nna4b4nsq - Mathbox for Steven Nguyen

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Theorem nna4b4nsq 40024

Description: Strengthening of Fermat's last theorem for exponent 4, where the sum is only assumed to be a square. (Contributed by SN, 23-Aug-2024.)

**Hypotheses**
Ref	Expression
nna4b4nsq.a	⊢ (𝜑 → 𝐴 ∈ ℕ)
nna4b4nsq.b	⊢ (𝜑 → 𝐵 ∈ ℕ)
nna4b4nsq.c	⊢ (𝜑 → 𝐶 ∈ ℕ)

**Assertion**
Ref	Expression
nna4b4nsq	⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) ≠ (𝐶↑2))

Proof of Theorem nna4b4nsq Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑖 𝑗 𝑘 𝑙 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nna4b4nsq.c	. 2 ⊢ (𝜑 → 𝐶 ∈ ℕ)
2		oveq1 7163	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑎↑4) = (𝐴↑4))
3	2	oveq1d 7171	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((𝑎↑4) + (𝑏↑4)) = ((𝐴↑4) + (𝑏↑4)))
4	3	eqeq1d 2760	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (((𝑎↑4) + (𝑏↑4)) = (𝑐↑2) ↔ ((𝐴↑4) + (𝑏↑4)) = (𝑐↑2)))
5		oveq1 7163	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑏↑4) = (𝐵↑4))
6	5	oveq2d 7172	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝐴↑4) + (𝑏↑4)) = ((𝐴↑4) + (𝐵↑4)))
7	6	eqeq1d 2760	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (((𝐴↑4) + (𝑏↑4)) = (𝑐↑2) ↔ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2)))
8		nna4b4nsq.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℕ)
9	8	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ℕ) ∧ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2)) → 𝐴 ∈ ℕ)
10		nna4b4nsq.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℕ)
11	10	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ℕ) ∧ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2)) → 𝐵 ∈ ℕ)
12		simpr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ℕ) ∧ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2)) → ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2))
13	4, 7, 9, 11, 12	2rspcedvdw 39732	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ ℕ) ∧ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2)) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))
14	13	ex 416	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ ℕ) → (((𝐴↑4) + (𝐵↑4)) = (𝑐↑2) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2)))
15	14	ss2rabdv 3982	. . . 4 ⊢ (𝜑 → {𝑐 ∈ ℕ ∣ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2)} ⊆ {𝑐 ∈ ℕ ∣ ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2)})
16		oveq1 7163	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑖 → (𝑓↑2) = (𝑖↑2))
17	16	eqeq2d 2769	. . . . . . . . . 10 ⊢ (𝑓 = 𝑖 → (((𝑔↑4) + (ℎ↑4)) = (𝑓↑2) ↔ ((𝑔↑4) + (ℎ↑4)) = (𝑖↑2)))
18	17	anbi2d 631	. . . . . . . . 9 ⊢ (𝑓 = 𝑖 → (((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2)) ↔ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑖↑2))))
19	18	anbi2d 631	. . . . . . . 8 ⊢ (𝑓 = 𝑖 → ((¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))) ↔ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑖↑2)))))
20	19	2rexbidv 3224	. . . . . . 7 ⊢ (𝑓 = 𝑖 → (∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))) ↔ ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑖↑2)))))
21		oveq1 7163	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑙 → (𝑓↑2) = (𝑙↑2))
22	21	eqeq2d 2769	. . . . . . . . . 10 ⊢ (𝑓 = 𝑙 → (((𝑔↑4) + (ℎ↑4)) = (𝑓↑2) ↔ ((𝑔↑4) + (ℎ↑4)) = (𝑙↑2)))
23	22	anbi2d 631	. . . . . . . . 9 ⊢ (𝑓 = 𝑙 → (((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2)) ↔ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑙↑2))))
24	23	anbi2d 631	. . . . . . . 8 ⊢ (𝑓 = 𝑙 → ((¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))) ↔ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑙↑2)))))
25	24	2rexbidv 3224	. . . . . . 7 ⊢ (𝑓 = 𝑙 → (∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))) ↔ ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑙↑2)))))
