| Step |
Hyp |
Ref |
Expression |
| 1 |
|
simpr |
⊢ ( ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) ∧ ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ) |
| 2 |
|
elmapi |
⊢ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) → 𝑎 : ( 1 ... 3 ) ⟶ ℕ0 ) |
| 3 |
|
df-3 |
⊢ 3 = ( 2 + 1 ) |
| 4 |
|
ssid |
⊢ ( 1 ... 3 ) ⊆ ( 1 ... 3 ) |
| 5 |
3 4
|
jm2.27dlem5 |
⊢ ( 1 ... 2 ) ⊆ ( 1 ... 3 ) |
| 6 |
|
2nn |
⊢ 2 ∈ ℕ |
| 7 |
6
|
jm2.27dlem3 |
⊢ 2 ∈ ( 1 ... 2 ) |
| 8 |
5 7
|
sselii |
⊢ 2 ∈ ( 1 ... 3 ) |
| 9 |
|
ffvelrn |
⊢ ( ( 𝑎 : ( 1 ... 3 ) ⟶ ℕ0 ∧ 2 ∈ ( 1 ... 3 ) ) → ( 𝑎 ‘ 2 ) ∈ ℕ0 ) |
| 10 |
2 8 9
|
sylancl |
⊢ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) → ( 𝑎 ‘ 2 ) ∈ ℕ0 ) |
| 11 |
10
|
adantr |
⊢ ( ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) ∧ ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑎 ‘ 2 ) ∈ ℕ0 ) |
| 12 |
|
3nn |
⊢ 3 ∈ ℕ |
| 13 |
12
|
jm2.27dlem3 |
⊢ 3 ∈ ( 1 ... 3 ) |
| 14 |
|
ffvelrn |
⊢ ( ( 𝑎 : ( 1 ... 3 ) ⟶ ℕ0 ∧ 3 ∈ ( 1 ... 3 ) ) → ( 𝑎 ‘ 3 ) ∈ ℕ0 ) |
| 15 |
2 13 14
|
sylancl |
⊢ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) → ( 𝑎 ‘ 3 ) ∈ ℕ0 ) |
| 16 |
15
|
adantr |
⊢ ( ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) ∧ ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑎 ‘ 3 ) ∈ ℕ0 ) |
| 17 |
|
rmxdiophlem |
⊢ ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑎 ‘ 2 ) ∈ ℕ0 ∧ ( 𝑎 ‘ 3 ) ∈ ℕ0 ) → ( ( 𝑎 ‘ 3 ) = ( ( 𝑎 ‘ 1 ) Xrm ( 𝑎 ‘ 2 ) ) ↔ ∃ 𝑏 ∈ ℕ0 ( 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
| 18 |
1 11 16 17
|
syl3anc |
⊢ ( ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) ∧ ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( ( 𝑎 ‘ 3 ) = ( ( 𝑎 ‘ 1 ) Xrm ( 𝑎 ‘ 2 ) ) ↔ ∃ 𝑏 ∈ ℕ0 ( 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
| 19 |
18
|
pm5.32da |
⊢ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) → ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑎 ‘ 3 ) = ( ( 𝑎 ‘ 1 ) Xrm ( 𝑎 ‘ 2 ) ) ) ↔ ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ∃ 𝑏 ∈ ℕ0 ( 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) ) |
| 20 |
|
anass |
⊢ ( ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ↔ ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
| 21 |
20
|
rexbii |
⊢ ( ∃ 𝑏 ∈ ℕ0 ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ↔ ∃ 𝑏 ∈ ℕ0 ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
| 22 |
|
r19.42v |
⊢ ( ∃ 𝑏 ∈ ℕ0 ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ↔ ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ∃ 𝑏 ∈ ℕ0 ( 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
| 23 |
21 22
|
bitr2i |
⊢ ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ∃ 𝑏 ∈ ℕ0 ( 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ↔ ∃ 𝑏 ∈ ℕ0 ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) |
| 24 |
19 23
|
bitrdi |
⊢ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) → ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑎 ‘ 3 ) = ( ( 𝑎 ‘ 1 ) Xrm ( 𝑎 ‘ 2 ) ) ) ↔ ∃ 𝑏 ∈ ℕ0 ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ) ) |
| 25 |
24
|
rabbiia |
⊢ { 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) ∣ ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑎 ‘ 3 ) = ( ( 𝑎 ‘ 1 ) Xrm ( 𝑎 ‘ 2 ) ) ) } = { 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) ∣ ∃ 𝑏 ∈ ℕ0 ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) } |
