Step |
Hyp |
Ref |
Expression |
1 |
|
smu01lem.1 |
⊢ ( 𝜑 → 𝐴 ⊆ ℕ0 ) |
2 |
|
smu01lem.2 |
⊢ ( 𝜑 → 𝐵 ⊆ ℕ0 ) |
3 |
|
smu01lem.3 |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) ) → ¬ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) ) |
4 |
|
smucl |
⊢ ( ( 𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0 ) → ( 𝐴 smul 𝐵 ) ⊆ ℕ0 ) |
5 |
1 2 4
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 smul 𝐵 ) ⊆ ℕ0 ) |
6 |
5
|
sseld |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝐴 smul 𝐵 ) → 𝑘 ∈ ℕ0 ) ) |
7 |
|
noel |
⊢ ¬ 𝑘 ∈ ∅ |
8 |
|
peano2nn0 |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑘 + 1 ) ∈ ℕ0 ) |
9 |
|
fveqeq2 |
⊢ ( 𝑥 = 0 → ( ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑥 ) = ∅ ↔ ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 0 ) = ∅ ) ) |
10 |
9
|
imbi2d |
⊢ ( 𝑥 = 0 → ( ( 𝜑 → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑥 ) = ∅ ) ↔ ( 𝜑 → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 0 ) = ∅ ) ) ) |
11 |
|
fveqeq2 |
⊢ ( 𝑥 = 𝑘 → ( ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑥 ) = ∅ ↔ ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑘 ) = ∅ ) ) |
12 |
11
|
imbi2d |
⊢ ( 𝑥 = 𝑘 → ( ( 𝜑 → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑥 ) = ∅ ) ↔ ( 𝜑 → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑘 ) = ∅ ) ) ) |
13 |
|
fveqeq2 |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑥 ) = ∅ ↔ ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ∅ ) ) |
14 |
13
|
imbi2d |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( ( 𝜑 → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑥 ) = ∅ ) ↔ ( 𝜑 → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ∅ ) ) ) |
15 |
|
eqid |
⊢ seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) = seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) |
16 |
1 2 15
|
smup0 |
⊢ ( 𝜑 → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 0 ) = ∅ ) |
17 |
|
oveq1 |
⊢ ( ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑘 ) = ∅ → ( ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑘 ) sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) } ) = ( ∅ sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) } ) ) |
18 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝐴 ⊆ ℕ0 ) |
19 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝐵 ⊆ ℕ0 ) |
20 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
21 |
18 19 15 20
|
smupp1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑘 ) sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) } ) ) |
22 |
3
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → ¬ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) ) |
23 |
22
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ∀ 𝑛 ∈ ℕ0 ¬ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) ) |
24 |
|
rabeq0 |
⊢ ( { 𝑛 ∈ ℕ0 ∣ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) } = ∅ ↔ ∀ 𝑛 ∈ ℕ0 ¬ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) ) |
25 |
23 24
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → { 𝑛 ∈ ℕ0 ∣ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) } = ∅ ) |
26 |
25
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ∅ sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) } ) = ( ∅ sadd ∅ ) ) |
27 |
|
0ss |
⊢ ∅ ⊆ ℕ0 |
28 |
|
sadid1 |
⊢ ( ∅ ⊆ ℕ0 → ( ∅ sadd ∅ ) = ∅ ) |
29 |
27 28
|
mp1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ∅ sadd ∅ ) = ∅ ) |
30 |
26 29
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ∅ = ( ∅ sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) } ) ) |
31 |
21 30
|
eqeq12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ∅ ↔ ( ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑘 ) sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) } ) = ( ∅ sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑘 ∈ 𝐴 ∧ ( 𝑛 − 𝑘 ) ∈ 𝐵 ) } ) ) ) |
32 |
17 31
|
syl5ibr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑘 ) = ∅ → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ∅ ) ) |
33 |
32
|
expcom |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝜑 → ( ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑘 ) = ∅ → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ∅ ) ) ) |
34 |
33
|
a2d |
⊢ ( 𝑘 ∈ ℕ0 → ( ( 𝜑 → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ 𝑘 ) = ∅ ) → ( 𝜑 → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ∅ ) ) ) |
35 |
10 12 14 14 16 34
|
nn0ind |
⊢ ( ( 𝑘 + 1 ) ∈ ℕ0 → ( 𝜑 → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ∅ ) ) |
36 |
8 35
|
syl |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝜑 → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ∅ ) ) |
37 |
36
|
impcom |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ∅ ) |
38 |
37
|
eleq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ∈ ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ↔ 𝑘 ∈ ∅ ) ) |
39 |
7 38
|
mtbiri |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ¬ 𝑘 ∈ ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) |
40 |
18 19 15 20
|
smuval |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ∈ ( 𝐴 smul 𝐵 ) ↔ 𝑘 ∈ ( seq 0 ( ( 𝑝 ∈ 𝒫 ℕ0 , 𝑚 ∈ ℕ0 ↦ ( 𝑝 sadd { 𝑛 ∈ ℕ0 ∣ ( 𝑚 ∈ 𝐴 ∧ ( 𝑛 − 𝑚 ) ∈ 𝐵 ) } ) ) , ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ∅ , ( 𝑛 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) ) |
41 |
39 40
|
mtbird |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ¬ 𝑘 ∈ ( 𝐴 smul 𝐵 ) ) |
42 |
41
|
ex |
⊢ ( 𝜑 → ( 𝑘 ∈ ℕ0 → ¬ 𝑘 ∈ ( 𝐴 smul 𝐵 ) ) ) |
43 |
6 42
|
syld |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝐴 smul 𝐵 ) → ¬ 𝑘 ∈ ( 𝐴 smul 𝐵 ) ) ) |
44 |
43
|
pm2.01d |
⊢ ( 𝜑 → ¬ 𝑘 ∈ ( 𝐴 smul 𝐵 ) ) |
45 |
44
|
eq0rdv |
⊢ ( 𝜑 → ( 𝐴 smul 𝐵 ) = ∅ ) |