| Step |
Hyp |
Ref |
Expression |
| 1 |
|
nna4b4nsq.a |
⊢ ( 𝜑 → 𝐴 ∈ ℕ ) |
| 2 |
|
nna4b4nsq.b |
⊢ ( 𝜑 → 𝐵 ∈ ℕ ) |
| 3 |
|
nna4b4nsq.c |
⊢ ( 𝜑 → 𝐶 ∈ ℕ ) |
| 4 |
|
oveq1 |
⊢ ( 𝑐 = 𝑓 → ( 𝑐 ↑ 2 ) = ( 𝑓 ↑ 2 ) ) |
| 5 |
4
|
eqeq2d |
⊢ ( 𝑐 = 𝑓 → ( ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ↔ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) |
| 6 |
5
|
anbi2d |
⊢ ( 𝑐 = 𝑓 → ( ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ↔ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) |
| 7 |
6
|
anbi2d |
⊢ ( 𝑐 = 𝑓 → ( ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ↔ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) ) |
| 8 |
7
|
2rexbidv |
⊢ ( 𝑐 = 𝑓 → ( ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ↔ ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) ) |
| 9 |
|
oveq1 |
⊢ ( 𝑐 = 𝑔 → ( 𝑐 ↑ 2 ) = ( 𝑔 ↑ 2 ) ) |
| 10 |
9
|
eqeq2d |
⊢ ( 𝑐 = 𝑔 → ( ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ↔ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑔 ↑ 2 ) ) ) |
| 11 |
10
|
anbi2d |
⊢ ( 𝑐 = 𝑔 → ( ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ↔ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑔 ↑ 2 ) ) ) ) |
| 12 |
11
|
anbi2d |
⊢ ( 𝑐 = 𝑔 → ( ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ↔ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑔 ↑ 2 ) ) ) ) ) |
| 13 |
12
|
2rexbidv |
⊢ ( 𝑐 = 𝑔 → ( ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ↔ ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑔 ↑ 2 ) ) ) ) ) |
| 14 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
| 15 |
14
|
eqimssi |
⊢ ℕ ⊆ ( ℤ≥ ‘ 1 ) |
| 16 |
15
|
a1i |
⊢ ( 𝜑 → ℕ ⊆ ( ℤ≥ ‘ 1 ) ) |
| 17 |
|
breq2 |
⊢ ( 𝑑 = ℎ → ( 2 ∥ 𝑑 ↔ 2 ∥ ℎ ) ) |
| 18 |
17
|
notbid |
⊢ ( 𝑑 = ℎ → ( ¬ 2 ∥ 𝑑 ↔ ¬ 2 ∥ ℎ ) ) |
| 19 |
|
oveq1 |
⊢ ( 𝑑 = ℎ → ( 𝑑 gcd 𝑒 ) = ( ℎ gcd 𝑒 ) ) |
| 20 |
19
|
eqeq1d |
⊢ ( 𝑑 = ℎ → ( ( 𝑑 gcd 𝑒 ) = 1 ↔ ( ℎ gcd 𝑒 ) = 1 ) ) |
| 21 |
|
oveq1 |
⊢ ( 𝑑 = ℎ → ( 𝑑 ↑ 4 ) = ( ℎ ↑ 4 ) ) |
| 22 |
21
|
oveq1d |
⊢ ( 𝑑 = ℎ → ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( ( ℎ ↑ 4 ) + ( 𝑒 ↑ 4 ) ) ) |
| 23 |
22
|
eqeq1d |
⊢ ( 𝑑 = ℎ → ( ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ↔ ( ( ℎ ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) |
| 24 |
20 23
|
anbi12d |
⊢ ( 𝑑 = ℎ → ( ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ↔ ( ( ℎ gcd 𝑒 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) |
| 25 |
18 24
|
anbi12d |
⊢ ( 𝑑 = ℎ → ( ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ↔ ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑒 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) ) |
| 26 |
|
oveq2 |
⊢ ( 𝑒 = 𝑖 → ( ℎ gcd 𝑒 ) = ( ℎ gcd 𝑖 ) ) |
| 27 |
26
|
eqeq1d |
⊢ ( 𝑒 = 𝑖 → ( ( ℎ gcd 𝑒 ) = 1 ↔ ( ℎ gcd 𝑖 ) = 1 ) ) |
| 28 |
|
oveq1 |
⊢ ( 𝑒 = 𝑖 → ( 𝑒 ↑ 4 ) = ( 𝑖 ↑ 4 ) ) |
| 29 |
28
|
oveq2d |
⊢ ( 𝑒 = 𝑖 → ( ( ℎ ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) ) |
| 30 |
29
|
eqeq1d |
⊢ ( 𝑒 = 𝑖 → ( ( ( ℎ ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ↔ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) |