| 26 |
|
3nn0 |
⊢ 3 ∈ ℕ0 |
| 27 |
|
vex |
⊢ 𝑐 ∈ V |
| 28 |
27
|
resex |
⊢ ( 𝑐 ↾ ( 1 ... 3 ) ) ∈ V |
| 29 |
|
fvex |
⊢ ( 𝑐 ‘ 4 ) ∈ V |
| 30 |
|
df-2 |
⊢ 2 = ( 1 + 1 ) |
| 31 |
30 5
|
jm2.27dlem5 |
⊢ ( 1 ... 1 ) ⊆ ( 1 ... 3 ) |
| 32 |
|
1nn |
⊢ 1 ∈ ℕ |
| 33 |
32
|
jm2.27dlem3 |
⊢ 1 ∈ ( 1 ... 1 ) |
| 34 |
31 33
|
sselii |
⊢ 1 ∈ ( 1 ... 3 ) |
| 35 |
34
|
jm2.27dlem1 |
⊢ ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) → ( 𝑎 ‘ 1 ) = ( 𝑐 ‘ 1 ) ) |
| 36 |
35
|
eleq1d |
⊢ ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) → ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
| 37 |
36
|
adantr |
⊢ ( ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) ∧ 𝑏 = ( 𝑐 ‘ 4 ) ) → ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
| 38 |
|
simpr |
⊢ ( ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) ∧ 𝑏 = ( 𝑐 ‘ 4 ) ) → 𝑏 = ( 𝑐 ‘ 4 ) ) |
| 39 |
8
|
jm2.27dlem1 |
⊢ ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) → ( 𝑎 ‘ 2 ) = ( 𝑐 ‘ 2 ) ) |
| 40 |
35 39
|
oveq12d |
⊢ ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) → ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) |
| 41 |
40
|
adantr |
⊢ ( ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) ∧ 𝑏 = ( 𝑐 ‘ 4 ) ) → ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) |
| 42 |
38 41
|
eqeq12d |
⊢ ( ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) ∧ 𝑏 = ( 𝑐 ‘ 4 ) ) → ( 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ↔ ( 𝑐 ‘ 4 ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) ) |
| 43 |
37 42
|
anbi12d |
⊢ ( ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) ∧ 𝑏 = ( 𝑐 ‘ 4 ) ) → ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ↔ ( ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑐 ‘ 4 ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) ) ) |
| 44 |
13
|
jm2.27dlem1 |
⊢ ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) → ( 𝑎 ‘ 3 ) = ( 𝑐 ‘ 3 ) ) |
| 45 |
44
|
oveq1d |
⊢ ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) → ( ( 𝑎 ‘ 3 ) ↑ 2 ) = ( ( 𝑐 ‘ 3 ) ↑ 2 ) ) |
| 46 |
45
|
adantr |
⊢ ( ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) ∧ 𝑏 = ( 𝑐 ‘ 4 ) ) → ( ( 𝑎 ‘ 3 ) ↑ 2 ) = ( ( 𝑐 ‘ 3 ) ↑ 2 ) ) |
| 47 |
35
|
oveq1d |
⊢ ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) → ( ( 𝑎 ‘ 1 ) ↑ 2 ) = ( ( 𝑐 ‘ 1 ) ↑ 2 ) ) |
| 48 |
47
|
oveq1d |
⊢ ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) → ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) = ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) ) |
| 49 |
|
oveq1 |
⊢ ( 𝑏 = ( 𝑐 ‘ 4 ) → ( 𝑏 ↑ 2 ) = ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) |
| 50 |
48 49
|
oveqan12d |
⊢ ( ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) ∧ 𝑏 = ( 𝑐 ‘ 4 ) ) → ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) = ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) |
| 51 |
46 50
|
oveq12d |
⊢ ( ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) ∧ 𝑏 = ( 𝑐 ‘ 4 ) ) → ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) ) |
| 52 |
51
|
eqeq1d |
