Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
2 |
|
psrbagconf1o.s |
⊢ 𝑆 = { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹 } |
3 |
|
psrbagleadd1.t |
⊢ 𝑇 = { 𝑧 ∈ 𝐷 ∣ 𝑧 ∘r ≤ ( 𝐹 ∘f + 𝐺 ) } |
4 |
|
elrabi |
⊢ ( 𝑋 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹 } → 𝑋 ∈ 𝐷 ) |
5 |
4 2
|
eleq2s |
⊢ ( 𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐷 ) |
6 |
5
|
3ad2ant3 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ∈ 𝐷 ) |
7 |
|
simp2 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝐺 ∈ 𝐷 ) |
8 |
1
|
psrbagaddcl |
⊢ ( ( 𝑋 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝑋 ∘f + 𝐺 ) ∈ 𝐷 ) |
9 |
6 7 8
|
syl2anc |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ( 𝑋 ∘f + 𝐺 ) ∈ 𝐷 ) |
10 |
1
|
psrbagf |
⊢ ( 𝑋 ∈ 𝐷 → 𝑋 : 𝐼 ⟶ ℕ0 ) |
11 |
6 10
|
syl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝑋 : 𝐼 ⟶ ℕ0 ) |
12 |
11
|
ffvelcdmda |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑥 ) ∈ ℕ0 ) |
13 |
12
|
nn0red |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑥 ) ∈ ℝ ) |
14 |
1
|
psrbagf |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 : 𝐼 ⟶ ℕ0 ) |
15 |
14
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝐹 : 𝐼 ⟶ ℕ0 ) |
16 |
15
|
ffvelcdmda |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℕ0 ) |
17 |
16
|
nn0red |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
18 |
1
|
psrbagf |
⊢ ( 𝐺 ∈ 𝐷 → 𝐺 : 𝐼 ⟶ ℕ0 ) |
19 |
18
|
3ad2ant2 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝐺 : 𝐼 ⟶ ℕ0 ) |
20 |
19
|
ffvelcdmda |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ∈ ℕ0 ) |
21 |
20
|
nn0red |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ∈ ℝ ) |
22 |
|
breq1 |
⊢ ( 𝑦 = 𝑋 → ( 𝑦 ∘r ≤ 𝐹 ↔ 𝑋 ∘r ≤ 𝐹 ) ) |
23 |
22 2
|
elrab2 |
⊢ ( 𝑋 ∈ 𝑆 ↔ ( 𝑋 ∈ 𝐷 ∧ 𝑋 ∘r ≤ 𝐹 ) ) |
24 |
23
|
simprbi |
⊢ ( 𝑋 ∈ 𝑆 → 𝑋 ∘r ≤ 𝐹 ) |
25 |
24
|
3ad2ant3 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ∘r ≤ 𝐹 ) |
26 |
10
|
ffnd |
⊢ ( 𝑋 ∈ 𝐷 → 𝑋 Fn 𝐼 ) |
27 |
5 26
|
syl |
⊢ ( 𝑋 ∈ 𝑆 → 𝑋 Fn 𝐼 ) |
28 |
27
|
3ad2ant3 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝑋 Fn 𝐼 ) |
29 |
14
|
ffnd |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼 ) |
30 |
29
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝐹 Fn 𝐼 ) |
31 |
|
id |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 ∈ 𝐷 ) |
32 |
31 29
|
fndmexd |
⊢ ( 𝐹 ∈ 𝐷 → 𝐼 ∈ V ) |
33 |
32
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝐼 ∈ V ) |
34 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
35 |
|
eqidd |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑥 ) = ( 𝑋 ‘ 𝑥 ) ) |
36 |
|
eqidd |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
37 |
28 30 33 33 34 35 36
|
ofrfval |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ( 𝑋 ∘r ≤ 𝐹 ↔ ∀ 𝑥 ∈ 𝐼 ( 𝑋 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) |
38 |
25 37
|
mpbid |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ∀ 𝑥 ∈ 𝐼 ( 𝑋 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
39 |
38
|
r19.21bi |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑥 ) ) |
40 |
13 17 21 39
|
leadd1dd |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑋 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ) |
41 |
40
|
ralrimiva |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ∀ 𝑥 ∈ 𝐼 ( ( 𝑋 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ) |
42 |
1
|
psrbagf |
⊢ ( ( 𝑋 ∘f + 𝐺 ) ∈ 𝐷 → ( 𝑋 ∘f + 𝐺 ) : 𝐼 ⟶ ℕ0 ) |
43 |
42
|
ffnd |
⊢ ( ( 𝑋 ∘f + 𝐺 ) ∈ 𝐷 → ( 𝑋 ∘f + 𝐺 ) Fn 𝐼 ) |
44 |
9 43
|
syl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ( 𝑋 ∘f + 𝐺 ) Fn 𝐼 ) |
45 |
1
|
psrbagaddcl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 ∘f + 𝐺 ) ∈ 𝐷 ) |
46 |
45
|
3adant3 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ( 𝐹 ∘f + 𝐺 ) ∈ 𝐷 ) |
47 |
1
|
psrbagf |
⊢ ( ( 𝐹 ∘f + 𝐺 ) ∈ 𝐷 → ( 𝐹 ∘f + 𝐺 ) : 𝐼 ⟶ ℕ0 ) |
48 |
47
|
ffnd |
⊢ ( ( 𝐹 ∘f + 𝐺 ) ∈ 𝐷 → ( 𝐹 ∘f + 𝐺 ) Fn 𝐼 ) |
49 |
46 48
|
syl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ( 𝐹 ∘f + 𝐺 ) Fn 𝐼 ) |
50 |
18
|
ffnd |
⊢ ( 𝐺 ∈ 𝐷 → 𝐺 Fn 𝐼 ) |
51 |
50
|
3ad2ant2 |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝐺 Fn 𝐼 ) |
52 |
|
eqidd |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
53 |
28 51 33 33 34 35 52
|
ofval |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑋 ∘f + 𝐺 ) ‘ 𝑥 ) = ( ( 𝑋 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ) |
54 |
30 51 33 33 34 36 52
|
ofval |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑥 ) = ( ( 𝐹 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ) |
55 |
44 49 33 33 34 53 54
|
ofrfval |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ( ( 𝑋 ∘f + 𝐺 ) ∘r ≤ ( 𝐹 ∘f + 𝐺 ) ↔ ∀ 𝑥 ∈ 𝐼 ( ( 𝑋 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ≤ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ) ) |
56 |
41 55
|
mpbird |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ( 𝑋 ∘f + 𝐺 ) ∘r ≤ ( 𝐹 ∘f + 𝐺 ) ) |
57 |
|
breq1 |
⊢ ( 𝑧 = ( 𝑋 ∘f + 𝐺 ) → ( 𝑧 ∘r ≤ ( 𝐹 ∘f + 𝐺 ) ↔ ( 𝑋 ∘f + 𝐺 ) ∘r ≤ ( 𝐹 ∘f + 𝐺 ) ) ) |
58 |
57 3
|
elrab2 |
⊢ ( ( 𝑋 ∘f + 𝐺 ) ∈ 𝑇 ↔ ( ( 𝑋 ∘f + 𝐺 ) ∈ 𝐷 ∧ ( 𝑋 ∘f + 𝐺 ) ∘r ≤ ( 𝐹 ∘f + 𝐺 ) ) ) |
59 |
9 56 58
|
sylanbrc |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ( 𝑋 ∘f + 𝐺 ) ∈ 𝑇 ) |