| 31 |
27 30
|
anbi12d |
⊢ ( 𝑒 = 𝑖 → ( ( ( ℎ gcd 𝑒 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ↔ ( ( ℎ gcd 𝑖 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) |
| 32 |
31
|
anbi2d |
⊢ ( 𝑒 = 𝑖 → ( ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑒 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ↔ ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑖 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) ) |
| 33 |
25 32
|
cbvrex2vw |
⊢ ( ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ↔ ∃ ℎ ∈ ℕ ∃ 𝑖 ∈ ℕ ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑖 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) |
| 34 |
|
simplrl |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 ∈ ℕ ) ∧ ( ℎ ∈ ℕ ∧ 𝑖 ∈ ℕ ) ) ∧ ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑖 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) → ℎ ∈ ℕ ) |
| 35 |
|
simplrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 ∈ ℕ ) ∧ ( ℎ ∈ ℕ ∧ 𝑖 ∈ ℕ ) ) ∧ ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑖 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) → 𝑖 ∈ ℕ ) |
| 36 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 ∈ ℕ ) ∧ ( ℎ ∈ ℕ ∧ 𝑖 ∈ ℕ ) ) ∧ ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑖 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) → 𝑓 ∈ ℕ ) |
| 37 |
|
simprl |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 ∈ ℕ ) ∧ ( ℎ ∈ ℕ ∧ 𝑖 ∈ ℕ ) ) ∧ ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑖 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) → ¬ 2 ∥ ℎ ) |
| 38 |
|
simprrl |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 ∈ ℕ ) ∧ ( ℎ ∈ ℕ ∧ 𝑖 ∈ ℕ ) ) ∧ ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑖 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) → ( ℎ gcd 𝑖 ) = 1 ) |
| 39 |
|
simprrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 ∈ ℕ ) ∧ ( ℎ ∈ ℕ ∧ 𝑖 ∈ ℕ ) ) ∧ ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑖 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) → ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) |
| 40 |
34 35 36 37 38 39
|
flt4lem7 |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 ∈ ℕ ) ∧ ( ℎ ∈ ℕ ∧ 𝑖 ∈ ℕ ) ) ∧ ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑖 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) → ∃ 𝑔 ∈ ℕ ( ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑔 ↑ 2 ) ) ) ∧ 𝑔 < 𝑓 ) ) |
| 41 |
40
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ℕ ) ∧ ( ℎ ∈ ℕ ∧ 𝑖 ∈ ℕ ) ) → ( ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑖 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) → ∃ 𝑔 ∈ ℕ ( ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑔 ↑ 2 ) ) ) ∧ 𝑔 < 𝑓 ) ) ) |
| 42 |
41
|
rexlimdvva |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ℕ ) → ( ∃ ℎ ∈ ℕ ∃ 𝑖 ∈ ℕ ( ¬ 2 ∥ ℎ ∧ ( ( ℎ gcd 𝑖 ) = 1 ∧ ( ( ℎ ↑ 4 ) + ( 𝑖 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) → ∃ 𝑔 ∈ ℕ ( ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑔 ↑ 2 ) ) ) ∧ 𝑔 < 𝑓 ) ) ) |