⊢ ( ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) ∧ 𝑏 = ( 𝑐 ‘ 4 ) ) → ( ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ↔ ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) = 1 ) ) |
| 53 |
43 52
|
anbi12d |
⊢ ( ( 𝑎 = ( 𝑐 ↾ ( 1 ... 3 ) ) ∧ 𝑏 = ( 𝑐 ‘ 4 ) ) → ( ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ↔ ( ( ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑐 ‘ 4 ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) ∧ ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) = 1 ) ) ) |
| 54 |
28 29 53
|
sbc2ie |
⊢ ( [ ( 𝑐 ↾ ( 1 ... 3 ) ) / 𝑎 ] [ ( 𝑐 ‘ 4 ) / 𝑏 ] ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) ↔ ( ( ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑐 ‘ 4 ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) ∧ ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) = 1 ) ) |
| 55 |
54
|
rabbii |
⊢ { 𝑐 ∈ ( ℕ0 ↑m ( 1 ... 4 ) ) ∣ [ ( 𝑐 ↾ ( 1 ... 3 ) ) / 𝑎 ] [ ( 𝑐 ‘ 4 ) / 𝑏 ] ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) } = { 𝑐 ∈ ( ℕ0 ↑m ( 1 ... 4 ) ) ∣ ( ( ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑐 ‘ 4 ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) ∧ ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) = 1 ) } |
| 56 |
|
4nn0 |
⊢ 4 ∈ ℕ0 |
| 57 |
|
rmydioph |
⊢ { 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) ∣ ( ( 𝑏 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑏 ‘ 3 ) = ( ( 𝑏 ‘ 1 ) Yrm ( 𝑏 ‘ 2 ) ) ) } ∈ ( Dioph ‘ 3 ) |
| 58 |
|
simp1 |
⊢ ( ( ( 𝑏 ‘ 1 ) = ( 𝑐 ‘ 1 ) ∧ ( 𝑏 ‘ 2 ) = ( 𝑐 ‘ 2 ) ∧ ( 𝑏 ‘ 3 ) = ( 𝑐 ‘ 4 ) ) → ( 𝑏 ‘ 1 ) = ( 𝑐 ‘ 1 ) ) |
| 59 |
58
|
eleq1d |
⊢ ( ( ( 𝑏 ‘ 1 ) = ( 𝑐 ‘ 1 ) ∧ ( 𝑏 ‘ 2 ) = ( 𝑐 ‘ 2 ) ∧ ( 𝑏 ‘ 3 ) = ( 𝑐 ‘ 4 ) ) → ( ( 𝑏 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
| 60 |
|
simp3 |
⊢ ( ( ( 𝑏 ‘ 1 ) = ( 𝑐 ‘ 1 ) ∧ ( 𝑏 ‘ 2 ) = ( 𝑐 ‘ 2 ) ∧ ( 𝑏 ‘ 3 ) = ( 𝑐 ‘ 4 ) ) → ( 𝑏 ‘ 3 ) = ( 𝑐 ‘ 4 ) ) |
| 61 |
|
simp2 |
⊢ ( ( ( 𝑏 ‘ 1 ) = ( 𝑐 ‘ 1 ) ∧ ( 𝑏 ‘ 2 ) = ( 𝑐 ‘ 2 ) ∧ ( 𝑏 ‘ 3 ) = ( 𝑐 ‘ 4 ) ) → ( 𝑏 ‘ 2 ) = ( 𝑐 ‘ 2 ) ) |
| 62 |
58 61
|
oveq12d |
⊢ ( ( ( 𝑏 ‘ 1 ) = ( 𝑐 ‘ 1 ) ∧ ( 𝑏 ‘ 2 ) = ( 𝑐 ‘ 2 ) ∧ ( 𝑏 ‘ 3 ) = ( 𝑐 ‘ 4 ) ) → ( ( 𝑏 ‘ 1 ) Yrm ( 𝑏 ‘ 2 ) ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) |
| 63 |
60 62
|
eqeq12d |
⊢ ( ( ( 𝑏 ‘ 1 ) = ( 𝑐 ‘ 1 ) ∧ ( 𝑏 ‘ 2 ) = ( 𝑐 ‘ 2 ) ∧ ( 𝑏 ‘ 3 ) = ( 𝑐 ‘ 4 ) ) → ( ( 𝑏 ‘ 3 ) = ( ( 𝑏 ‘ 1 ) Yrm ( 𝑏 ‘ 2 ) ) ↔ ( 𝑐 ‘ 4 ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) ) |
| 64 |
59 63
|
anbi12d |
⊢ ( ( ( 𝑏 ‘ 1 ) = ( 𝑐 ‘ 1 ) ∧ ( 𝑏 ‘ 2 ) = ( 𝑐 ‘ 2 ) ∧ ( 𝑏 ‘ 3 ) = ( 𝑐 ‘ 4 ) ) → ( ( ( 𝑏 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑏 ‘ 3 ) = ( ( 𝑏 ‘ 1 ) Yrm ( 𝑏 ‘ 2 ) ) ) ↔ ( ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑐 ‘ 4 ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) ) ) |
| 65 |
|
df-4 |
⊢ 4 = ( 3 + 1 ) |
| 66 |
|
ssid |
⊢ ( 1 ... 4 ) ⊆ ( 1 ... 4 ) |
| 67 |
65 66
|
jm2.27dlem5 |
⊢ ( 1 ... 3 ) ⊆ ( 1 ... 4 ) |
| 68 |
3 67
|
jm2.27dlem5 |
⊢ ( 1 ... 2 ) ⊆ ( 1 ... 4 ) |
| 69 |
30 68
|
jm2.27dlem5 |
⊢ ( 1 ... 1 ) ⊆ ( 1 ... 4 ) |
| 70 |
69 33
|
sselii |
⊢ 1 ∈ ( 1 ... 4 ) |
| 71 |
68 7
|
sselii |
⊢ 2 ∈ ( 1 ... 4 ) |
| 72 |
|
4nn |
⊢ 4 ∈ ℕ |
| 73 |
72
|
jm2.27dlem3 |
⊢ 4 ∈ ( 1 ... 4 ) |
| 74 |
64 70 71 73
|
rabren3dioph |
