 𝑚 𝑃 𝑘 ) ) ∈ V ) → ( 𝑚 𝐻 𝑘 ) = ( 𝐶 ‘ ( 𝑚 𝑃 𝑘 ) ) ) |
54 |
49 50 52 53
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → ( 𝑚 𝐻 𝑘 ) = ( 𝐶 ‘ ( 𝑚 𝑃 𝑘 ) ) ) |
55 |
|
simp1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → 𝜑 ) |
56 |
|
eqid |
⊢ { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } = { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } |
57 |
56 2
|
rabexd |
⊢ ( 𝜑 → { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } ∈ V ) |
58 |
55 57
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } ∈ V ) |
59 |
4
|
ovmpt4g |
⊢ ( ( 𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ ∧ { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } ∈ V ) → ( 𝑚 𝑃 𝑘 ) = { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } ) |
60 |
49 50 58 59
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → ( 𝑚 𝑃 𝑘 ) = { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } ) |
61 |
|
ssrab2 |
⊢ { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } ⊆ 𝑆 |
62 |
60 61
|
eqsstrdi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → ( 𝑚 𝑃 𝑘 ) ⊆ 𝑆 ) |
63 |
57
|
ralrimivw |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ℕ { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } ∈ V ) |
64 |
63
|
ralrimivw |
⊢ ( 𝜑 → ∀ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ℕ { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } ∈ V ) |
65 |
64
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → ∀ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ℕ { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } ∈ V ) |
66 |
4
|
elrnmpoid |
⊢ ( ( 𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ ∧ ∀ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ℕ { 𝑠 ∈ 𝑆 ∣ { 𝑥 ∈ dom ( 𝐹 ‘ 𝑚 ) ∣ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) < ( 𝐴 + ( 1 / 𝑘 ) ) } = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑚 ) ) } ∈ V ) → ( 𝑚 𝑃 𝑘 ) ∈ ran 𝑃 ) |
67 |
49 50 65 66
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → ( 𝑚 𝑃 𝑘 ) ∈ ran 𝑃 ) |
68 |
|
ovex |
⊢ ( 𝑚 𝑃 𝑘 ) ∈ V |
69 |
|
eleq1 |
⊢ ( 𝑟 = ( 𝑚 𝑃 𝑘 ) → ( 𝑟 ∈ ran 𝑃 ↔ ( 𝑚 𝑃 𝑘 ) ∈ ran 𝑃 ) ) |
70 |
69
|
anbi2d |
⊢ ( 𝑟 = ( 𝑚 𝑃 𝑘 ) → ( ( 𝜑 ∧ 𝑟 ∈ ran 𝑃 ) ↔ ( 𝜑 ∧ ( 𝑚 𝑃 𝑘 ) ∈ ran 𝑃 ) ) ) |
71 |
|
fveq2 |
⊢ ( 𝑟 = ( 𝑚 𝑃 𝑘 ) → ( 𝐶 ‘ 𝑟 ) = ( 𝐶 ‘ ( 𝑚 𝑃 𝑘 ) ) ) |
72 |
|
id |
⊢ ( 𝑟 = ( 𝑚 𝑃 𝑘 ) → 𝑟 = ( 𝑚 𝑃 𝑘 ) ) |
73 |
71 72
|
eleq12d |
⊢ ( 𝑟 = ( 𝑚 𝑃 𝑘 ) → ( ( 𝐶 ‘ 𝑟 ) ∈ 𝑟 ↔ ( 𝐶 ‘ ( 𝑚 𝑃 𝑘 ) ) ∈ ( 𝑚 𝑃 𝑘 ) ) ) |
74 |
70 73
|
imbi12d |
⊢ ( 𝑟 = ( 𝑚 𝑃 𝑘 ) → ( ( ( 𝜑 ∧ 𝑟 ∈ ran 𝑃 ) → ( 𝐶 ‘ 𝑟 ) ∈ 𝑟 ) ↔ ( ( 𝜑 ∧ ( 𝑚 𝑃 𝑘 ) ∈ ran 𝑃 ) → ( 𝐶 ‘ ( 𝑚 𝑃 𝑘 ) ) ∈ ( 𝑚 𝑃 𝑘 ) ) ) ) |
75 |
68 74 7
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑚 𝑃 𝑘 ) ∈ ran 𝑃 ) → ( 𝐶 ‘ ( 𝑚 𝑃 𝑘 ) ) ∈ ( 𝑚 𝑃 𝑘 ) ) |
76 |
55 67 75
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → ( 𝐶 ‘ ( 𝑚 𝑃 𝑘 ) ) ∈ ( 𝑚 𝑃 𝑘 ) ) |
77 |
62 76
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → ( 𝐶 ‘ ( 𝑚 𝑃 𝑘 ) ) ∈ 𝑆 ) |
78 |
54 77
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍 ) → ( 𝑚 𝐻 𝑘 ) ∈ 𝑆 ) |
79 |
42 44 48 78
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝑚 𝐻 𝑘 ) ∈ 𝑆 ) |
80 |
36 39 40 79
|
saliincl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ 𝑛 ∈ 𝑍 ) → ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( 𝑚 𝐻 𝑘 ) ∈ 𝑆 ) |
81 |
33 35 80
|
saliuncl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( 𝑚 𝐻 𝑘 ) ∈ 𝑆 ) |
82 |
2 30 32 81
|
saliincl |
⊢ ( 𝜑 → ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( 𝑚 𝐻 𝑘 ) ∈ 𝑆 ) |
83 |
6 82
|
eqeltrid |
⊢ ( 𝜑 → 𝐼 ∈ 𝑆 ) |
84 |
|
incom |
⊢ ( 𝐷 ∩ 𝐼 ) = ( 𝐼 ∩ 𝐷 ) |
85 |
2 28 83 84
|
elrestd |
⊢ ( 𝜑 → ( 𝐷 ∩ 𝐼 ) ∈ ( 𝑆 ↾t 𝐷 ) ) |