| 43 |
33 42
|
biimtrid |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ℕ ) → ( ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) → ∃ 𝑔 ∈ ℕ ( ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑔 ↑ 2 ) ) ) ∧ 𝑔 < 𝑓 ) ) ) |
| 44 |
43
|
impr |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ ℕ ∧ ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑓 ↑ 2 ) ) ) ) ) → ∃ 𝑔 ∈ ℕ ( ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑔 ↑ 2 ) ) ) ∧ 𝑔 < 𝑓 ) ) |
| 45 |
8 13 16 44
|
infdesc |
⊢ ( 𝜑 → { 𝑐 ∈ ℕ ∣ ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) } = ∅ ) |
| 46 |
|
rabeq0 |
⊢ ( { 𝑐 ∈ ℕ ∣ ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) } = ∅ ↔ ∀ 𝑐 ∈ ℕ ¬ ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 47 |
45 46
|
sylib |
⊢ ( 𝜑 → ∀ 𝑐 ∈ ℕ ¬ ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 48 |
|
ralnex |
⊢ ( ∀ 𝑐 ∈ ℕ ¬ ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ↔ ¬ ∃ 𝑐 ∈ ℕ ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 49 |
47 48
|
sylib |
⊢ ( 𝜑 → ¬ ∃ 𝑐 ∈ ℕ ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 50 |
|
breq2 |
⊢ ( 𝑑 = 𝑎 → ( 2 ∥ 𝑑 ↔ 2 ∥ 𝑎 ) ) |
| 51 |
50
|
notbid |
⊢ ( 𝑑 = 𝑎 → ( ¬ 2 ∥ 𝑑 ↔ ¬ 2 ∥ 𝑎 ) ) |
| 52 |
|
oveq1 |
⊢ ( 𝑑 = 𝑎 → ( 𝑑 gcd 𝑒 ) = ( 𝑎 gcd 𝑒 ) ) |
| 53 |
52
|
eqeq1d |
⊢ ( 𝑑 = 𝑎 → ( ( 𝑑 gcd 𝑒 ) = 1 ↔ ( 𝑎 gcd 𝑒 ) = 1 ) ) |
| 54 |
|
oveq1 |
⊢ ( 𝑑 = 𝑎 → ( 𝑑 ↑ 4 ) = ( 𝑎 ↑ 4 ) ) |
| 55 |
54
|
oveq1d |
⊢ ( 𝑑 = 𝑎 → ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( ( 𝑎 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) ) |
| 56 |
55
|
eqeq1d |
⊢ ( 𝑑 = 𝑎 → ( ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ↔ ( ( 𝑎 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) |
| 57 |
53 56
|
anbi12d |
⊢ ( 𝑑 = 𝑎 → ( ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ↔ ( ( 𝑎 gcd 𝑒 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 58 |
51 57
|
anbi12d |
⊢ ( 𝑑 = 𝑎 → ( ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ↔ ( ¬ 2 ∥ 𝑎 ∧ ( ( 𝑎 gcd 𝑒 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) ) |
| 59 |
|
oveq2 |
⊢ ( 𝑒 = 𝑏 → ( 𝑎 gcd 𝑒 ) = ( 𝑎 gcd 𝑏 ) ) |
| 60 |
59
|
eqeq1d |
⊢ ( 𝑒 = 𝑏 → ( ( 𝑎 gcd 𝑒 ) = 1 ↔ ( 𝑎 gcd 𝑏 ) = 1 ) ) |
| 61 |
|
oveq1 |
⊢ ( 𝑒 = 𝑏 → ( 𝑒 ↑ 4 ) = ( 𝑏 ↑ 4 ) ) |
| 62 |
61
|
oveq2d |
⊢ ( 𝑒 = 𝑏 → ( ( 𝑎 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) ) |
| 63 |
62
|
eqeq1d |
⊢ ( 𝑒 = 𝑏 → ( ( ( 𝑎 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ↔ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) |
| 64 |
60 63
|
anbi12d |
⊢ ( 𝑒 = 𝑏 → ( ( ( 𝑎 gcd 𝑒 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ↔ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 65 |
64
|
anbi2d |
⊢ ( 𝑒 = 𝑏 → ( ( ¬ 2 ∥ 𝑎 ∧ ( ( 𝑎 gcd 𝑒 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ↔ ( ¬ 2 ∥ 𝑎 ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) ) |
| 66 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) → 𝑎 ∈ ℕ ) |
| 67 |
66
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑎 ) → 𝑎 ∈ ℕ ) |