⊢ ( ( 4 ∈ ℕ0 ∧ { 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) ∣ ( ( 𝑏 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑏 ‘ 3 ) = ( ( 𝑏 ‘ 1 ) Yrm ( 𝑏 ‘ 2 ) ) ) } ∈ ( Dioph ‘ 3 ) ) → { 𝑐 ∈ ( ℕ0 ↑m ( 1 ... 4 ) ) ∣ ( ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑐 ‘ 4 ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) } ∈ ( Dioph ‘ 4 ) ) |
| 75 |
56 57 74
|
mp2an |
⊢ { 𝑐 ∈ ( ℕ0 ↑m ( 1 ... 4 ) ) ∣ ( ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑐 ‘ 4 ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) } ∈ ( Dioph ‘ 4 ) |
| 76 |
|
ovex |
⊢ ( 1 ... 4 ) ∈ V |
| 77 |
67 13
|
sselii |
⊢ 3 ∈ ( 1 ... 4 ) |
| 78 |
|
mzpproj |
⊢ ( ( ( 1 ... 4 ) ∈ V ∧ 3 ∈ ( 1 ... 4 ) ) → ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( 𝑐 ‘ 3 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) |
| 79 |
76 77 78
|
mp2an |
⊢ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( 𝑐 ‘ 3 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) |
| 80 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
| 81 |
|
mzpexpmpt |
⊢ ( ( ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( 𝑐 ‘ 3 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ∧ 2 ∈ ℕ0 ) → ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( 𝑐 ‘ 3 ) ↑ 2 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) |
| 82 |
79 80 81
|
mp2an |
⊢ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( 𝑐 ‘ 3 ) ↑ 2 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) |
| 83 |
|
mzpproj |
⊢ ( ( ( 1 ... 4 ) ∈ V ∧ 1 ∈ ( 1 ... 4 ) ) → ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( 𝑐 ‘ 1 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) |
| 84 |
76 70 83
|
mp2an |
⊢ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( 𝑐 ‘ 1 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) |
| 85 |
|
mzpexpmpt |
⊢ ( ( ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( 𝑐 ‘ 1 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ∧ 2 ∈ ℕ0 ) → ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( 𝑐 ‘ 1 ) ↑ 2 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) |
| 86 |
84 80 85
|
mp2an |
⊢ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( 𝑐 ‘ 1 ) ↑ 2 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) |
| 87 |
|
1z |
⊢ 1 ∈ ℤ |
| 88 |
|
mzpconstmpt |
⊢ ( ( ( 1 ... 4 ) ∈ V ∧ 1 ∈ ℤ ) → ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ 1 ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) |
| 89 |
76 87 88
|
mp2an |
⊢ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ 1 ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) |
| 90 |
|
mzpsubmpt |
⊢ ( ( ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( 𝑐 ‘ 1 ) ↑ 2 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ∧ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ 1 ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) → ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) |
| 91 |
86 89 90
|
mp2an |
⊢ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) |
| 92 |
|
mzpproj |
⊢ ( ( ( 1 ... 4 ) ∈ V ∧ 4 ∈ ( 1 ... 4 ) ) → ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( 𝑐 ‘ 4 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) |
| 93 |
76 73 92
|
mp2an |
⊢ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( 𝑐 ‘ 4 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) |
| 94 |
|
