Theorem flt4lem5a

Description: Part 1 of Equation 1 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html . (Contributed by SN, 22-Aug-2024)

	Ref	Expression
Hypotheses	flt4lem5a.m	⊢ 𝑀 = ( ( ( √ ‘ ( 𝐶 + ( 𝐵 ↑ 2 ) ) ) + ( √ ‘ ( 𝐶 − ( 𝐵 ↑ 2 ) ) ) ) / 2 )
	flt4lem5a.n	⊢ 𝑁 = ( ( ( √ ‘ ( 𝐶 + ( 𝐵 ↑ 2 ) ) ) − ( √ ‘ ( 𝐶 − ( 𝐵 ↑ 2 ) ) ) ) / 2 )
	flt4lem5a.r	⊢ 𝑅 = ( ( ( √ ‘ ( 𝑀 + 𝑁 ) ) + ( √ ‘ ( 𝑀 − 𝑁 ) ) ) / 2 )
	flt4lem5a.s	⊢ 𝑆 = ( ( ( √ ‘ ( 𝑀 + 𝑁 ) ) − ( √ ‘ ( 𝑀 − 𝑁 ) ) ) / 2 )
	flt4lem5a.a	⊢ ( 𝜑 → 𝐴 ∈ ℕ )
	flt4lem5a.b	⊢ ( 𝜑 → 𝐵 ∈ ℕ )
	flt4lem5a.c	⊢ ( 𝜑 → 𝐶 ∈ ℕ )
	flt4lem5a.1	⊢ ( 𝜑 → ¬ 2 ∥ 𝐴 )
	flt4lem5a.2	⊢ ( 𝜑 → ( 𝐴 gcd 𝐶 ) = 1 )
	flt4lem5a.3	⊢ ( 𝜑 → ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) )
Assertion	flt4lem5a	⊢ ( 𝜑 → ( ( 𝐴 ↑ 2 ) + ( 𝑁 ↑ 2 ) ) = ( 𝑀 ↑ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		flt4lem5a.m	⊢ 𝑀 = ( ( ( √ ‘ ( 𝐶 + ( 𝐵 ↑ 2 ) ) ) + ( √ ‘ ( 𝐶 − ( 𝐵 ↑ 2 ) ) ) ) / 2 )
2		flt4lem5a.n	⊢ 𝑁 = ( ( ( √ ‘ ( 𝐶 + ( 𝐵 ↑ 2 ) ) ) − ( √ ‘ ( 𝐶 − ( 𝐵 ↑ 2 ) ) ) ) / 2 )
3		flt4lem5a.r	⊢ 𝑅 = ( ( ( √ ‘ ( 𝑀 + 𝑁 ) ) + ( √ ‘ ( 𝑀 − 𝑁 ) ) ) / 2 )
4		flt4lem5a.s	⊢ 𝑆 = ( ( ( √ ‘ ( 𝑀 + 𝑁 ) ) − ( √ ‘ ( 𝑀 − 𝑁 ) ) ) / 2 )
5		flt4lem5a.a	⊢ ( 𝜑 → 𝐴 ∈ ℕ )
6		flt4lem5a.b	⊢ ( 𝜑 → 𝐵 ∈ ℕ )
7		flt4lem5a.c	⊢ ( 𝜑 → 𝐶 ∈ ℕ )
8		flt4lem5a.1	⊢ ( 𝜑 → ¬ 2 ∥ 𝐴 )
9		flt4lem5a.2	⊢ ( 𝜑 → ( 𝐴 gcd 𝐶 ) = 1 )
10		flt4lem5a.3	⊢ ( 𝜑 → ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( 𝐶 ↑ 2 ) )
11	5	nnsqcld	⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) ∈ ℕ )
12	6	nnsqcld	⊢ ( 𝜑 → ( 𝐵 ↑ 2 ) ∈ ℕ )
13		2prm	⊢ 2 ∈ ℙ
14	5	nnzd	⊢ ( 𝜑 → 𝐴 ∈ ℤ )
15		prmdvdssq	⊢ ( ( 2 ∈ ℙ ∧ 𝐴 ∈ ℤ ) → ( 2 ∥ 𝐴 ↔ 2 ∥ ( 𝐴 ↑ 2 ) ) )
16	13 14 15	sylancr	⊢ ( 𝜑 → ( 2 ∥ 𝐴 ↔ 2 ∥ ( 𝐴 ↑ 2 ) ) )
17	8 16	mtbid	⊢ ( 𝜑 → ¬ 2 ∥ ( 𝐴 ↑ 2 ) )
18		2nn	⊢ 2 ∈ ℕ
19	18	a1i	⊢ ( 𝜑 → 2 ∈ ℕ )
20		rplpwr	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ ∧ 2 ∈ ℕ ) → ( ( 𝐴 gcd 𝐶 ) = 1 → ( ( 𝐴 ↑ 2 ) gcd 𝐶 ) = 1 ) )
21	5 7 19 20	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 gcd 𝐶 ) = 1 → ( ( 𝐴 ↑ 2 ) gcd 𝐶 ) = 1 ) )
22	9 21	mpd	⊢ ( 𝜑 → ( ( 𝐴 ↑ 2 ) gcd 𝐶 ) = 1 )
23	5	nncnd	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
24	23	flt4lem	⊢ ( 𝜑 → ( 𝐴 ↑ 4 ) = ( ( 𝐴 ↑ 2 ) ↑ 2 ) )
25	6	nncnd	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
26	25	flt4lem	⊢ ( 𝜑 → ( 𝐵 ↑ 4 ) = ( ( 𝐵 ↑ 2 ) ↑ 2 ) )
27	24 26	oveq12d	⊢ ( 𝜑 → ( ( 𝐴 ↑ 4 ) + ( 𝐵 ↑ 4 ) ) = ( ( ( 𝐴 ↑ 2 ) ↑ 2 ) + ( ( 𝐵 ↑ 2 ) ↑ 2 ) ) )
28	27 10	eqtr3d	⊢ ( 𝜑 → ( ( ( 𝐴 ↑ 2 ) ↑ 2 ) + ( ( 𝐵 ↑ 2 ) ↑ 2 ) ) = ( 𝐶 ↑ 2 ) )
29	11 12 7 17 22 28	flt4lem1	⊢ ( 𝜑 → ( ( ( 𝐴 ↑ 2 ) ∈ ℕ ∧ ( 𝐵 ↑ 2 ) ∈ ℕ ∧ 𝐶 ∈ ℕ ) ∧ ( ( ( 𝐴 ↑ 2 ) ↑ 2 ) + ( ( 𝐵 ↑ 2 ) ↑ 2 ) ) = ( 𝐶 ↑ 2 ) ∧ ( ( ( 𝐴 ↑ 2 ) gcd ( 𝐵 ↑ 2 ) ) = 1 ∧ ¬ 2 ∥ ( 𝐴 ↑ 2 ) ) ) )
30	1	pythagtriplem11	⊢ ( ( ( ( 𝐴 ↑ 2 ) ∈ ℕ ∧ ( 𝐵 ↑ 2 ) ∈ ℕ ∧ 𝐶 ∈ ℕ ) ∧ ( ( ( 𝐴 ↑ 2 ) ↑ 2 ) + ( ( 𝐵 ↑ 2 ) ↑ 2 ) ) = ( 𝐶 ↑ 2 ) ∧ ( ( ( 𝐴 ↑ 2 ) gcd ( 𝐵 ↑ 2 ) ) = 1 ∧ ¬ 2 ∥ ( 𝐴 ↑ 2 ) ) ) → 𝑀 ∈ ℕ )
31	29 30	syl	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
32	31	nnsqcld	⊢ ( 𝜑 → ( 𝑀 ↑ 2 ) ∈ ℕ )
33	32	nncnd	⊢ ( 𝜑 → ( 𝑀 ↑ 2 ) ∈ ℂ )
34	2	pythagtriplem13	⊢ ( ( ( ( 𝐴 ↑ 2 ) ∈ ℕ ∧ ( 𝐵 ↑ 2 ) ∈ ℕ ∧ 𝐶 ∈ ℕ ) ∧ ( ( ( 𝐴 ↑ 2 ) ↑ 2 ) + ( ( 𝐵 ↑ 2 ) ↑ 2 ) ) = ( 𝐶 ↑ 2 ) ∧ ( ( ( 𝐴 ↑ 2 ) gcd ( 𝐵 ↑ 2 ) ) = 1 ∧ ¬ 2 ∥ ( 𝐴 ↑ 2 ) ) ) → 𝑁 ∈ ℕ )
35	29 34	syl	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
36	35	nnsqcld	⊢ ( 𝜑 → ( 𝑁 ↑ 2 ) ∈ ℕ )
37	36	nncnd	⊢ ( 𝜑 → ( 𝑁 ↑ 2 ) ∈ ℂ )
38	1 2	pythagtriplem15	⊢ ( ( ( ( 𝐴 ↑ 2 ) ∈ ℕ ∧ ( 𝐵 ↑ 2 ) ∈ ℕ ∧ 𝐶 ∈ ℕ ) ∧ ( ( ( 𝐴 ↑ 2 ) ↑ 2 ) + ( ( 𝐵 ↑ 2 ) ↑ 2 ) ) = ( 𝐶 ↑ 2 ) ∧ ( ( ( 𝐴 ↑ 2 ) gcd ( 𝐵 ↑ 2 ) ) = 1 ∧ ¬ 2 ∥ ( 𝐴 ↑ 2 ) ) ) → ( 𝐴 ↑ 2 ) = ( ( 𝑀 ↑ 2 ) − ( 𝑁 ↑ 2 ) ) )
39	29 38	syl	⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) = ( ( 𝑀 ↑ 2 ) − ( 𝑁 ↑ 2 ) ) )
40	33 37 39	mvrrsubd	⊢ ( 𝜑 → ( ( 𝐴 ↑ 2 ) + ( 𝑁 ↑ 2 ) ) = ( 𝑀 ↑ 2 ) )