Step |
Hyp |
Ref |
Expression |
1 |
|
sadcl |
⊢ ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ) → ( 𝐴 sadd 𝐵 ) ⊆ ℕ0 ) |
2 |
|
sadcl |
⊢ ( ( ( 𝐴 sadd 𝐵 ) ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) → ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) ⊆ ℕ0 ) |
3 |
1 2
|
stoic3 |
⊢ ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) → ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) ⊆ ℕ0 ) |
4 |
3
|
sseld |
⊢ ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) → ( 𝑘 ∈ ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) → 𝑘 ∈ ℕ0 ) ) |
5 |
|
simp1 |
⊢ ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) → 𝐴 ⊆ ℕ0 ) |
6 |
|
sadcl |
⊢ ( ( 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) → ( 𝐵 sadd 𝐶 ) ⊆ ℕ0 ) |
7 |
6
|
3adant1 |
⊢ ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) → ( 𝐵 sadd 𝐶 ) ⊆ ℕ0 ) |
8 |
|
sadcl |
⊢ ( ( 𝐴 ⊆ ℕ0 ∧ ( 𝐵 sadd 𝐶 ) ⊆ ℕ0 ) → ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ⊆ ℕ0 ) |
9 |
5 7 8
|
syl2anc |
⊢ ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) → ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ⊆ ℕ0 ) |
10 |
9
|
sseld |
⊢ ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) → ( 𝑘 ∈ ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) → 𝑘 ∈ ℕ0 ) ) |
11 |
|
simpl1 |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝐴 ⊆ ℕ0 ) |
12 |
|
simpl2 |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝐵 ⊆ ℕ0 ) |
13 |
|
simpl3 |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝐶 ⊆ ℕ0 ) |
14 |
|
simpr |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
15 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
16 |
15
|
a1i |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 1 ∈ ℕ0 ) |
17 |
14 16
|
nn0addcld |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 + 1 ) ∈ ℕ0 ) |
18 |
11 12 13 17
|
sadasslem |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) ∩ ( 0 ..^ ( 𝑘 + 1 ) ) ) = ( ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ∩ ( 0 ..^ ( 𝑘 + 1 ) ) ) ) |
19 |
18
|
eleq2d |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ∈ ( ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) ∩ ( 0 ..^ ( 𝑘 + 1 ) ) ) ↔ 𝑘 ∈ ( ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ∩ ( 0 ..^ ( 𝑘 + 1 ) ) ) ) ) |
20 |
|
elin |
⊢ ( 𝑘 ∈ ( ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) ∩ ( 0 ..^ ( 𝑘 + 1 ) ) ) ↔ ( 𝑘 ∈ ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) ∧ 𝑘 ∈ ( 0 ..^ ( 𝑘 + 1 ) ) ) ) |
21 |
|
elin |
⊢ ( 𝑘 ∈ ( ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ∩ ( 0 ..^ ( 𝑘 + 1 ) ) ) ↔ ( 𝑘 ∈ ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( 𝑘 + 1 ) ) ) ) |
22 |
19 20 21
|
3bitr3g |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑘 ∈ ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) ∧ 𝑘 ∈ ( 0 ..^ ( 𝑘 + 1 ) ) ) ↔ ( 𝑘 ∈ ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( 𝑘 + 1 ) ) ) ) ) |
23 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
24 |
14 23
|
eleqtrdi |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ( ℤ≥ ‘ 0 ) ) |
25 |
|
eluzfz2 |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 0 ) → 𝑘 ∈ ( 0 ... 𝑘 ) ) |
26 |
24 25
|
syl |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ( 0 ... 𝑘 ) ) |
27 |
14
|
nn0zd |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℤ ) |
28 |
|
fzval3 |
⊢ ( 𝑘 ∈ ℤ → ( 0 ... 𝑘 ) = ( 0 ..^ ( 𝑘 + 1 ) ) ) |
29 |
27 28
|
syl |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 0 ... 𝑘 ) = ( 0 ..^ ( 𝑘 + 1 ) ) ) |
30 |
26 29
|
eleqtrd |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ( 0 ..^ ( 𝑘 + 1 ) ) ) |
31 |
30
|
biantrud |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ∈ ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) ↔ ( 𝑘 ∈ ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) ∧ 𝑘 ∈ ( 0 ..^ ( 𝑘 + 1 ) ) ) ) ) |
32 |
30
|
biantrud |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ∈ ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ↔ ( 𝑘 ∈ ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( 𝑘 + 1 ) ) ) ) ) |
33 |
22 31 32
|
3bitr4d |
⊢ ( ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ∈ ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) ↔ 𝑘 ∈ ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ) ) |
34 |
33
|
ex |
⊢ ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) → ( 𝑘 ∈ ℕ0 → ( 𝑘 ∈ ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) ↔ 𝑘 ∈ ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ) ) ) |
35 |
4 10 34
|
pm5.21ndd |
⊢ ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) → ( 𝑘 ∈ ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) ↔ 𝑘 ∈ ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ) ) |
36 |
35
|
eqrdv |
⊢ ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0 ) → ( ( 𝐴 sadd 𝐵 ) sadd 𝐶 ) = ( 𝐴 sadd ( 𝐵 sadd 𝐶 ) ) ) |