| 68 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) → 𝑏 ∈ ℕ ) |
| 69 |
68
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑎 ) → 𝑏 ∈ ℕ ) |
| 70 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) → ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) |
| 71 |
70
|
anim1ci |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑎 ) → ( ¬ 2 ∥ 𝑎 ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 72 |
58 65 67 69 71
|
2rspcedvdw |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑎 ) → ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 73 |
|
breq2 |
⊢ ( 𝑑 = 𝑏 → ( 2 ∥ 𝑑 ↔ 2 ∥ 𝑏 ) ) |
| 74 |
73
|
notbid |
⊢ ( 𝑑 = 𝑏 → ( ¬ 2 ∥ 𝑑 ↔ ¬ 2 ∥ 𝑏 ) ) |
| 75 |
|
oveq1 |
⊢ ( 𝑑 = 𝑏 → ( 𝑑 gcd 𝑒 ) = ( 𝑏 gcd 𝑒 ) ) |
| 76 |
75
|
eqeq1d |
⊢ ( 𝑑 = 𝑏 → ( ( 𝑑 gcd 𝑒 ) = 1 ↔ ( 𝑏 gcd 𝑒 ) = 1 ) ) |
| 77 |
|
oveq1 |
⊢ ( 𝑑 = 𝑏 → ( 𝑑 ↑ 4 ) = ( 𝑏 ↑ 4 ) ) |
| 78 |
77
|
oveq1d |
⊢ ( 𝑑 = 𝑏 → ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( ( 𝑏 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) ) |
| 79 |
78
|
eqeq1d |
⊢ ( 𝑑 = 𝑏 → ( ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ↔ ( ( 𝑏 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) |
| 80 |
76 79
|
anbi12d |
⊢ ( 𝑑 = 𝑏 → ( ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ↔ ( ( 𝑏 gcd 𝑒 ) = 1 ∧ ( ( 𝑏 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 81 |
74 80
|
anbi12d |
⊢ ( 𝑑 = 𝑏 → ( ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ↔ ( ¬ 2 ∥ 𝑏 ∧ ( ( 𝑏 gcd 𝑒 ) = 1 ∧ ( ( 𝑏 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) ) |
| 82 |
|
oveq2 |
⊢ ( 𝑒 = 𝑎 → ( 𝑏 gcd 𝑒 ) = ( 𝑏 gcd 𝑎 ) ) |
| 83 |
82
|
eqeq1d |
⊢ ( 𝑒 = 𝑎 → ( ( 𝑏 gcd 𝑒 ) = 1 ↔ ( 𝑏 gcd 𝑎 ) = 1 ) ) |
| 84 |
|
oveq1 |
⊢ ( 𝑒 = 𝑎 → ( 𝑒 ↑ 4 ) = ( 𝑎 ↑ 4 ) ) |
| 85 |
84
|
oveq2d |
⊢ ( 𝑒 = 𝑎 → ( ( 𝑏 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( ( 𝑏 ↑ 4 ) + ( 𝑎 ↑ 4 ) ) ) |
| 86 |
85
|
eqeq1d |
⊢ ( 𝑒 = 𝑎 → ( ( ( 𝑏 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ↔ ( ( 𝑏 ↑ 4 ) + ( 𝑎 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) |
| 87 |
83 86
|
anbi12d |
⊢ ( 𝑒 = 𝑎 → ( ( ( 𝑏 gcd 𝑒 ) = 1 ∧ ( ( 𝑏 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ↔ ( ( 𝑏 gcd 𝑎 ) = 1 ∧ ( ( 𝑏 ↑ 4 ) + ( 𝑎 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 88 |
87
|
anbi2d |
⊢ ( 𝑒 = 𝑎 → ( ( ¬ 2 ∥ 𝑏 ∧ ( ( 𝑏 gcd 𝑒 ) = 1 ∧ ( ( 𝑏 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ↔ ( ¬ 2 ∥ 𝑏 ∧ ( ( 𝑏 gcd 𝑎 ) = 1 ∧ ( ( 𝑏 ↑ 4 ) + ( 𝑎 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) ) |
| 89 |
68
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → 𝑏 ∈ ℕ ) |
| 90 |
66
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → 𝑎 ∈ ℕ ) |
| 91 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ¬ 2 ∥ 𝑏 ) |
| 92 |
89
|
nnzd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → 𝑏 ∈ ℤ ) |
| 93 |
90
|
nnzd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → 𝑎 ∈ ℤ ) |
| 94 |
92 93
|
gcdcomd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ( 𝑏 gcd 𝑎 ) = ( 𝑎 gcd 𝑏 ) ) |
| 95 |
|
simplrl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ( 𝑎 gcd 𝑏 ) = 1 ) |
| 96 |
94 95
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ( 𝑏 gcd 𝑎 ) = 1 ) |
| 97 |
|
4nn0 |
⊢ 4 ∈ ℕ0 |
| 98 |
97
|
a1i |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → 4 ∈ ℕ0 ) |
| 99 |
89 98
|
nnexpcld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ( 𝑏 ↑ 4 ) ∈ ℕ ) |
| 100 |
99
|
nncnd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ( 𝑏 ↑ 4 ) ∈ ℂ ) |
| 101 |
90 98
|
nnexpcld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ( 𝑎 ↑ 4 ) ∈ ℕ ) |
| 102 |
101
|
nncnd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ( 𝑎 ↑ 4 ) ∈ ℂ ) |
| 103 |
100 102
|
addcomd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ( ( 𝑏 ↑ 4 ) + ( 𝑎 ↑ 4 ) ) = ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) ) |
| 104 |
|
simplrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) |
| 105 |
103 104
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ( ( 𝑏 ↑ 4 ) + ( 𝑎 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) |
| 106 |
91 96 105
|
jca32 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ( ¬ 2 ∥ 𝑏 ∧ ( ( 𝑏 gcd 𝑎 ) = 1 ∧ ( ( 𝑏 ↑ 4 ) + ( 𝑎 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 107 |
81 88 89 90 106
|
2rspcedvdw |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ ¬ 2 ∥ 𝑏 ) → ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 108 |
66
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ 2 ∥ 𝑎 ) → 𝑎 ∈ ℕ ) |
| 109 |
68
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ 2 ∥ 𝑎 ) → 𝑏 ∈ ℕ ) |
| 110 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ 2 ∥ 𝑎 ) → 2 ∥ 𝑎 ) |
| 111 |
|
simplrl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ 2 ∥ 𝑎 ) → ( 𝑎 gcd 𝑏 ) = 1 ) |
| 112 |
108 109 110 111
|
flt4lem2 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ∧ 2 ∥ 𝑎 ) → ¬ 2 ∥ 𝑏 ) |
| 113 |
112
|
ex |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) → ( 2 ∥ 𝑎 → ¬ 2 ∥ 𝑏 ) ) |
| 114 |
|
imor |
⊢ ( ( 2 ∥ 𝑎 → ¬ 2 ∥ 𝑏 ) ↔ ( ¬ 2 ∥ 𝑎 ∨ ¬ 2 ∥ 𝑏 ) ) |
| 115 |
113 114
|
sylib |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) → ( ¬ 2 ∥ 𝑎 ∨ ¬ 2 ∥ 𝑏 ) ) |
| 116 |
72 107 115
|
mpjaodan |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) ∧ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) → ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 117 |
116
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑏 ∈ ℕ ∧ 𝑎 ∈ ℕ ) ) → ( ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) → ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) ) |
| 118 |
117
|
rexlimdvva |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ℕ ) → ( ∃ 𝑏 ∈ ℕ ∃ 𝑎 ∈ ℕ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) → ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) ) |
| 119 |
118
|
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑐 ∈ ℕ ∃ 𝑏 ∈ ℕ ∃ 𝑎 ∈ ℕ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) → ∃ 𝑐 ∈ ℕ ∃ 𝑑 ∈ ℕ ∃ 𝑒 ∈ ℕ ( ¬ 2 ∥ 𝑑 ∧ ( ( 𝑑 gcd 𝑒 ) = 1 ∧ ( ( 𝑑 ↑ 4 ) + ( 𝑒 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) ) |