26		nnuz 12334	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
27	26	eqimssi 3952	. . . . . . . 8 ⊢ ℕ ⊆ (ℤ_≥‘1)
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ ⊆ (ℤ_≥‘1))
29		breq2 5040	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑗 → (2 ∥ 𝑔 ↔ 2 ∥ 𝑗))
30	29	notbid 321	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑗 → (¬ 2 ∥ 𝑔 ↔ ¬ 2 ∥ 𝑗))
31		oveq1 7163	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑗 → (𝑔 gcd ℎ) = (𝑗 gcd ℎ))
32	31	eqeq1d 2760	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑗 → ((𝑔 gcd ℎ) = 1 ↔ (𝑗 gcd ℎ) = 1))
33		oveq1 7163	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑗 → (𝑔↑4) = (𝑗↑4))
34	33	oveq1d 7171	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑗 → ((𝑔↑4) + (ℎ↑4)) = ((𝑗↑4) + (ℎ↑4)))
35	34	eqeq1d 2760	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑗 → (((𝑔↑4) + (ℎ↑4)) = (𝑖↑2) ↔ ((𝑗↑4) + (ℎ↑4)) = (𝑖↑2)))
36	32, 35	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑗 → (((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑖↑2)) ↔ ((𝑗 gcd ℎ) = 1 ∧ ((𝑗↑4) + (ℎ↑4)) = (𝑖↑2))))
37	30, 36	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑔 = 𝑗 → ((¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑖↑2))) ↔ (¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd ℎ) = 1 ∧ ((𝑗↑4) + (ℎ↑4)) = (𝑖↑2)))))
38		oveq2 7164	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑘 → (𝑗 gcd ℎ) = (𝑗 gcd 𝑘))
39	38	eqeq1d 2760	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑘 → ((𝑗 gcd ℎ) = 1 ↔ (𝑗 gcd 𝑘) = 1))
40		oveq1 7163	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑘 → (ℎ↑4) = (𝑘↑4))
41	40	oveq2d 7172	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑘 → ((𝑗↑4) + (ℎ↑4)) = ((𝑗↑4) + (𝑘↑4)))
42	41	eqeq1d 2760	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑘 → (((𝑗↑4) + (ℎ↑4)) = (𝑖↑2) ↔ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2)))
43	39, 42	anbi12d 633	. . . . . . . . . . 11 ⊢ (ℎ = 𝑘 → (((𝑗 gcd ℎ) = 1 ∧ ((𝑗↑4) + (ℎ↑4)) = (𝑖↑2)) ↔ ((𝑗 gcd 𝑘) = 1 ∧ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2))))
44	43	anbi2d 631	. . . . . . . . . 10 ⊢ (ℎ = 𝑘 → ((¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd ℎ) = 1 ∧ ((𝑗↑4) + (ℎ↑4)) = (𝑖↑2))) ↔ (¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd 𝑘) = 1 ∧ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2)))))
45	37, 44	cbvrex2vw 3374	. . . . . . . . 9 ⊢ (∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑖↑2))) ↔ ∃𝑗 ∈ ℕ ∃𝑘 ∈ ℕ (¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd 𝑘) = 1 ∧ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2))))
46		simplrl 776	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ)) ∧ (¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd 𝑘) = 1 ∧ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2)))) → 𝑗 ∈ ℕ)
47		simplrr 777	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ)) ∧ (¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd 𝑘) = 1 ∧ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2)))) → 𝑘 ∈ ℕ)
48		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ)) ∧ (¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd 𝑘) = 1 ∧ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2)))) → 𝑖 ∈ ℕ)
49		simprl 770	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ)) ∧ (¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd 𝑘) = 1 ∧ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2)))) → ¬ 2 ∥ 𝑗)
50		simprrl 780	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ)) ∧ (¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd 𝑘) = 1 ∧ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2)))) → (𝑗 gcd 𝑘) = 1)
51		simprrr 781	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ)) ∧ (¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd 𝑘) = 1 ∧ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2)))) → ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2))
