Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
2 |
|
simpr1 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐹 ∈ 𝐷 ) |
3 |
1
|
psrbag |
⊢ ( 𝐼 ∈ 𝑉 → ( 𝐹 ∈ 𝐷 ↔ ( 𝐹 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) ) |
4 |
3
|
adantr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐹 ∈ 𝐷 ↔ ( 𝐹 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) ) |
5 |
2 4
|
mpbid |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐹 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) |
6 |
5
|
simpld |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐹 : 𝐼 ⟶ ℕ0 ) |
7 |
6
|
ffnd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐹 Fn 𝐼 ) |
8 |
|
simpr2 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐺 : 𝐼 ⟶ ℕ0 ) |
9 |
8
|
ffnd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐺 Fn 𝐼 ) |
10 |
|
simpl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐼 ∈ 𝑉 ) |
11 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
12 |
7 9 10 10 11
|
offn |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐹 ∘f − 𝐺 ) Fn 𝐼 ) |
13 |
|
eqidd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
14 |
|
eqidd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
15 |
7 9 10 10 11 13 14
|
ofval |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ∘f − 𝐺 ) ‘ 𝑥 ) = ( ( 𝐹 ‘ 𝑥 ) − ( 𝐺 ‘ 𝑥 ) ) ) |
16 |
|
simpr3 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐺 ∘r ≤ 𝐹 ) |
17 |
9 7 10 10 11 14 13
|
ofrfval |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐺 ∘r ≤ 𝐹 ↔ ∀ 𝑥 ∈ 𝐼 ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
18 |
16 17
|
mpbid |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ∀ 𝑥 ∈ 𝐼 ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
19 |
18
|
r19.21bi |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
20 |
8
|
ffvelrnda |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ∈ ℕ0 ) |
21 |
6
|
ffvelrnda |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℕ0 ) |
22 |
|
nn0sub |
⊢ ( ( ( 𝐺 ‘ 𝑥 ) ∈ ℕ0 ∧ ( 𝐹 ‘ 𝑥 ) ∈ ℕ0 ) → ( ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ↔ ( ( 𝐹 ‘ 𝑥 ) − ( 𝐺 ‘ 𝑥 ) ) ∈ ℕ0 ) ) |
23 |
20 21 22
|
syl2anc |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐺 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ↔ ( ( 𝐹 ‘ 𝑥 ) − ( 𝐺 ‘ 𝑥 ) ) ∈ ℕ0 ) ) |
24 |
19 23
|
mpbid |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑥 ) − ( 𝐺 ‘ 𝑥 ) ) ∈ ℕ0 ) |
25 |
15 24
|
eqeltrd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ∘f − 𝐺 ) ‘ 𝑥 ) ∈ ℕ0 ) |
26 |
25
|
ralrimiva |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ∘f − 𝐺 ) ‘ 𝑥 ) ∈ ℕ0 ) |
27 |
|
ffnfv |
⊢ ( ( 𝐹 ∘f − 𝐺 ) : 𝐼 ⟶ ℕ0 ↔ ( ( 𝐹 ∘f − 𝐺 ) Fn 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ∘f − 𝐺 ) ‘ 𝑥 ) ∈ ℕ0 ) ) |
28 |
12 26 27
|
sylanbrc |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐹 ∘f − 𝐺 ) : 𝐼 ⟶ ℕ0 ) |
29 |
5
|
simprd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( ◡ 𝐹 “ ℕ ) ∈ Fin ) |
30 |
20
|
nn0ge0d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → 0 ≤ ( 𝐺 ‘ 𝑥 ) ) |
31 |
|
nn0re |
⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ ℕ0 → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
32 |
|
nn0re |
⊢ ( ( 𝐺 ‘ 𝑥 ) ∈ ℕ0 → ( 𝐺 ‘ 𝑥 ) ∈ ℝ ) |
33 |
|
subge02 |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) ∈ ℝ ∧ ( 𝐺 ‘ 𝑥 ) ∈ ℝ ) → ( 0 ≤ ( 𝐺 ‘ 𝑥 ) ↔ ( ( 𝐹 ‘ 𝑥 ) − ( 𝐺 ‘ 𝑥 ) ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
34 |
31 32 33
|
syl2an |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) ∈ ℕ0 ∧ ( 𝐺 ‘ 𝑥 ) ∈ ℕ0 ) → ( 0 ≤ ( 𝐺 ‘ 𝑥 ) ↔ ( ( 𝐹 ‘ 𝑥 ) − ( 𝐺 ‘ 𝑥 ) ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
35 |
21 20 34
|
syl2anc |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 0 ≤ ( 𝐺 ‘ 𝑥 ) ↔ ( ( 𝐹 ‘ 𝑥 ) − ( 𝐺 ‘ 𝑥 ) ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
36 |
30 35
|
mpbid |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑥 ) − ( 𝐺 ‘ 𝑥 ) ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
37 |
36
|
ralrimiva |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ‘ 𝑥 ) − ( 𝐺 ‘ 𝑥 ) ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
38 |
12 7 10 10 11 15 13
|
ofrfval |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( ( 𝐹 ∘f − 𝐺 ) ∘r ≤ 𝐹 ↔ ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ‘ 𝑥 ) − ( 𝐺 ‘ 𝑥 ) ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
39 |
37 38
|
mpbird |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐹 ∘f − 𝐺 ) ∘r ≤ 𝐹 ) |
40 |
1
|
psrbaglesuppOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ ( 𝐹 ∘f − 𝐺 ) : 𝐼 ⟶ ℕ0 ∧ ( 𝐹 ∘f − 𝐺 ) ∘r ≤ 𝐹 ) ) → ( ◡ ( 𝐹 ∘f − 𝐺 ) “ ℕ ) ⊆ ( ◡ 𝐹 “ ℕ ) ) |
41 |
10 2 28 39 40
|
syl13anc |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( ◡ ( 𝐹 ∘f − 𝐺 ) “ ℕ ) ⊆ ( ◡ 𝐹 “ ℕ ) ) |
42 |
29 41
|
ssfid |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( ◡ ( 𝐹 ∘f − 𝐺 ) “ ℕ ) ∈ Fin ) |
43 |
1
|
psrbag |
⊢ ( 𝐼 ∈ 𝑉 → ( ( 𝐹 ∘f − 𝐺 ) ∈ 𝐷 ↔ ( ( 𝐹 ∘f − 𝐺 ) : 𝐼 ⟶ ℕ0 ∧ ( ◡ ( 𝐹 ∘f − 𝐺 ) “ ℕ ) ∈ Fin ) ) ) |
44 |
43
|
adantr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( ( 𝐹 ∘f − 𝐺 ) ∈ 𝐷 ↔ ( ( 𝐹 ∘f − 𝐺 ) : 𝐼 ⟶ ℕ0 ∧ ( ◡ ( 𝐹 ∘f − 𝐺 ) “ ℕ ) ∈ Fin ) ) ) |
45 |
28 42 44
|
mpbir2and |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐹 ∘f − 𝐺 ) ∈ 𝐷 ) |
46 |
45 39
|
jca |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( ( 𝐹 ∘f − 𝐺 ) ∈ 𝐷 ∧ ( 𝐹 ∘f − 𝐺 ) ∘r ≤ 𝐹 ) ) |