| 120 |
49 119
|
mtod |
⊢ ( 𝜑 → ¬ ∃ 𝑐 ∈ ℕ ∃ 𝑏 ∈ ℕ ∃ 𝑎 ∈ ℕ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) |
| 121 |
|
rexcom13 |
⊢ ( ∃ 𝑐 ∈ ℕ ∃ 𝑏 ∈ ℕ ∃ 𝑎 ∈ ℕ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ↔ ∃ 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ ∃ 𝑐 ∈ ℕ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) |
| 122 |
120 121
|
sylnib |
⊢ ( 𝜑 → ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ ∃ 𝑐 ∈ ℕ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) |
| 123 |
|
oveq1 |
⊢ ( 𝑎 = ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) → ( 𝑎 gcd 𝑏 ) = ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd 𝑏 ) ) |
| 124 |
123
|
eqeq1d |
⊢ ( 𝑎 = ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) → ( ( 𝑎 gcd 𝑏 ) = 1 ↔ ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd 𝑏 ) = 1 ) ) |
| 125 |
|
oveq1 |
⊢ ( 𝑎 = ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) → ( 𝑎 ↑ 4 ) = ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) ) |
| 126 |
125
|
oveq1d |
⊢ ( 𝑎 = ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) → ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( 𝑏 ↑ 4 ) ) ) |
| 127 |
126
|
eqeq1d |
⊢ ( 𝑎 = ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) → ( ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ↔ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) |
| 128 |
124 127
|
anbi12d |
⊢ ( 𝑎 = ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) → ( ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ↔ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd 𝑏 ) = 1 ∧ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 129 |
|
oveq2 |
⊢ ( 𝑏 = ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) → ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd 𝑏 ) = ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ) ) |
| 130 |
129
|
eqeq1d |
⊢ ( 𝑏 = ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) → ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd 𝑏 ) = 1 ↔ ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ) = 1 ) ) |
| 131 |
|
oveq1 |
⊢ ( 𝑏 = ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) → ( 𝑏 ↑ 4 ) = ( ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) ) |
| 132 |
131
|
oveq2d |
⊢ ( 𝑏 = ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) → ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) ) ) |
| 133 |
132
|
eqeq1d |
⊢ ( 𝑏 = ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) → ( ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ↔ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) |
| 134 |
130 133
|
anbi12d |
⊢ ( 𝑏 = ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) → ( ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd 𝑏 ) = 1 ∧ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ↔ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ) = 1 ∧ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) ) |
| 135 |
|
oveq1 |
⊢ ( 𝑐 = ( 𝐶 / ( ( 𝐴 gcd 𝐵 ) ↑ 2 ) ) → ( 𝑐 ↑ 2 ) = ( ( 𝐶 / ( ( 𝐴 gcd 𝐵 ) ↑ 2 ) ) ↑ 2 ) ) |