52	46, 47, 48, 49, 50, 51	flt4lem7 40023	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ)) ∧ (¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd 𝑘) = 1 ∧ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2)))) → ∃𝑙 ∈ ℕ (∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑙↑2))) ∧ 𝑙 < 𝑖))
53	52	ex 416	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ (𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ)) → ((¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd 𝑘) = 1 ∧ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2))) → ∃𝑙 ∈ ℕ (∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑙↑2))) ∧ 𝑙 < 𝑖)))
54	53	rexlimdvva 3218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (∃𝑗 ∈ ℕ ∃𝑘 ∈ ℕ (¬ 2 ∥ 𝑗 ∧ ((𝑗 gcd 𝑘) = 1 ∧ ((𝑗↑4) + (𝑘↑4)) = (𝑖↑2))) → ∃𝑙 ∈ ℕ (∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑙↑2))) ∧ 𝑙 < 𝑖)))
55	45, 54	syl5bi 245	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑖↑2))) → ∃𝑙 ∈ ℕ (∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑙↑2))) ∧ 𝑙 < 𝑖)))
56	55	impr 458	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ ℕ ∧ ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑖↑2))))) → ∃𝑙 ∈ ℕ (∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑙↑2))) ∧ 𝑙 < 𝑖))
57	20, 25, 28, 56	infdesc 40007	. . . . . 6 ⊢ (𝜑 → {𝑓 ∈ ℕ ∣ ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2)))} = ∅)
58		breq2 5040	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑑 → (2 ∥ 𝑔 ↔ 2 ∥ 𝑑))
59	58	notbid 321	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑑 → (¬ 2 ∥ 𝑔 ↔ ¬ 2 ∥ 𝑑))
60		oveq1 7163	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑑 → (𝑔 gcd ℎ) = (𝑑 gcd ℎ))
61	60	eqeq1d 2760	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑑 → ((𝑔 gcd ℎ) = 1 ↔ (𝑑 gcd ℎ) = 1))
62		oveq1 7163	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑑 → (𝑔↑4) = (𝑑↑4))
63	62	oveq1d 7171	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑑 → ((𝑔↑4) + (ℎ↑4)) = ((𝑑↑4) + (ℎ↑4)))
64	63	eqeq1d 2760	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑑 → (((𝑔↑4) + (ℎ↑4)) = (𝑓↑2) ↔ ((𝑑↑4) + (ℎ↑4)) = (𝑓↑2)))
65	61, 64	anbi12d 633	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑑 → (((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2)) ↔ ((𝑑 gcd ℎ) = 1 ∧ ((𝑑↑4) + (ℎ↑4)) = (𝑓↑2))))
66	59, 65	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑑 → ((¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))) ↔ (¬ 2 ∥ 𝑑 ∧ ((𝑑 gcd ℎ) = 1 ∧ ((𝑑↑4) + (ℎ↑4)) = (𝑓↑2)))))
67		oveq2 7164	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = 𝑒 → (𝑑 gcd ℎ) = (𝑑 gcd 𝑒))
68	67	eqeq1d 2760	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝑒 → ((𝑑 gcd ℎ) = 1 ↔ (𝑑 gcd 𝑒) = 1))
69		oveq1 7163	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 𝑒 → (ℎ↑4) = (𝑒↑4))
70	69	oveq2d 7172	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = 𝑒 → ((𝑑↑4) + (ℎ↑4)) = ((𝑑↑4) + (𝑒↑4)))
71	70	eqeq1d 2760	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝑒 → (((𝑑↑4) + (ℎ↑4)) = (𝑓↑2) ↔ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)))
72	68, 71	anbi12d 633	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑒 → (((𝑑 gcd ℎ) = 1 ∧ ((𝑑↑4) + (ℎ↑4)) = (𝑓↑2)) ↔ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))))
73	72	anbi2d 631	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑒 → ((¬ 2 ∥ 𝑑 ∧ ((𝑑 gcd ℎ) = 1 ∧ ((𝑑↑4) + (ℎ↑4)) = (𝑓↑2))) ↔ (¬ 2 ∥ 𝑑 ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)))))
74		simprl 770	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) → 𝑑 ∈ ℕ)
75	74	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑑) → 𝑑 ∈ ℕ)
76		simprr 772	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) → 𝑒 ∈ ℕ)
77	76	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑑) → 𝑒 ∈ ℕ)
78		simpr 488	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑑) → ¬ 2 ∥ 𝑑)
