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 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ) |
12 |
11 8
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
13 |
12
|
dmeqd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
14 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑍 |
15 |
14
|
nfcri |
⊢ Ⅎ 𝑥 𝑛 ∈ 𝑍 |
16 |
2 15
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) |
17 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
18 |
7
|
3expa |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) |
19 |
16 17 18
|
dmmptdf |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 ) |
20 |
13 19
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝐴 = dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) |
21 |
1 20
|
iineq2d |
⊢ ( 𝜑 → ∩ 𝑛 ∈ 𝑍 𝐴 = ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) |
22 |
2 21
|
rabeqd |
⊢ ( 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 } ) |
23 |
|
nfv |
⊢ Ⅎ 𝑦 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) |
24 |
3 23
|
nfan |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) |
25 |
|
nfii1 |
⊢ Ⅎ 𝑛 ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) |
26 |
25
|
nfcri |
⊢ Ⅎ 𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) |
27 |
1 26
|
nfan |
⊢ Ⅎ 𝑛 ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) |
28 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝜑 ) |
29 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 ) |
30 |
|
eliinid |
⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) |
31 |
30
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) |
32 |
13 19
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = 𝐴 ) |
33 |
32
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) = 𝐴 ) |
34 |
31 33
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 ) |
35 |
12
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ) |
36 |
35
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ) |
37 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
38 |
|
fvmpt4 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 ) |
39 |
37 7 38
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 ) |
40 |
36 39
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) |
41 |
40
|
breq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 ≤ 𝐵 ↔ 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) ) |
42 |
28 29 34 41
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛 ∈ 𝑍 ) → ( 𝑦 ≤ 𝐵 ↔ 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) ) |
43 |
27 42
|
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 ) ) ) |
44 |
24 43
|
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 ) ) ) |
45 |
2 44
|
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 ) } ) |
46 |
22 45
|
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 ) } ) |
47 |
9 46
|
eqtrid |
⊢ ( 𝜑 → 𝐷 = { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ) |
48 |
|
nfcv |
⊢ Ⅎ 𝑛 ℝ |
49 |
|
nfra1 |
⊢ Ⅎ 𝑛 ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 |
50 |
48 49
|
nfrexw |
⊢ Ⅎ 𝑛 ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 |
51 |
|
nfii1 |
⊢ Ⅎ 𝑛 ∩ 𝑛 ∈ 𝑍 𝐴 |
52 |
50 51
|
nfrabw |
⊢ Ⅎ 𝑛 { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } |
53 |
9 52
|
nfcxfr |
⊢ Ⅎ 𝑛 𝐷 |
54 |
53
|
nfcri |
⊢ Ⅎ 𝑛 𝑥 ∈ 𝐷 |
55 |
1 54
|
nfan |
⊢ Ⅎ 𝑛 ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) |
56 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝜑 ) |
57 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 ) |
58 |
|
rabidim1 |
⊢ ( 𝑥 ∈ { 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 } → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ) |
59 |
58 9
|
eleq2s |
⊢ ( 𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ) |
60 |
|
eliinid |
⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 ) |
61 |
59 60
|
sylan |
⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 ) |
62 |
61
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑥 ∈ 𝐴 ) |
63 |
56 57 62 40
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝑛 ∈ 𝑍 ) → 𝐵 = ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) |
64 |
55 63
|
mpteq2da |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) ) |
65 |
64
|
rneqd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) ) |
66 |
65
|
infeq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = inf ( ran ( 𝑛 ∈ 𝑍 ↦ ( ( ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) |
67 |
2 47 66
|
mpteq12da |
⊢ ( 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 , < ) ) ) |
68 |
10 67
|
eqtrid |
⊢ ( 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 , < ) ) ) |
69 |
|
nfmpt1 |
⊢ Ⅎ 𝑛 ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
70 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
71 |
14 70
|
nfmpt |
⊢ Ⅎ 𝑥 ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
72 |
1 8
|
fmptd2f |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) : 𝑍 ⟶ ( SMblFn ‘ 𝑆 ) ) |
73 |
|
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 ) } |
74 |
|
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 , < ) ) |
75 |
69 71 4 5 6 72 73 74
|
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 ) ) |
76 |
68 75
|
eqeltrd |
⊢ ( 𝜑 → 𝐺 ∈ ( SMblFn ‘ 𝑆 ) ) |