Step |
Hyp |
Ref |
Expression |
1 |
|
smfinfmpt.n |
⊢ Ⅎ 𝑛 𝜑 |
2 |
|
smfinfmpt.x |
⊢ Ⅎ 𝑥 𝜑 |
3 |
|
smfinfmpt.y |
⊢ Ⅎ 𝑦 𝜑 |
4 |
|
smfinfmpt.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
5 |
|
smfinfmpt.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
6 |
|
smfinfmpt.s |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |
7 |
|
smfinfmpt.b |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) |
8 |
|
smfinfmpt.f |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
9 |
|
smfinfmpt.d |
⊢ 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } |
10 |
|
smfinfmpt.g |
⊢ 𝐺 = ( 𝑥 ∈ 𝐷 ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) ) |
11 |
10
|
a1i |
⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐷 ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) ) ) |
12 |
9
|
a1i |
⊢ ( 𝜑 → 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } ) |
13 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ) |
14 |
13 8
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
15 |
14
|
dmeqd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
16 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑛 |
17 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑍 |
18 |
16 17
|
nfel |
⊢ Ⅎ 𝑥 𝑛 ∈ 𝑍 |
19 |
2 18
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) |
20 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
21 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑆 ∈ SAlg ) |
22 |
7
|
3expa |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) |
23 |
19 21 22 8
|
smffmpt |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℝ ) |
24 |
23
|
fvmptelrn |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
25 |
19 20 24
|
dmmptdf |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 ) |
26 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝐴 = 𝐴 ) |
27 |
15 25 26
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝐴 = dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) |
28 |
1 27
|
iineq2d |
⊢ ( 𝜑 → ∩ 𝑛 ∈ 𝑍 𝐴 = ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) |
29 |
|
nfcv |
⊢ Ⅎ 𝑥 ∩ 𝑛 ∈ 𝑍 𝐴 |
30 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
31 |
17 30
|
nfmpt |
⊢ Ⅎ 𝑥 ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
32 |
31 16
|
nffv |
⊢ Ⅎ 𝑥 ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) |
33 |
32
|
nfdm |
⊢ Ⅎ 𝑥 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) |
34 |
17 33
|
nfiin |
⊢ Ⅎ 𝑥 ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) |
35 |
29 34
|
rabeqf |
⊢ ( ∩ 𝑛 ∈ 𝑍 𝐴 = ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) → { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } ) |
36 |
28 35
|
syl |
⊢ ( 𝜑 → { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } ) |
37 |
|
nfv |
⊢ Ⅎ 𝑦 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) |
38 |
3 37
|
nfan |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) |
39 |
|
nfcv |
⊢ Ⅎ 𝑛 𝑥 |
40 |
|
nfii1 |
⊢ Ⅎ 𝑛 ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) |
41 |
39 40
|
nfel |
⊢ Ⅎ 𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) |
42 |
1 41
|
nfan |
⊢ Ⅎ 𝑛 ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) |
43 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝜑 ) |
44 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 ) |
45 |
|
eliinid |
⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) |
46 |
45
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) |
47 |
27
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = 𝐴 ) |
48 |
47
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = 𝐴 ) |
49 |
46 48
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 ) |
50 |
14
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ) |
51 |
50
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ) |
52 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
53 |
20
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 ) |
54 |
52 7 53
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 ) |
55 |
51 54
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) |
56 |
55
|
breq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 ≤ 𝐵 ↔ 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) ) |
57 |
43 44 49 56
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → ( 𝑦 ≤ 𝐵 ↔ 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) ) |
58 |
42 57
|
ralbida |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) → ( ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ↔ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) ) |
59 |
38 58
|
rexbid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) → ( ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) ) |
60 |
2 59
|
rabbida |
⊢ ( 𝜑 → { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ) |
61 |
36 60
|
eqtrd |
⊢ ( 𝜑 → { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ) |
62 |
12 61
|
eqtrd |
⊢ ( 𝜑 → 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ) |
63 |
2 62
|
alrimi |
⊢ ( 𝜑 → ∀ 𝑥 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ) |
64 |
|
nfcv |
⊢ Ⅎ 𝑛 ℝ |
65 |
|
nfra1 |
⊢ Ⅎ 𝑛 ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 |
66 |
64 65
|
nfrex |
⊢ Ⅎ 𝑛 ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 |
67 |
|
nfii1 |
⊢ Ⅎ 𝑛 ∩ 𝑛 ∈ 𝑍 𝐴 |
68 |
66 67
|
nfrabw |
⊢ Ⅎ 𝑛 { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } |
69 |
9 68
|
nfcxfr |
⊢ Ⅎ 𝑛 𝐷 |
70 |
39 69
|
nfel |
⊢ Ⅎ 𝑛 𝑥 ∈ 𝐷 |
71 |
1 70
|
nfan |
⊢ Ⅎ 𝑛 ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) |
72 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝜑 ) |
73 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 ) |
74 |
9
|
eleq2i |
⊢ ( 𝑥 ∈ 𝐷 ↔ 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } ) |
75 |
74
|
biimpi |
⊢ ( 𝑥 ∈ 𝐷 → 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } ) |
76 |
|
rabidim1 |
⊢ ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ) |
77 |
75 76
|
syl |
⊢ ( 𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ) |
78 |
77
|
adantr |
⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ) |
79 |
|
simpr |
⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 ) |
80 |
|
eliinid |
⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 ) |
81 |
78 79 80
|
syl2anc |
⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 ) |
82 |
81
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 ) |
83 |
55
|
idi |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) |
84 |
72 73 82 83
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝐵 = ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) |
85 |
71 84
|
mpteq2da |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) ) |
86 |
85
|
rneqd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) ) |
87 |
86
|
infeq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) |
88 |
87
|
ex |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐷 → inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) ) |
89 |
2 88
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐷 inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) |
90 |
|
mpteq12f |
⊢ ( ( ∀ 𝑥 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ∧ ∀ 𝑥 ∈ 𝐷 inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) → ( 𝑥 ∈ 𝐷 ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) ) = ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) ) |
91 |
63 89 90
|
syl2anc |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐷 ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) ) = ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) ) |
92 |
11 91
|
eqtrd |
⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) ) |
93 |
|
nfmpt1 |
⊢ Ⅎ 𝑛 ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
94 |
|
eqid |
⊢ ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
95 |
1 8 94
|
fmptdf |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) : 𝑍 ⟶ ( SMblFn ‘ 𝑆 ) ) |
96 |
|
eqid |
⊢ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } |
97 |
|
eqid |
⊢ ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) = ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) |
98 |
93 31 4 5 6 95 96 97
|
smfinf |
⊢ ( 𝜑 → ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) ∈ ( SMblFn ‘ 𝑆 ) ) |
99 |
92 98
|
eqeltrd |
⊢ ( 𝜑 → 𝐺 ∈ ( SMblFn ‘ 𝑆 ) ) |