| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fargshift.g |
⊢ 𝐺 = ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↦ ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) |
| 2 |
|
f1f |
⊢ ( 𝐹 : ( 1 ... 𝑁 ) –1-1→ dom 𝐸 → 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) |
| 3 |
1
|
fargshiftf |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) → 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) |
| 4 |
2 3
|
sylan2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –1-1→ dom 𝐸 ) → 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) |
| 5 |
|
ffn |
⊢ ( 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 → 𝐹 Fn ( 1 ... 𝑁 ) ) |
| 6 |
|
fseq1hash |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 Fn ( 1 ... 𝑁 ) ) → ( ♯ ‘ 𝐹 ) = 𝑁 ) |
| 7 |
5 6
|
sylan2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) → ( ♯ ‘ 𝐹 ) = 𝑁 ) |
| 8 |
2 7
|
sylan2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –1-1→ dom 𝐸 ) → ( ♯ ‘ 𝐹 ) = 𝑁 ) |
| 9 |
|
eleq1 |
⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0 ) ) |
| 10 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( 1 ... ( ♯ ‘ 𝐹 ) ) = ( 1 ... 𝑁 ) ) |
| 11 |
|
f1eq2 |
⊢ ( ( 1 ... ( ♯ ‘ 𝐹 ) ) = ( 1 ... 𝑁 ) → ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐸 ↔ 𝐹 : ( 1 ... 𝑁 ) –1-1→ dom 𝐸 ) ) |
| 12 |
10 11
|
syl |
⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐸 ↔ 𝐹 : ( 1 ... 𝑁 ) –1-1→ dom 𝐸 ) ) |
| 13 |
9 12
|
anbi12d |
⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐸 ) ↔ ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –1-1→ dom 𝐸 ) ) ) |
| 14 |
|
dff13 |
⊢ ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐸 ↔ ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ∀ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∀ 𝑙 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) ) ) |
| 15 |
|
fz0add1fz1 |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) |
| 16 |
|
fz0add1fz1 |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) |
| 17 |
15 16
|
anim12dan |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) |
| 18 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑦 + 1 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ ( 𝑦 + 1 ) ) ) |
| 19 |
18
|
eqeq1d |
⊢ ( 𝑘 = ( 𝑦 + 1 ) → ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) ↔ ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ 𝑙 ) ) ) |
| 20 |
|
eqeq1 |
⊢ ( 𝑘 = ( 𝑦 + 1 ) → ( 𝑘 = 𝑙 ↔ ( 𝑦 + 1 ) = 𝑙 ) ) |
| 21 |
19 20
|
imbi12d |
⊢ ( 𝑘 = ( 𝑦 + 1 ) → ( ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) ↔ ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ 𝑙 ) → ( 𝑦 + 1 ) = 𝑙 ) ) ) |
| 22 |
|
fveq2 |
⊢ ( 𝑙 = ( 𝑧 + 1 ) → ( 𝐹 ‘ 𝑙 ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) |
| 23 |
22
|
eqeq2d |
⊢ ( 𝑙 = ( 𝑧 + 1 ) → ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ 𝑙 ) ↔ ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) ) |
| 24 |
|
eqeq2 |
⊢ ( 𝑙 = ( 𝑧 + 1 ) → ( ( 𝑦 + 1 ) = 𝑙 ↔ ( 𝑦 + 1 ) = ( 𝑧 + 1 ) ) ) |
| 25 |
23 24
|
imbi12d |
⊢ ( 𝑙 = ( 𝑧 + 1 ) → ( ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ 𝑙 ) → ( 𝑦 + 1 ) = 𝑙 ) ↔ ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) → ( 𝑦 + 1 ) = ( 𝑧 + 1 ) ) ) ) |
| 26 |
21 25
|
rspc2v |
⊢ ( ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) → ( ∀ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∀ 𝑙 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) → ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) → ( 𝑦 + 1 ) = ( 𝑧 + 1 ) ) ) ) |
| 27 |
26
|
adantl |
⊢ ( ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) → ( ∀ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∀ 𝑙 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) → ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) → ( 𝑦 + 1 ) = ( 𝑧 + 1 ) ) ) ) |
| 28 |
1
|
fargshiftfv |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) → ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑦 + 1 ) ) ) ) |
| 29 |
28
|
expcom |
⊢ ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 → ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑦 + 1 ) ) ) ) ) |
| 30 |
29
|
com13 |
⊢ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑦 + 1 ) ) ) ) ) |
| 31 |
30
|
adantr |
⊢ ( ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑦 + 1 ) ) ) ) ) |
| 32 |
31
|
impcom |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑦 + 1 ) ) ) ) |
| 33 |
32
|
impcom |
⊢ ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑦 + 1 ) ) ) |
| 34 |
1
|
fargshiftfv |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) → ( 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) ) |
| 35 |
34
|
expcom |
⊢ ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 → ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) ) ) |
| 36 |
35
|
com13 |
⊢ ( 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) ) ) |
| 37 |
36
|
adantl |
⊢ ( ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) ) ) |
| 38 |
37
|
impcom |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) ) |
| 39 |
38
|
impcom |
⊢ ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) |
| 40 |
33 39
|
eqeq12d |
⊢ ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) ↔ ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) ) |
| 41 |
40
|
adantr |
⊢ ( ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) ↔ ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) ) |
| 42 |
41
|
adantr |
⊢ ( ( ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) ∧ ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) → ( 𝑦 + 1 ) = ( 𝑧 + 1 ) ) ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) ↔ ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) ) |
| 43 |
|
elfzoelz |
⊢ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → 𝑦 ∈ ℤ ) |
| 44 |
