Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
2 |
|
frnnn0supp |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ) → ( 𝐺 supp 0 ) = ( ◡ 𝐺 “ ℕ ) ) |
3 |
2
|
3ad2antr2 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐺 supp 0 ) = ( ◡ 𝐺 “ ℕ ) ) |
4 |
|
simpr2 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐺 : 𝐼 ⟶ ℕ0 ) |
5 |
|
eldifi |
⊢ ( 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) → 𝑥 ∈ 𝐼 ) |
6 |
|
simpr3 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐺 ∘r ≤ 𝐹 ) |
7 |
4
|
ffnd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐺 Fn 𝐼 ) |
8 |
1
|
psrbagfOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) → 𝐹 : 𝐼 ⟶ ℕ0 ) |
9 |
8
|
3ad2antr1 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐹 : 𝐼 ⟶ ℕ0 ) |
10 |
9
|
ffnd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐹 Fn 𝐼 ) |
11 |
|
simpl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐼 ∈ 𝑉 ) |
12 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
13 |
|
eqidd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
14 |
|
eqidd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
15 |
7 10 11 11 12 13 14
|
ofrfval |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐺 ∘r ≤ 𝐹 ↔ ∀ 𝑥 ∈ 𝐼 ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
16 |
6 15
|
mpbid |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ∀ 𝑥 ∈ 𝐼 ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
17 |
16
|
r19.21bi |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
18 |
5 17
|
sylan2 |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
19 |
11 9
|
jca |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ ℕ0 ) ) |
20 |
|
frnnn0supp |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 supp 0 ) = ( ◡ 𝐹 “ ℕ ) ) |
21 |
|
eqimss |
⊢ ( ( 𝐹 supp 0 ) = ( ◡ 𝐹 “ ℕ ) → ( 𝐹 supp 0 ) ⊆ ( ◡ 𝐹 “ ℕ ) ) |
22 |
19 20 21
|
3syl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐹 supp 0 ) ⊆ ( ◡ 𝐹 “ ℕ ) ) |
23 |
|
c0ex |
⊢ 0 ∈ V |
24 |
23
|
a1i |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 0 ∈ V ) |
25 |
9 22 11 24
|
suppssr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
26 |
18 25
|
breqtrd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( 𝐺 ‘ 𝑥 ) ≤ 0 ) |
27 |
|
ffvelrn |
⊢ ( ( 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ∈ ℕ0 ) |
28 |
4 5 27
|
syl2an |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( 𝐺 ‘ 𝑥 ) ∈ ℕ0 ) |
29 |
28
|
nn0ge0d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → 0 ≤ ( 𝐺 ‘ 𝑥 ) ) |
30 |
28
|
nn0red |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( 𝐺 ‘ 𝑥 ) ∈ ℝ ) |
31 |
|
0re |
⊢ 0 ∈ ℝ |
32 |
|
letri3 |
⊢ ( ( ( 𝐺 ‘ 𝑥 ) ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( 𝐺 ‘ 𝑥 ) = 0 ↔ ( ( 𝐺 ‘ 𝑥 ) ≤ 0 ∧ 0 ≤ ( 𝐺 ‘ 𝑥 ) ) ) ) |
33 |
30 31 32
|
sylancl |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( ( 𝐺 ‘ 𝑥 ) = 0 ↔ ( ( 𝐺 ‘ 𝑥 ) ≤ 0 ∧ 0 ≤ ( 𝐺 ‘ 𝑥 ) ) ) ) |
34 |
26 29 33
|
mpbir2and |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( 𝐺 ‘ 𝑥 ) = 0 ) |
35 |
4 34
|
suppss |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐺 supp 0 ) ⊆ ( ◡ 𝐹 “ ℕ ) ) |
36 |
3 35
|
eqsstrrd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( ◡ 𝐺 “ ℕ ) ⊆ ( ◡ 𝐹 “ ℕ ) ) |