79		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑑) → ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)))
80	78, 79	jca 515	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑑) → (¬ 2 ∥ 𝑑 ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))))
81	66, 73, 75, 77, 80	2rspcedvdw 39732	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑑) → ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))))
82		breq2 5040	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑒 → (2 ∥ 𝑔 ↔ 2 ∥ 𝑒))
83	82	notbid 321	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑒 → (¬ 2 ∥ 𝑔 ↔ ¬ 2 ∥ 𝑒))
84		oveq1 7163	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑒 → (𝑔 gcd ℎ) = (𝑒 gcd ℎ))
85	84	eqeq1d 2760	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑒 → ((𝑔 gcd ℎ) = 1 ↔ (𝑒 gcd ℎ) = 1))
86		oveq1 7163	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑒 → (𝑔↑4) = (𝑒↑4))
87	86	oveq1d 7171	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑒 → ((𝑔↑4) + (ℎ↑4)) = ((𝑒↑4) + (ℎ↑4)))
88	87	eqeq1d 2760	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑒 → (((𝑔↑4) + (ℎ↑4)) = (𝑓↑2) ↔ ((𝑒↑4) + (ℎ↑4)) = (𝑓↑2)))
89	85, 88	anbi12d 633	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑒 → (((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2)) ↔ ((𝑒 gcd ℎ) = 1 ∧ ((𝑒↑4) + (ℎ↑4)) = (𝑓↑2))))
90	83, 89	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑒 → ((¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))) ↔ (¬ 2 ∥ 𝑒 ∧ ((𝑒 gcd ℎ) = 1 ∧ ((𝑒↑4) + (ℎ↑4)) = (𝑓↑2)))))
91		oveq2 7164	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = 𝑑 → (𝑒 gcd ℎ) = (𝑒 gcd 𝑑))
92	91	eqeq1d 2760	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝑑 → ((𝑒 gcd ℎ) = 1 ↔ (𝑒 gcd 𝑑) = 1))
93		oveq1 7163	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 𝑑 → (ℎ↑4) = (𝑑↑4))
94	93	oveq2d 7172	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = 𝑑 → ((𝑒↑4) + (ℎ↑4)) = ((𝑒↑4) + (𝑑↑4)))
95	94	eqeq1d 2760	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝑑 → (((𝑒↑4) + (ℎ↑4)) = (𝑓↑2) ↔ ((𝑒↑4) + (𝑑↑4)) = (𝑓↑2)))
96	92, 95	anbi12d 633	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑑 → (((𝑒 gcd ℎ) = 1 ∧ ((𝑒↑4) + (ℎ↑4)) = (𝑓↑2)) ↔ ((𝑒 gcd 𝑑) = 1 ∧ ((𝑒↑4) + (𝑑↑4)) = (𝑓↑2))))
97	96	anbi2d 631	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑑 → ((¬ 2 ∥ 𝑒 ∧ ((𝑒 gcd ℎ) = 1 ∧ ((𝑒↑4) + (ℎ↑4)) = (𝑓↑2))) ↔ (¬ 2 ∥ 𝑒 ∧ ((𝑒 gcd 𝑑) = 1 ∧ ((𝑒↑4) + (𝑑↑4)) = (𝑓↑2)))))
98	76	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → 𝑒 ∈ ℕ)
99	74	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → 𝑑 ∈ ℕ)
100		simpr 488	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → ¬ 2 ∥ 𝑒)
101	98	nnzd 12138	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → 𝑒 ∈ ℤ)
102	99	nnzd 12138	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → 𝑑 ∈ ℤ)
103	101, 102	gcdcomd 15926	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → (𝑒 gcd 𝑑) = (𝑑 gcd 𝑒))
104		simplrl 776	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → (𝑑 gcd 𝑒) = 1)
105	103, 104	eqtrd 2793	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → (𝑒 gcd 𝑑) = 1)
106		4nn0 11966	. . . . . . . . . . . . . . . . . . . 20 ⊢ 4 ∈ ℕ₀
107	106	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → 4 ∈ ℕ₀)
108	98, 107	nnexpcld 13669	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → (𝑒↑4) ∈ ℕ)
109	108	nncnd 11703	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → (𝑒↑4) ∈ ℂ)
110	99, 107	nnexpcld 13669	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → (𝑑↑4) ∈ ℕ)
111	110	nncnd 11703	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → (𝑑↑4) ∈ ℂ)
112	109, 111	addcomd 10893	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → ((𝑒↑4) + (𝑑↑4)) = ((𝑑↑4) + (𝑒↑4)))
113		simplrr 777	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))
114	112, 113	eqtrd 2793	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → ((𝑒↑4) + (𝑑↑4)) = (𝑓↑2))
