Step |
Hyp |
Ref |
Expression |
1 |
|
opelxp |
⊢ ( 〈 ∅ , 〈 𝐹 , 𝐿 〉 〉 ∈ ( V × ( ℤ × ℤ ) ) ↔ ( ∅ ∈ V ∧ 〈 𝐹 , 𝐿 〉 ∈ ( ℤ × ℤ ) ) ) |
2 |
|
opelxp |
⊢ ( 〈 𝐹 , 𝐿 〉 ∈ ( ℤ × ℤ ) ↔ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) |
3 |
|
swrdval |
⊢ ( ( ∅ ∈ V ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) → ( ∅ substr 〈 𝐹 , 𝐿 〉 ) = if ( ( 𝐹 ..^ 𝐿 ) ⊆ dom ∅ , ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) , ∅ ) ) |
4 |
|
fzonlt0 |
⊢ ( ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) → ( ¬ 𝐹 < 𝐿 ↔ ( 𝐹 ..^ 𝐿 ) = ∅ ) ) |
5 |
4
|
biimprd |
⊢ ( ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) → ( ( 𝐹 ..^ 𝐿 ) = ∅ → ¬ 𝐹 < 𝐿 ) ) |
6 |
5
|
con2d |
⊢ ( ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) → ( 𝐹 < 𝐿 → ¬ ( 𝐹 ..^ 𝐿 ) = ∅ ) ) |
7 |
6
|
impcom |
⊢ ( ( 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → ¬ ( 𝐹 ..^ 𝐿 ) = ∅ ) |
8 |
|
ss0 |
⊢ ( ( 𝐹 ..^ 𝐿 ) ⊆ ∅ → ( 𝐹 ..^ 𝐿 ) = ∅ ) |
9 |
7 8
|
nsyl |
⊢ ( ( 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → ¬ ( 𝐹 ..^ 𝐿 ) ⊆ ∅ ) |
10 |
|
dm0 |
⊢ dom ∅ = ∅ |
11 |
10
|
a1i |
⊢ ( ( 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → dom ∅ = ∅ ) |
12 |
11
|
sseq2d |
⊢ ( ( 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → ( ( 𝐹 ..^ 𝐿 ) ⊆ dom ∅ ↔ ( 𝐹 ..^ 𝐿 ) ⊆ ∅ ) ) |
13 |
9 12
|
mtbird |
⊢ ( ( 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → ¬ ( 𝐹 ..^ 𝐿 ) ⊆ dom ∅ ) |
14 |
13
|
iffalsed |
⊢ ( ( 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → if ( ( 𝐹 ..^ 𝐿 ) ⊆ dom ∅ , ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) , ∅ ) = ∅ ) |
15 |
|
ssidd |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → ∅ ⊆ ∅ ) |
16 |
4
|
biimpac |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → ( 𝐹 ..^ 𝐿 ) = ∅ ) |
17 |
10
|
a1i |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → dom ∅ = ∅ ) |
18 |
15 16 17
|
3sstr4d |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → ( 𝐹 ..^ 𝐿 ) ⊆ dom ∅ ) |
19 |
18
|
iftrued |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → if ( ( 𝐹 ..^ 𝐿 ) ⊆ dom ∅ , ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) , ∅ ) = ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) ) |
20 |
|
zre |
⊢ ( 𝐿 ∈ ℤ → 𝐿 ∈ ℝ ) |
21 |
|
zre |
⊢ ( 𝐹 ∈ ℤ → 𝐹 ∈ ℝ ) |
22 |
|
lenlt |
⊢ ( ( 𝐿 ∈ ℝ ∧ 𝐹 ∈ ℝ ) → ( 𝐿 ≤ 𝐹 ↔ ¬ 𝐹 < 𝐿 ) ) |
23 |
22
|
bicomd |
⊢ ( ( 𝐿 ∈ ℝ ∧ 𝐹 ∈ ℝ ) → ( ¬ 𝐹 < 𝐿 ↔ 𝐿 ≤ 𝐹 ) ) |
24 |
20 21 23
|
syl2anr |
⊢ ( ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) → ( ¬ 𝐹 < 𝐿 ↔ 𝐿 ≤ 𝐹 ) ) |
25 |
|
fzo0n |
⊢ ( ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) → ( 𝐿 ≤ 𝐹 ↔ ( 0 ..^ ( 𝐿 − 𝐹 ) ) = ∅ ) ) |
26 |
24 25
|
bitrd |
⊢ ( ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) → ( ¬ 𝐹 < 𝐿 ↔ ( 0 ..^ ( 𝐿 − 𝐹 ) ) = ∅ ) ) |
27 |
26
|
biimpac |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → ( 0 ..^ ( 𝐿 − 𝐹 ) ) = ∅ ) |
28 |
27
|
mpteq1d |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) = ( 𝑥 ∈ ∅ ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) ) |
29 |
28
|
dmeqd |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → dom ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) = dom ( 𝑥 ∈ ∅ ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) ) |
