Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
2 |
|
df-ofr |
⊢ ∘r ≤ = { 〈 𝑎 , 𝑏 〉 ∣ ∀ 𝑐 ∈ ( dom 𝑎 ∩ dom 𝑏 ) ( 𝑎 ‘ 𝑐 ) ≤ ( 𝑏 ‘ 𝑐 ) } |
3 |
2
|
relopabiv |
⊢ Rel ∘r ≤ |
4 |
3
|
brrelex1i |
⊢ ( 𝐺 ∘r ≤ 𝐹 → 𝐺 ∈ V ) |
5 |
4
|
3ad2ant3 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → 𝐺 ∈ V ) |
6 |
|
simp2 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → 𝐺 : 𝐼 ⟶ ℕ0 ) |
7 |
|
frnnn0suppg |
⊢ ( ( 𝐺 ∈ V ∧ 𝐺 : 𝐼 ⟶ ℕ0 ) → ( 𝐺 supp 0 ) = ( ◡ 𝐺 “ ℕ ) ) |
8 |
5 6 7
|
syl2anc |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → ( 𝐺 supp 0 ) = ( ◡ 𝐺 “ ℕ ) ) |
9 |
|
eldifi |
⊢ ( 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) → 𝑥 ∈ 𝐼 ) |
10 |
|
simp3 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → 𝐺 ∘r ≤ 𝐹 ) |
11 |
6
|
ffnd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → 𝐺 Fn 𝐼 ) |
12 |
1
|
psrbagf |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 : 𝐼 ⟶ ℕ0 ) |
13 |
12
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → 𝐹 : 𝐼 ⟶ ℕ0 ) |
14 |
13
|
ffnd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → 𝐹 Fn 𝐼 ) |
15 |
|
simp1 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → 𝐹 ∈ 𝐷 ) |
16 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
17 |
|
eqidd |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
18 |
|
eqidd |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
19 |
11 14 5 15 16 17 18
|
ofrfvalg |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → ( 𝐺 ∘r ≤ 𝐹 ↔ ∀ 𝑥 ∈ 𝐼 ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
20 |
10 19
|
mpbid |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → ∀ 𝑥 ∈ 𝐼 ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
21 |
20
|
r19.21bi |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
22 |
9 21
|
sylan2 |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
23 |
|
frnnn0suppg |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 supp 0 ) = ( ◡ 𝐹 “ ℕ ) ) |
24 |
15 13 23
|
syl2anc |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → ( 𝐹 supp 0 ) = ( ◡ 𝐹 “ ℕ ) ) |
25 |
|
eqimss |
⊢ ( ( 𝐹 supp 0 ) = ( ◡ 𝐹 “ ℕ ) → ( 𝐹 supp 0 ) ⊆ ( ◡ 𝐹 “ ℕ ) ) |
26 |
24 25
|
syl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → ( 𝐹 supp 0 ) ⊆ ( ◡ 𝐹 “ ℕ ) ) |
27 |
|
c0ex |
⊢ 0 ∈ V |
28 |
27
|
a1i |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → 0 ∈ V ) |
29 |
13 26 15 28
|
suppssrg |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
30 |
22 29
|
breqtrd |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( 𝐺 ‘ 𝑥 ) ≤ 0 ) |
31 |
|
ffvelrn |
⊢ ( ( 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ∈ ℕ0 ) |
32 |
6 9 31
|
syl2an |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( 𝐺 ‘ 𝑥 ) ∈ ℕ0 ) |
33 |
32
|
nn0ge0d |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → 0 ≤ ( 𝐺 ‘ 𝑥 ) ) |
34 |
32
|
nn0red |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( 𝐺 ‘ 𝑥 ) ∈ ℝ ) |
35 |
|
0re |
⊢ 0 ∈ ℝ |
36 |
|
letri3 |
⊢ ( ( ( 𝐺 ‘ 𝑥 ) ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( 𝐺 ‘ 𝑥 ) = 0 ↔ ( ( 𝐺 ‘ 𝑥 ) ≤ 0 ∧ 0 ≤ ( 𝐺 ‘ 𝑥 ) ) ) ) |
37 |
34 35 36
|
sylancl |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( ( 𝐺 ‘ 𝑥 ) = 0 ↔ ( ( 𝐺 ‘ 𝑥 ) ≤ 0 ∧ 0 ≤ ( 𝐺 ‘ 𝑥 ) ) ) ) |
38 |
30 33 37
|
mpbir2and |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ◡ 𝐹 “ ℕ ) ) ) → ( 𝐺 ‘ 𝑥 ) = 0 ) |
39 |
6 38
|
suppss |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → ( 𝐺 supp 0 ) ⊆ ( ◡ 𝐹 “ ℕ ) ) |
40 |
8 39
|
eqsstrrd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) → ( ◡ 𝐺 “ ℕ ) ⊆ ( ◡ 𝐹 “ ℕ ) ) |