115	100, 105, 114	jca32 519	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → (¬ 2 ∥ 𝑒 ∧ ((𝑒 gcd 𝑑) = 1 ∧ ((𝑒↑4) + (𝑑↑4)) = (𝑓↑2))))
116	90, 97, 98, 99, 115	2rspcedvdw 39732	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ ¬ 2 ∥ 𝑒) → ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))))
117	74	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → 𝑑 ∈ ℕ)
118	117	nnsqcld 13668	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → (𝑑↑2) ∈ ℕ)
119	76	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → 𝑒 ∈ ℕ)
120	119	nnsqcld 13668	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → (𝑒↑2) ∈ ℕ)
121		simp-4r 783	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → 𝑓 ∈ ℕ)
122		2z 12066	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℤ
123		simplrl 776	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) → 𝑑 ∈ ℕ)
124	123	nnzd 12138	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) → 𝑑 ∈ ℤ)
125		2nn 11760	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℕ
126	125	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) → 2 ∈ ℕ)
127		dvdsexp2im 15741	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 2 ∈ ℕ) → (2 ∥ 𝑑 → 2 ∥ (𝑑↑2)))
128	122, 124, 126, 127	mp3an2i 1463	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) → (2 ∥ 𝑑 → 2 ∥ (𝑑↑2)))
129	128	imp 410	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → 2 ∥ (𝑑↑2))
130		2nn0 11964	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℕ₀
131	130	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → 2 ∈ ℕ₀)
132	117	nncnd 11703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → 𝑑 ∈ ℂ)
133	132	flt4lem 40009	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → (𝑑↑4) = ((𝑑↑2)↑2))
134	119	nncnd 11703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → 𝑒 ∈ ℂ)
135	134	flt4lem 40009	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → (𝑒↑4) = ((𝑒↑2)↑2))
136	133, 135	oveq12d 7174	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → ((𝑑↑4) + (𝑒↑4)) = (((𝑑↑2)↑2) + ((𝑒↑2)↑2)))
137		simplrr 777	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))
138	136, 137	eqtr3d 2795	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → (((𝑑↑2)↑2) + ((𝑒↑2)↑2)) = (𝑓↑2))
139		simplrl 776	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → (𝑑 gcd 𝑒) = 1)
140	125	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → 2 ∈ ℕ)
141		rppwr 15973	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ ∧ 2 ∈ ℕ) → ((𝑑 gcd 𝑒) = 1 → ((𝑑↑2) gcd (𝑒↑2)) = 1))
142	117, 119, 140, 141	syl3anc 1368	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → ((𝑑 gcd 𝑒) = 1 → ((𝑑↑2) gcd (𝑒↑2)) = 1))
143	139, 142	mpd 15	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → ((𝑑↑2) gcd (𝑒↑2)) = 1)
144	118, 120, 121, 131, 138, 143	fltaccoprm 40004	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → ((𝑑↑2) gcd 𝑓) = 1)
145	118, 120, 121, 129, 144, 138	flt4lem2 40011	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → ¬ 2 ∥ (𝑒↑2))
146	119	nnzd 12138	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → 𝑒 ∈ ℤ)
147		dvdsexp2im 15741	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℤ ∧ 𝑒 ∈ ℤ ∧ 2 ∈ ℕ) → (2 ∥ 𝑒 → 2 ∥ (𝑒↑2)))
148	122, 146, 140, 147	mp3an2i 1463	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → (2 ∥ 𝑒 → 2 ∥ (𝑒↑2)))
149	145, 148	mtod 201	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) ∧ 2 ∥ 𝑑) → ¬ 2 ∥ 𝑒)
150	149	ex 416	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) → (2 ∥ 𝑑 → ¬ 2 ∥ 𝑒))
151		imor 850	. . . . . . . . . . . . . 14 ⊢ ((2 ∥ 𝑑 → ¬ 2 ∥ 𝑒) ↔ (¬ 2 ∥ 𝑑 ∨ ¬ 2 ∥ 𝑒))
152	150, 151	sylib 221	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) → (¬ 2 ∥ 𝑑 ∨ ¬ 2 ∥ 𝑒))
153	81, 116, 152	mpjaodan 956	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) ∧ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))) → ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))))