43
|
zcnd |
⊢ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → 𝑦 ∈ ℂ ) |
| 45 |
44
|
adantr |
⊢ ( ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑦 ∈ ℂ ) |
| 46 |
45
|
adantl |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → 𝑦 ∈ ℂ ) |
| 47 |
|
elfzoelz |
⊢ ( 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → 𝑧 ∈ ℤ ) |
| 48 |
47
|
zcnd |
⊢ ( 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → 𝑧 ∈ ℂ ) |
| 49 |
48
|
adantl |
⊢ ( ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑧 ∈ ℂ ) |
| 50 |
49
|
adantl |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → 𝑧 ∈ ℂ ) |
| 51 |
|
1cnd |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → 1 ∈ ℂ ) |
| 52 |
46 50 51
|
3jca |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ ) ) |
| 53 |
52
|
adantl |
⊢ ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) → ( 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ ) ) |
| 54 |
53
|
adantr |
⊢ ( ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ ) ) |
| 55 |
|
addcan2 |
⊢ ( ( 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑦 + 1 ) = ( 𝑧 + 1 ) ↔ 𝑦 = 𝑧 ) ) |
| 56 |
54 55
|
syl |
⊢ ( ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑦 + 1 ) = ( 𝑧 + 1 ) ↔ 𝑦 = 𝑧 ) ) |
| 57 |
56
|
imbi2d |
⊢ ( ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) → ( 𝑦 + 1 ) = ( 𝑧 + 1 ) ) ↔ ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) → 𝑦 = 𝑧 ) ) ) |
| 58 |
57
|
biimpa |
⊢ ( ( ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) ∧ ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) → ( 𝑦 + 1 ) = ( 𝑧 + 1 ) ) ) → ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) → 𝑦 = 𝑧 ) ) |
| 59 |
42 58
|
sylbid |
⊢ ( ( ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) ∧ ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) → ( 𝑦 + 1 ) = ( 𝑧 + 1 ) ) ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) |
| 60 |
59
|
ex |
⊢ ( ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝐹 ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ‘ ( 𝑧 + 1 ) ) → ( 𝑦 + 1 ) = ( 𝑧 + 1 ) ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) |
| 61 |
27 60
|
syld |
⊢ ( ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) → ( ∀ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∀ 𝑙 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) |
| 62 |
61
|
exp31 |
⊢ ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 → ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) → ( ∀ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∀ 𝑙 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) ) ) |
| 63 |
62
|
com24 |
⊢ ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 → ( ∀ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∀ 𝑙 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) → ( ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) ) ) |
| 64 |
63
|
imp |
⊢ ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ∀ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∀ 𝑙 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) ) → ( ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) ) |
| 65 |
64
|
com13 |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝑦 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∧ ( 𝑧 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ∀ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∀ 𝑙 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) ) |
| 66 |
17 65
|
mpd |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ∀ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∀ 𝑙 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) |
| 67 |
66
|
expcom |
⊢ ( ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ∀ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∀ 𝑙 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) ) |
| 68 |
67
|
com13 |
⊢ ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ∀ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∀ 𝑙 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) ) → ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( ( 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) ) |
| 69 |
68
|
ralrimdvv |
⊢ ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ∀ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ∀ 𝑙 ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑙 ) → 𝑘 = 𝑙 ) ) → ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ∀ 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∀ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) |
| 70 |
14 69
|
sylbi |
⊢ ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐸 → ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ∀ 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∀ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) |
| 71 |
70
|
impcom |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐸 ) → ∀ 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∀ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) |
| 72 |
13 71
|
syl6bir |
⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –1-1→ dom 𝐸 ) → ∀ 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∀ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) |
| 73 |
8 72
|
mpcom |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –1-1→ dom 𝐸 ) → ∀ 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∀ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) |
| 74 |
|
dff13 |
⊢ ( 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐸 ↔ ( 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ∀ 𝑦 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∀ 𝑧 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) |
| 75 |
4 73 74
|
sylanbrc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –1-1→ dom 𝐸 ) → 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐸 ) |