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 |
|
eqid |
⊢ ( 𝑥 ∈ 𝑆 ↦ ( 𝐹 ∘f − 𝑥 ) ) = ( 𝑥 ∈ 𝑆 ↦ ( 𝐹 ∘f − 𝑥 ) ) |
4 |
1 2
|
psrbagconcl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑥 ∈ 𝑆 ) → ( 𝐹 ∘f − 𝑥 ) ∈ 𝑆 ) |
5 |
1 2
|
psrbagconcl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑧 ∈ 𝑆 ) → ( 𝐹 ∘f − 𝑧 ) ∈ 𝑆 ) |
6 |
1
|
psrbagf |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 : 𝐼 ⟶ ℕ0 ) |
7 |
6
|
adantr |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝐹 : 𝐼 ⟶ ℕ0 ) |
8 |
7
|
ffvelrnda |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑛 ) ∈ ℕ0 ) |
9 |
2
|
ssrab3 |
⊢ 𝑆 ⊆ 𝐷 |
10 |
9
|
sseli |
⊢ ( 𝑧 ∈ 𝑆 → 𝑧 ∈ 𝐷 ) |
11 |
10
|
adantl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑧 ∈ 𝑆 ) → 𝑧 ∈ 𝐷 ) |
12 |
1
|
psrbagf |
⊢ ( 𝑧 ∈ 𝐷 → 𝑧 : 𝐼 ⟶ ℕ0 ) |
13 |
11 12
|
syl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑧 ∈ 𝑆 ) → 𝑧 : 𝐼 ⟶ ℕ0 ) |
14 |
13
|
adantrl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝑧 : 𝐼 ⟶ ℕ0 ) |
15 |
14
|
ffvelrnda |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝑧 ‘ 𝑛 ) ∈ ℕ0 ) |
16 |
|
simprl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝑥 ∈ 𝑆 ) |
17 |
9 16
|
sselid |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝑥 ∈ 𝐷 ) |
18 |
1
|
psrbagf |
⊢ ( 𝑥 ∈ 𝐷 → 𝑥 : 𝐼 ⟶ ℕ0 ) |
19 |
17 18
|
syl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝑥 : 𝐼 ⟶ ℕ0 ) |
20 |
19
|
ffvelrnda |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝑥 ‘ 𝑛 ) ∈ ℕ0 ) |
21 |
|
nn0cn |
⊢ ( ( 𝐹 ‘ 𝑛 ) ∈ ℕ0 → ( 𝐹 ‘ 𝑛 ) ∈ ℂ ) |
22 |
|
nn0cn |
⊢ ( ( 𝑧 ‘ 𝑛 ) ∈ ℕ0 → ( 𝑧 ‘ 𝑛 ) ∈ ℂ ) |
23 |
|
nn0cn |
⊢ ( ( 𝑥 ‘ 𝑛 ) ∈ ℕ0 → ( 𝑥 ‘ 𝑛 ) ∈ ℂ ) |
24 |
|
subsub23 |
⊢ ( ( ( 𝐹 ‘ 𝑛 ) ∈ ℂ ∧ ( 𝑧 ‘ 𝑛 ) ∈ ℂ ∧ ( 𝑥 ‘ 𝑛 ) ∈ ℂ ) → ( ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) = ( 𝑥 ‘ 𝑛 ) ↔ ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) = ( 𝑧 ‘ 𝑛 ) ) ) |
25 |
21 22 23 24
|
syl3an |
⊢ ( ( ( 𝐹 ‘ 𝑛 ) ∈ ℕ0 ∧ ( 𝑧 ‘ 𝑛 ) ∈ ℕ0 ∧ ( 𝑥 ‘ 𝑛 ) ∈ ℕ0 ) → ( ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) = ( 𝑥 ‘ 𝑛 ) ↔ ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) = ( 𝑧 ‘ 𝑛 ) ) ) |
26 |
8 15 20 25
|
syl3anc |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) = ( 𝑥 ‘ 𝑛 ) ↔ ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) = ( 𝑧 ‘ 𝑛 ) ) ) |
27 |
|
eqcom |
⊢ ( ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) ↔ ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) = ( 𝑥 ‘ 𝑛 ) ) |
28 |
|
eqcom |
⊢ ( ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) ↔ ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) = ( 𝑧 ‘ 𝑛 ) ) |
29 |
26 27 28
|
3bitr4g |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) ↔ ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) ) ) |
30 |
6
|
ffnd |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼 ) |
31 |
30
|
adantr |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝐹 Fn 𝐼 ) |
32 |
13
|
ffnd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑧 ∈ 𝑆 ) → 𝑧 Fn 𝐼 ) |
33 |
32
|
adantrl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝑧 Fn 𝐼 ) |
34 |
19
|
ffnd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝑥 Fn 𝐼 ) |
35 |
16 34
|