154	153	ex 416	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ 𝑒 ∈ ℕ)) → (((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)) → ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2)))))
155	154	rexlimdvva 3218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → (∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)) → ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2)))))
156	155	reximdva 3198	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑓 ∈ ℕ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)) → ∃𝑓 ∈ ℕ ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2)))))
157	156	con3d 155	. . . . . . . 8 ⊢ (𝜑 → (¬ ∃𝑓 ∈ ℕ ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))) → ¬ ∃𝑓 ∈ ℕ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))))
158		ralnex 3163	. . . . . . . 8 ⊢ (∀𝑓 ∈ ℕ ¬ ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))) ↔ ¬ ∃𝑓 ∈ ℕ ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))))
159		ralnex 3163	. . . . . . . 8 ⊢ (∀𝑓 ∈ ℕ ¬ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)) ↔ ¬ ∃𝑓 ∈ ℕ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)))
160	157, 158, 159	3imtr4g 299	. . . . . . 7 ⊢ (𝜑 → (∀𝑓 ∈ ℕ ¬ ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))) → ∀𝑓 ∈ ℕ ¬ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))))
161		rabeq0 4283	. . . . . . 7 ⊢ ({𝑓 ∈ ℕ ∣ ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2)))} = ∅ ↔ ∀𝑓 ∈ ℕ ¬ ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2))))
162		rabeq0 4283	. . . . . . 7 ⊢ ({𝑓 ∈ ℕ ∣ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))} = ∅ ↔ ∀𝑓 ∈ ℕ ¬ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)))
163	160, 161, 162	3imtr4g 299	. . . . . 6 ⊢ (𝜑 → ({𝑓 ∈ ℕ ∣ ∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑓↑2)))} = ∅ → {𝑓 ∈ ℕ ∣ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))} = ∅))
164	57, 163	mpd 15	. . . . 5 ⊢ (𝜑 → {𝑓 ∈ ℕ ∣ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))} = ∅)
165		oveq1 7163	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑐 / ((𝑎 gcd 𝑏)↑2)) → (𝑓↑2) = ((𝑐 / ((𝑎 gcd 𝑏)↑2))↑2))
166	165	eqeq2d 2769	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑐 / ((𝑎 gcd 𝑏)↑2)) → (((𝑑↑4) + (𝑒↑4)) = (𝑓↑2) ↔ ((𝑑↑4) + (𝑒↑4)) = ((𝑐 / ((𝑎 gcd 𝑏)↑2))↑2)))
167	166	anbi2d 631	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑐 / ((𝑎 gcd 𝑏)↑2)) → (((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)) ↔ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = ((𝑐 / ((𝑎 gcd 𝑏)↑2))↑2))))
168		oveq1 7163	. . . . . . . . . . . . 13 ⊢ (𝑑 = (𝑎 / (𝑎 gcd 𝑏)) → (𝑑 gcd 𝑒) = ((𝑎 / (𝑎 gcd 𝑏)) gcd 𝑒))
169	168	eqeq1d 2760	. . . . . . . . . . . 12 ⊢ (𝑑 = (𝑎 / (𝑎 gcd 𝑏)) → ((𝑑 gcd 𝑒) = 1 ↔ ((𝑎 / (𝑎 gcd 𝑏)) gcd 𝑒) = 1))
170		oveq1 7163	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝑎 / (𝑎 gcd 𝑏)) → (𝑑↑4) = ((𝑎 / (𝑎 gcd 𝑏))↑4))
171	170	oveq1d 7171	. . . . . . . . . . . . 13 ⊢ (𝑑 = (𝑎 / (𝑎 gcd 𝑏)) → ((𝑑↑4) + (𝑒↑4)) = (((𝑎 / (𝑎 gcd 𝑏))↑4) + (𝑒↑4)))
172	171	eqeq1d 2760	. . . . . . . . . . . 12 ⊢ (𝑑 = (𝑎 / (𝑎 gcd 𝑏)) → (((𝑑↑4) + (𝑒↑4)) = ((𝑐 / ((𝑎 gcd 𝑏)↑2))↑2) ↔ (((𝑎 / (𝑎 gcd 𝑏))↑4) + (𝑒↑4)) = ((𝑐 / ((𝑎 gcd 𝑏)↑2))↑2)))
173	169, 172	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑑 = (𝑎 / (𝑎 gcd 𝑏)) → (((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = ((𝑐 / ((𝑎 gcd 𝑏)↑2))↑2)) ↔ (((𝑎 / (𝑎 gcd 𝑏)) gcd 𝑒) = 1 ∧ (((𝑎 / (𝑎 gcd 𝑏))↑4) + (𝑒↑4)) = ((𝑐 / ((𝑎 gcd 𝑏)↑2))↑2))))
174		oveq2 7164	. . . . . . . . . . . . 13 ⊢ (𝑒 = (𝑏 / (𝑎 gcd 𝑏)) → ((𝑎 / (𝑎 gcd 𝑏)) gcd 𝑒) = ((𝑎 / (𝑎 gcd 𝑏)) gcd (𝑏 / (𝑎 gcd 𝑏))))
175	174	eqeq1d 2760	. . . . . . . . . . . 12 ⊢ (𝑒 = (𝑏 / (𝑎 gcd 𝑏)) → (((𝑎 / (𝑎 gcd 𝑏)) gcd 𝑒) = 1 ↔ ((𝑎 / (𝑎 gcd 𝑏)) gcd (𝑏 / (𝑎 gcd 𝑏))) = 1))
176		oveq1 7163	. . . . . . . . . . . . . 14 ⊢ (𝑒 = (𝑏 / (𝑎 gcd 𝑏)) → (𝑒↑4) = ((𝑏 / (𝑎 gcd 𝑏))↑4))
177	176	oveq2d 7172	. . . . . . . . . . . . 13 ⊢ (𝑒 = (𝑏 / (𝑎 gcd 𝑏)) → (((𝑎 / (𝑎 gcd 𝑏))↑4) + (𝑒↑4)) = (((𝑎 / (𝑎 gcd 𝑏))↑4) + ((𝑏 / (𝑎 gcd 𝑏))↑4)))
178	177	eqeq1d 2760	. . . . . . . . . . . 12 ⊢ (𝑒 = (𝑏 / (𝑎 gcd 𝑏)) → ((((𝑎 / (𝑎 gcd 𝑏))↑4) + (𝑒↑4)) = ((𝑐 / ((𝑎 gcd 𝑏)↑2))↑2) ↔ (((𝑎 / (𝑎 gcd 𝑏))↑4) + ((𝑏 / (𝑎 gcd 𝑏))↑4)) = ((𝑐 / ((𝑎 gcd 𝑏)↑2))↑2)))
179	175, 178	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑒 = (𝑏 / (𝑎 gcd 𝑏)) → ((((𝑎 / (𝑎 gcd 𝑏)) gcd 𝑒) = 1 ∧ (((𝑎 / (𝑎 gcd 𝑏))↑4) + (𝑒↑4)) = ((𝑐 / ((𝑎 gcd 𝑏)↑2))↑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))))
180		simplrr 777	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → 𝑎 ∈ ℕ)
181		simprl 770	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → 𝑏 ∈ ℕ)
182		simplrl 776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → 𝑐 ∈ ℕ)
183		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))
184	180, 181, 182, 183	flt4lem6 40022	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((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)))
185	184	simpld 498	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → ((𝑎 / (𝑎 gcd 𝑏)) ∈ ℕ ∧ (𝑏 / (𝑎 gcd 𝑏)) ∈ ℕ ∧ (𝑐 / ((𝑎 gcd 𝑏)↑2)) ∈ ℕ))
186	185	simp3d 1141	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → (𝑐 / ((𝑎 gcd 𝑏)↑2)) ∈ ℕ)
187	185	simp1d 1139	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → (𝑎 / (𝑎 gcd 𝑏)) ∈ ℕ)
188	185	simp2d 1140	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → (𝑏 / (𝑎 gcd 𝑏)) ∈ ℕ)
189	180	nnzd 12138	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → 𝑎 ∈ ℤ)
190	181	nnzd 12138	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → 𝑏 ∈ ℤ)
191	181	nnne0d 11737	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → 𝑏 ≠ 0)
192		divgcdcoprm0 16074	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑏 ≠ 0) → ((𝑎 / (𝑎 gcd 𝑏)) gcd (𝑏 / (𝑎 gcd 𝑏))) = 1)
193	189, 190, 191, 192	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → ((𝑎 / (𝑎 gcd 𝑏)) gcd (𝑏 / (𝑎 gcd 𝑏))) = 1)
194	184	simprd 499	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → (((𝑎 / (𝑎 gcd 𝑏))↑4) + ((𝑏 / (𝑎 gcd 𝑏))↑4)) = ((𝑐 / ((𝑎 gcd 𝑏)↑2))↑2))
195	193, 194	jca 515	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((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)))
196	167, 173, 179, 186, 187, 188, 195	3rspcedvdw 39733	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) ∧ (𝑏 ∈ ℕ ∧ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))) → ∃𝑓 ∈ ℕ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)))
197	196	rexlimdvaa 3209	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ℕ ∧ 𝑎 ∈ ℕ)) → (∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2) → ∃𝑓 ∈ ℕ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))))
198	197	rexlimdvva 3218	. . . . . . . 8 ⊢ (𝜑 → (∃𝑐 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2) → ∃𝑓 ∈ ℕ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))))
199	198	con3d 155	. . . . . . 7 ⊢ (𝜑 → (¬ ∃𝑓 ∈ ℕ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)) → ¬ ∃𝑐 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2)))
200		ralnex 3163	. . . . . . 7 ⊢ (∀𝑐 ∈ ℕ ¬ ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2) ↔ ¬ ∃𝑐 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))
201	199, 159, 200	3imtr4g 299	. . . . . 6 ⊢ (𝜑 → (∀𝑓 ∈ ℕ ¬ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2)) → ∀𝑐 ∈ ℕ ¬ ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2)))
202		rabeq0 4283	. . . . . 6 ⊢ ({𝑐 ∈ ℕ ∣ ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2)} = ∅ ↔ ∀𝑐 ∈ ℕ ¬ ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2))
203	201, 162, 202	3imtr4g 299	. . . . 5 ⊢ (𝜑 → ({𝑓 ∈ ℕ ∣ ∃𝑑 ∈ ℕ ∃𝑒 ∈ ℕ ((𝑑 gcd 𝑒) = 1 ∧ ((𝑑↑4) + (𝑒↑4)) = (𝑓↑2))} = ∅ → {𝑐 ∈ ℕ ∣ ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2)} = ∅))
204	164, 203	mpd 15	. . . 4 ⊢ (𝜑 → {𝑐 ∈ ℕ ∣ ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2)} = ∅)