mzpexpmpt |
⊢ ( ( ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( 𝑐 ‘ 4 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ∧ 2 ∈ ℕ0 ) → ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) |
| 95 |
93 80 94
|
mp2an |
⊢ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) |
| 96 |
|
mzpmulmpt |
⊢ ( ( ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ∧ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) → ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) |
| 97 |
91 95 96
|
mp2an |
⊢ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) |
| 98 |
|
mzpsubmpt |
⊢ ( ( ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( 𝑐 ‘ 3 ) ↑ 2 ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ∧ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) → ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) |
| 99 |
82 97 98
|
mp2an |
⊢ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) |
| 100 |
|
eqrabdioph |
⊢ ( ( 4 ∈ ℕ0 ∧ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ∧ ( 𝑐 ∈ ( ℤ ↑m ( 1 ... 4 ) ) ↦ 1 ) ∈ ( mzPoly ‘ ( 1 ... 4 ) ) ) → { 𝑐 ∈ ( ℕ0 ↑m ( 1 ... 4 ) ) ∣ ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) = 1 } ∈ ( Dioph ‘ 4 ) ) |
| 101 |
56 99 89 100
|
mp3an |
⊢ { 𝑐 ∈ ( ℕ0 ↑m ( 1 ... 4 ) ) ∣ ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) = 1 } ∈ ( Dioph ‘ 4 ) |
| 102 |
|
anrabdioph |
⊢ ( ( { 𝑐 ∈ ( ℕ0 ↑m ( 1 ... 4 ) ) ∣ ( ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑐 ‘ 4 ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) } ∈ ( Dioph ‘ 4 ) ∧ { 𝑐 ∈ ( ℕ0 ↑m ( 1 ... 4 ) ) ∣ ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) = 1 } ∈ ( Dioph ‘ 4 ) ) → { 𝑐 ∈ ( ℕ0 ↑m ( 1 ... 4 ) ) ∣ ( ( ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑐 ‘ 4 ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) ∧ ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) = 1 ) } ∈ ( Dioph ‘ 4 ) ) |
| 103 |
75 101 102
|
mp2an |
⊢ { 𝑐 ∈ ( ℕ0 ↑m ( 1 ... 4 ) ) ∣ ( ( ( 𝑐 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑐 ‘ 4 ) = ( ( 𝑐 ‘ 1 ) Yrm ( 𝑐 ‘ 2 ) ) ) ∧ ( ( ( 𝑐 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑐 ‘ 1 ) ↑ 2 ) − 1 ) · ( ( 𝑐 ‘ 4 ) ↑ 2 ) ) ) = 1 ) } ∈ ( Dioph ‘ 4 ) |
| 104 |
55 103
|
eqeltri |
⊢ { 𝑐 ∈ ( ℕ0 ↑m ( 1 ... 4 ) ) ∣ [ ( 𝑐 ↾ ( 1 ... 3 ) ) / 𝑎 ] [ ( 𝑐 ‘ 4 ) / 𝑏 ] ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) } ∈ ( Dioph ‘ 4 ) |
| 105 |
65
|
rexfrabdioph |
⊢ ( ( 3 ∈ ℕ0 ∧ { 𝑐 ∈ ( ℕ0 ↑m ( 1 ... 4 ) ) ∣ [ ( 𝑐 ↾ ( 1 ... 3 ) ) / 𝑎 ] [ ( 𝑐 ‘ 4 ) / 𝑏 ] ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) } ∈ ( Dioph ‘ 4 ) ) → { 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) ∣ ∃ 𝑏 ∈ ℕ0 ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) } ∈ ( Dioph ‘ 3 ) ) |
| 106 |
26 104 105
|
mp2an |
⊢ { 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) ∣ ∃ 𝑏 ∈ ℕ0 ( ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑏 = ( ( 𝑎 ‘ 1 ) Yrm ( 𝑎 ‘ 2 ) ) ) ∧ ( ( ( 𝑎 ‘ 3 ) ↑ 2 ) − ( ( ( ( 𝑎 ‘ 1 ) ↑ 2 ) − 1 ) · ( 𝑏 ↑ 2 ) ) ) = 1 ) } ∈ ( Dioph ‘ 3 ) |
| 107 |
25 106
|
eqeltri |
⊢ { 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 3 ) ) ∣ ( ( 𝑎 ‘ 1 ) ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝑎 ‘ 3 ) = ( ( 𝑎 ‘ 1 ) Xrm ( 𝑎 ‘ 2 ) ) ) } ∈ ( Dioph ‘ 3 ) |