fndmexd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝐼 ∈ V ) |
36 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
37 |
|
eqidd |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑛 ) ) |
38 |
|
eqidd |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝑧 ‘ 𝑛 ) = ( 𝑧 ‘ 𝑛 ) ) |
39 |
31 33 35 35 36 37 38
|
ofval |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( 𝐹 ∘f − 𝑧 ) ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) ) |
40 |
39
|
eqeq2d |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑧 ) ‘ 𝑛 ) ↔ ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) ) ) |
41 |
|
eqidd |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝑥 ‘ 𝑛 ) = ( 𝑥 ‘ 𝑛 ) ) |
42 |
31 34 35 35 36 37 41
|
ofval |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( 𝐹 ∘f − 𝑥 ) ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) ) |
43 |
42
|
eqeq2d |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑥 ) ‘ 𝑛 ) ↔ ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) ) ) |
44 |
29 40 43
|
3bitr4d |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑧 ) ‘ 𝑛 ) ↔ ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑥 ) ‘ 𝑛 ) ) ) |
45 |
44
|
ralbidva |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ∀ 𝑛 ∈ 𝐼 ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑧 ) ‘ 𝑛 ) ↔ ∀ 𝑛 ∈ 𝐼 ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑥 ) ‘ 𝑛 ) ) ) |
46 |
5
|
adantrl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑧 ) ∈ 𝑆 ) |
47 |
9 46
|
sselid |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑧 ) ∈ 𝐷 ) |
48 |
1
|
psrbagf |
⊢ ( ( 𝐹 ∘f − 𝑧 ) ∈ 𝐷 → ( 𝐹 ∘f − 𝑧 ) : 𝐼 ⟶ ℕ0 ) |
49 |
47 48
|
syl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑧 ) : 𝐼 ⟶ ℕ0 ) |
50 |
49
|
ffnd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑧 ) Fn 𝐼 ) |
51 |
|
eqfnfv |
⊢ ( ( 𝑥 Fn 𝐼 ∧ ( 𝐹 ∘f − 𝑧 ) Fn 𝐼 ) → ( 𝑥 = ( 𝐹 ∘f − 𝑧 ) ↔ ∀ 𝑛 ∈ 𝐼 ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑧 ) ‘ 𝑛 ) ) ) |
52 |
34 50 51
|
syl2anc |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 = ( 𝐹 ∘f − 𝑧 ) ↔ ∀ 𝑛 ∈ 𝐼 ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑧 ) ‘ 𝑛 ) ) ) |
53 |
9 4
|
sselid |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑥 ∈ 𝑆 ) → ( 𝐹 ∘f − 𝑥 ) ∈ 𝐷 ) |
54 |
1
|
psrbagf |
⊢ ( ( 𝐹 ∘f − 𝑥 ) ∈ 𝐷 → ( 𝐹 ∘f − 𝑥 ) : 𝐼 ⟶ ℕ0 ) |
55 |
53 54
|
syl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑥 ∈ 𝑆 ) → ( 𝐹 ∘f − 𝑥 ) : 𝐼 ⟶ ℕ0 ) |
56 |
55
|
ffnd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑥 ∈ 𝑆 ) → ( 𝐹 ∘f − 𝑥 ) Fn 𝐼 ) |
57 |
56
|
adantrr |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑥 ) Fn 𝐼 ) |
58 |
|
eqfnfv |
⊢ ( ( 𝑧 Fn 𝐼 ∧ ( 𝐹 ∘f − 𝑥 ) Fn 𝐼 ) → ( 𝑧 = ( 𝐹 ∘f − 𝑥 ) ↔ ∀ 𝑛 ∈ 𝐼 ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑥 ) ‘ 𝑛 ) ) ) |
59 |
33 57 58
|
syl2anc |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑧 = ( 𝐹 ∘f − 𝑥 ) ↔ ∀ 𝑛 ∈ 𝐼 ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑥 ) ‘ 𝑛 ) ) ) |
60 |
45 52 59
|
3bitr4d |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 = ( 𝐹 ∘f − 𝑧 ) ↔ 𝑧 = ( 𝐹 ∘f − 𝑥 ) ) ) |
61 |
3 4 5 60
|
f1o2d |
⊢ ( 𝐹 ∈ 𝐷 → ( 𝑥 ∈ 𝑆 ↦ ( 𝐹 ∘f − 𝑥 ) ) : 𝑆 –1-1-onto→ 𝑆 ) |