205		sseq0 4298	. . . 4 ⊢ (({𝑐 ∈ ℕ ∣ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2)} ⊆ {𝑐 ∈ ℕ ∣ ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2)} ∧ {𝑐 ∈ ℕ ∣ ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑4) + (𝑏↑4)) = (𝑐↑2)} = ∅) → {𝑐 ∈ ℕ ∣ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2)} = ∅)
206	15, 204, 205	syl2anc 587	. . 3 ⊢ (𝜑 → {𝑐 ∈ ℕ ∣ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2)} = ∅)
207		rabeq0 4283	. . 3 ⊢ ({𝑐 ∈ ℕ ∣ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2)} = ∅ ↔ ∀𝑐 ∈ ℕ ¬ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2))
208	206, 207	sylib 221	. 2 ⊢ (𝜑 → ∀𝑐 ∈ ℕ ¬ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2))
209		oveq1 7163	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐↑2) = (𝐶↑2))
210	209	eqeq2d 2769	. . . 4 ⊢ (𝑐 = 𝐶 → (((𝐴↑4) + (𝐵↑4)) = (𝑐↑2) ↔ ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2)))
211	210	necon3bbid 2988	. . 3 ⊢ (𝑐 = 𝐶 → (¬ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2) ↔ ((𝐴↑4) + (𝐵↑4)) ≠ (𝐶↑2)))
212	211	rspcv 3538	. 2 ⊢ (𝐶 ∈ ℕ → (∀𝑐 ∈ ℕ ¬ ((𝐴↑4) + (𝐵↑4)) = (𝑐↑2) → ((𝐴↑4) + (𝐵↑4)) ≠ (𝐶↑2)))
213	1, 208, 212	sylc 65	1 ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) ≠ (𝐶↑2))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ∀wral 3070 ∃wrex 3071 {crab 3074 ⊆ wss 3860 ∅c0 4227 class class class wbr 5036 ‘cfv 6340 (class class class)co 7156 0cc0 10588 1c1 10589 + caddc 10591 < clt 10726 / cdiv 11348 ℕcn 11687 2c2 11742 4c4 11744 ℕ₀cn0 11947 ℤcz 12033 ℤ_≥cuz 12295 ↑cexp 13492 ∥ cdvds 15668 gcd cgcd 15906

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-inf2 9150 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 ax-pre-sup 10666 ax-addf 10667 ax-mulf 10668

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-se 5488 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7411 df-om 7586 df-1st 7699 df-2nd 7700 df-supp 7842 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-2o 8119 df-er 8305 df-map 8424 df-pm 8425 df-ixp 8493 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-fsupp 8880 df-fi 8921 df-sup 8952 df-inf 8953 df-oi 9020 df-card 9414 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-div 11349 df-nn 11688 df-2 11750 df-3 11751 df-4 11752 df-5 11753 df-6 11754 df-7 11755 df-8 11756 df-9 11757 df-n0 11948 df-z 12034 df-dec 12151 df-uz 12296 df-q 12402 df-rp 12444 df-xneg 12561 df-xadd 12562 df-xmul 12563 df-ioo 12796 df-ioc 12797 df-ico 12798 df-icc 12799 df-fz 12953 df-fzo 13096 df-fl 13224 df-mod 13300 df-seq 13432 df-exp 13493 df-fac 13697 df-bc 13726 df-hash 13754 df-shft 14487 df-cj 14519 df-re 14520 df-im 14521 df-sqrt 14655 df-abs 14656 df-limsup 14889 df-clim 14906 df-rlim 14907 df-sum 15104 df-ef 15482 df-sin 15484 df-cos 15485 df-pi 15487 df-dvds 15669 df-gcd 15907 df-prm 16081 df-numer 16143 df-denom 16144 df-struct 16556 df-ndx 16557 df-slot 16558 df-base 16560 df-sets 16561 df-ress 16562 df-plusg 16649 df-mulr 16650 df-starv 16651 df-sca 16652 df-vsca 16653 df-ip 16654 df-tset 16655 df-ple 16656 df-ds 16658 df-unif 16659 df-hom 16660 df-cco 16661 df-rest 16767 df-topn 16768 df-0g 16786 df-gsum 16787 df-topgen 16788 df-pt 16789 df-prds 16792 df-xrs 16846 df-qtop 16851 df-imas 16852 df-xps 16854 df-mre 16928 df-mrc 16929 df-acs 16931 df-mgm 17931 df-sgrp 17980 df-mnd 17991 df-submnd 18036 df-mulg 18305 df-cntz 18527 df-cmn 18988 df-psmet 20171 df-xmet 20172 df-met 20173 df-bl 20174 df-mopn 20175 df-fbas 20176 df-fg 20177 df-cnfld 20180 df-top 21607 df-topon 21624 df-topsp 21646 df-bases 21659 df-cld 21732 df-ntr 21733 df-cls 21734 df-nei 21811 df-lp 21849 df-perf 21850 df-cn 21940 df-cnp 21941 df-haus 22028 df-tx 22275 df-hmeo 22468 df-fil 22559 df-fm 22651 df-flim 22652 df-flf 22653 df-xms 23035 df-ms 23036 df-tms 23037 df-cncf 23592 df-limc 24578 df-dv 24579 df-log 25260 df-cxp 25261

This theorem is referenced by: (None)

W3C validator