| 136 |
135
|
eqeq2d |
⊢ ( 𝑐 = ( 𝐶 / ( ( 𝐴 gcd 𝐵 ) ↑ 2 ) ) → ( ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ↔ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) ) = ( ( 𝐶 / ( ( 𝐴 gcd 𝐵 ) ↑ 2 ) ) ↑ 2 ) ) ) |
| 137 |
136
|
anbi2d |
⊢ ( 𝑐 = ( 𝐶 / ( ( 𝐴 gcd 𝐵 ) ↑ 2 ) ) → ( ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ) = 1 ∧ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ↔ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ) = 1 ∧ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) ) = ( ( 𝐶 / ( ( 𝐴 gcd 𝐵 ) ↑ 2 ) ) ↑ 2 ) ) ) ) |
| 138 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → 𝐴 ∈ ℕ ) |
| 139 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → 𝐵 ∈ ℕ ) |
| 140 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → 𝐶 ∈ ℕ ) |
| 141 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) |
| 142 |
138 139 140 141
|
flt4lem6 |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ∈ ℕ ∧ ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ∈ ℕ ∧ ( 𝐶 / ( ( 𝐴 gcd 𝐵 ) ↑ 2 ) ) ∈ ℕ ) ∧ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) ) = ( ( 𝐶 / ( ( 𝐴 gcd 𝐵 ) ↑ 2 ) ) ↑ 2 ) ) ) |
| 143 |
142
|
simpld |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ∈ ℕ ∧ ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ∈ ℕ ∧ ( 𝐶 / ( ( 𝐴 gcd 𝐵 ) ↑ 2 ) ) ∈ ℕ ) ) |
| 144 |
143
|
simp1d |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ∈ ℕ ) |
| 145 |
143
|
simp2d |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ∈ ℕ ) |
| 146 |
143
|
simp3d |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → ( 𝐶 / ( ( 𝐴 gcd 𝐵 ) ↑ 2 ) ) ∈ ℕ ) |
| 147 |
1
|
nnzd |
⊢ ( 𝜑 → 𝐴 ∈ ℤ ) |
| 148 |
147
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → 𝐴 ∈ ℤ ) |
| 149 |
2
|
nnzd |
⊢ ( 𝜑 → 𝐵 ∈ ℤ ) |
| 150 |
149
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → 𝐵 ∈ ℤ ) |
| 151 |
2
|
nnne0d |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
| 152 |
151
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → 𝐵 ≠ 0 ) |
| 153 |
|
divgcdcoprm0 |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ) = 1 ) |
| 154 |
148 150 152 153
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ) = 1 ) |
| 155 |
142
|
simprd |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) ) = ( ( 𝐶 / ( ( 𝐴 gcd 𝐵 ) ↑ 2 ) ) ↑ 2 ) ) |
| 156 |
154 155
|
jca |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) gcd ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ) = 1 ∧ ( ( ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) + ( ( 𝐵 / ( 𝐴 gcd 𝐵 ) ) ↑ 4 ) ) = ( ( 𝐶 / ( ( 𝐴 gcd 𝐵 ) ↑ 2 ) ) ↑ 2 ) ) ) |
| 157 |
128 134 137 144 145 146 156
|
3rspcedvdw |
⊢ ( ( 𝜑 ∧ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) → ∃ 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ ∃ 𝑐 ∈ ℕ ( ( 𝑎 gcd 𝑏 ) = 1 ∧ ( ( 𝑎 ↑ 4 ) + ( 𝑏 ↑ 4 ) ) = ( 𝑐 ↑ 2 ) ) ) |
| 158 |
122 157
|
mtand |
⊢ ( 𝜑 → ¬ ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) ) |
| 159 |
158
|
neqned |
⊢ ( 𝜑 → ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) ≠ ( 𝐶 ↑ 2 ) ) |