30 |
|
ral0 |
⊢ ∀ 𝑥 ∈ ∅ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ∈ V |
31 |
|
dmmptg |
⊢ ( ∀ 𝑥 ∈ ∅ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ∈ V → dom ( 𝑥 ∈ ∅ ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) = ∅ ) |
32 |
30 31
|
mp1i |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → dom ( 𝑥 ∈ ∅ ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) = ∅ ) |
33 |
29 32
|
eqtrd |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → dom ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) = ∅ ) |
34 |
|
mptrel |
⊢ Rel ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) |
35 |
|
reldm0 |
⊢ ( Rel ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) → ( ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) = ∅ ↔ dom ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) = ∅ ) ) |
36 |
34 35
|
mp1i |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → ( ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) = ∅ ↔ dom ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) = ∅ ) ) |
37 |
33 36
|
mpbird |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) = ∅ ) |
38 |
19 37
|
eqtrd |
⊢ ( ( ¬ 𝐹 < 𝐿 ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → if ( ( 𝐹 ..^ 𝐿 ) ⊆ dom ∅ , ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) , ∅ ) = ∅ ) |
39 |
14 38
|
pm2.61ian |
⊢ ( ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) → if ( ( 𝐹 ..^ 𝐿 ) ⊆ dom ∅ , ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) , ∅ ) = ∅ ) |
40 |
39
|
3adant1 |
⊢ ( ( ∅ ∈ V ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) → if ( ( 𝐹 ..^ 𝐿 ) ⊆ dom ∅ , ( 𝑥 ∈ ( 0 ..^ ( 𝐿 − 𝐹 ) ) ↦ ( ∅ ‘ ( 𝑥 + 𝐹 ) ) ) , ∅ ) = ∅ ) |
41 |
3 40
|
eqtrd |
⊢ ( ( ∅ ∈ V ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) → ( ∅ substr 〈 𝐹 , 𝐿 〉 ) = ∅ ) |
42 |
41
|
3expb |
⊢ ( ( ∅ ∈ V ∧ ( 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ ) ) → ( ∅ substr 〈 𝐹 , 𝐿 〉 ) = ∅ ) |
43 |
2 42
|
sylan2b |
⊢ ( ( ∅ ∈ V ∧ 〈 𝐹 , 𝐿 〉 ∈ ( ℤ × ℤ ) ) → ( ∅ substr 〈 𝐹 , 𝐿 〉 ) = ∅ ) |
44 |
1 43
|
sylbi |
⊢ ( 〈 ∅ , 〈 𝐹 , 𝐿 〉 〉 ∈ ( V × ( ℤ × ℤ ) ) → ( ∅ substr 〈 𝐹 , 𝐿 〉 ) = ∅ ) |
45 |
|
df-substr |
⊢ substr = ( 𝑠 ∈ V , 𝑏 ∈ ( ℤ × ℤ ) ↦ if ( ( ( 1st ‘ 𝑏 ) ..^ ( 2nd ‘ 𝑏 ) ) ⊆ dom 𝑠 , ( 𝑧 ∈ ( 0 ..^ ( ( 2nd ‘ 𝑏 ) − ( 1st ‘ 𝑏 ) ) ) ↦ ( 𝑠 ‘ ( 𝑧 + ( 1st ‘ 𝑏 ) ) ) ) , ∅ ) ) |
46 |
|
ovex |
⊢ ( 0 ..^ ( ( 2nd ‘ 𝑏 ) − ( 1st ‘ 𝑏 ) ) ) ∈ V |
47 |
46
|
mptex |
⊢ ( 𝑧 ∈ ( 0 ..^ ( ( 2nd ‘ 𝑏 ) − ( 1st ‘ 𝑏 ) ) ) ↦ ( 𝑠 ‘ ( 𝑧 + ( 1st ‘ 𝑏 ) ) ) ) ∈ V |
48 |
|
0ex |
⊢ ∅ ∈ V |
49 |
47 48
|
ifex |
⊢ if ( ( ( 1st ‘ 𝑏 ) ..^ ( 2nd ‘ 𝑏 ) ) ⊆ dom 𝑠 , ( 𝑧 ∈ ( 0 ..^ ( ( 2nd ‘ 𝑏 ) − ( 1st ‘ 𝑏 ) ) ) ↦ ( 𝑠 ‘ ( 𝑧 + ( 1st ‘ 𝑏 ) ) ) ) , ∅ ) ∈ V |
50 |
45 49
|
dmmpo |
⊢ dom substr = ( V × ( ℤ × ℤ ) ) |
51 |
44 50
|
eleq2s |
⊢ ( 〈 ∅ , 〈 𝐹 , 𝐿 〉 〉 ∈ dom substr → ( ∅ substr 〈 𝐹 , 𝐿 〉 ) = ∅ ) |
52 |
|
df-ov |
⊢ ( ∅ substr 〈 𝐹 , 𝐿 〉 ) = ( substr ‘ 〈 ∅ , 〈 𝐹 , 𝐿 〉 〉 ) |
53 |
|
ndmfv |
⊢ ( ¬ 〈 ∅ , 〈 𝐹 , 𝐿 〉 〉 ∈ dom substr → ( substr ‘ 〈 ∅ , 〈 𝐹 , 𝐿 〉 〉 ) = ∅ ) |
54 |
52 53
|
eqtrid |
⊢ ( ¬ 〈 ∅ , 〈 𝐹 , 𝐿 〉 〉 ∈ dom substr → ( ∅ substr 〈 𝐹 , 𝐿 〉 ) = ∅ ) |
55 |
51 54
|
pm2.61i |
⊢ ( ∅ substr 〈 𝐹 , 𝐿 〉 ) = ∅ |