Theorem 2sqlem6

Description: Lemma for 2sq . If a number that is a sum of two squares is divisible by a number whose prime divisors are all sums of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015)

	Ref	Expression
Hypotheses	2sq.1	⊢ 𝑆 = ran ( 𝑤 ∈ ℤ[i] ↦ ( ( abs ‘ 𝑤 ) ↑ 2 ) )
	2sqlem6.1	⊢ ( 𝜑 → 𝐴 ∈ ℕ )
	2sqlem6.2	⊢ ( 𝜑 → 𝐵 ∈ ℕ )
	2sqlem6.3	⊢ ( 𝜑 → ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆 ) )
	2sqlem6.4	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ 𝑆 )
Assertion	2sqlem6	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		2sq.1	⊢ 𝑆 = ran ( 𝑤 ∈ ℤ[i] ↦ ( ( abs ‘ 𝑤 ) ↑ 2 ) )
2		2sqlem6.1	⊢ ( 𝜑 → 𝐴 ∈ ℕ )
3		2sqlem6.2	⊢ ( 𝜑 → 𝐵 ∈ ℕ )
4		2sqlem6.3	⊢ ( 𝜑 → ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆 ) )
5		2sqlem6.4	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ 𝑆 )
6		breq2	⊢ ( 𝑥 = 1 → ( 𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 1 ) )
7	6	imbi1d	⊢ ( 𝑥 = 1 → ( ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ( 𝑝 ∥ 1 → 𝑝 ∈ 𝑆 ) ) )
8	7	ralbidv	⊢ ( 𝑥 = 1 → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 1 → 𝑝 ∈ 𝑆 ) ) )
9		oveq2	⊢ ( 𝑥 = 1 → ( 𝑚 · 𝑥 ) = ( 𝑚 · 1 ) )
10	9	eleq1d	⊢ ( 𝑥 = 1 → ( ( 𝑚 · 𝑥 ) ∈ 𝑆 ↔ ( 𝑚 · 1 ) ∈ 𝑆 ) )
11	10	imbi1d	⊢ ( 𝑥 = 1 → ( ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝑚 · 1 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
12	11	ralbidv	⊢ ( 𝑥 = 1 → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 1 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
13	8 12	imbi12d	⊢ ( 𝑥 = 1 → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 1 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 1 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) )
14		breq2	⊢ ( 𝑥 = 𝑦 → ( 𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝑦 ) )
15	14	imbi1d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ) )
16	15	ralbidv	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ) )
17		oveq2	⊢ ( 𝑥 = 𝑦 → ( 𝑚 · 𝑥 ) = ( 𝑚 · 𝑦 ) )
18	17	eleq1d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑚 · 𝑥 ) ∈ 𝑆 ↔ ( 𝑚 · 𝑦 ) ∈ 𝑆 ) )
19	18	imbi1d	⊢ ( 𝑥 = 𝑦 → ( ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
20	19	ralbidv	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
21	16 20	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) )
22		breq2	⊢ ( 𝑥 = 𝑧 → ( 𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝑧 ) )
23	22	imbi1d	⊢ ( 𝑥 = 𝑧 → ( ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) )
24	23	ralbidv	⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) )
25		oveq2	⊢ ( 𝑥 = 𝑧 → ( 𝑚 · 𝑥 ) = ( 𝑚 · 𝑧 ) )
26	25	eleq1d	⊢ ( 𝑥 = 𝑧 → ( ( 𝑚 · 𝑥 ) ∈ 𝑆 ↔ ( 𝑚 · 𝑧 ) ∈ 𝑆 ) )
27	26	imbi1d	⊢ ( 𝑥 = 𝑧 → ( ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
28	27	ralbidv	⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
29	24 28	imbi12d	⊢ ( 𝑥 = 𝑧 → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) )
30		breq2	⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( 𝑝 ∥ 𝑥 ↔ 𝑝 ∥ ( 𝑦 · 𝑧 ) ) )
31	30	imbi1d	⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) )
32	31	ralbidv	⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) )
33		oveq2	⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( 𝑚 · 𝑥 ) = ( 𝑚 · ( 𝑦 · 𝑧 ) ) )
34	33	eleq1d	⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( ( 𝑚 · 𝑥 ) ∈ 𝑆 ↔ ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 ) )
35	34	imbi1d	⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
36	35	ralbidv	⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
37	32 36	imbi12d	⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) )
38		breq2	⊢ ( 𝑥 = 𝐵 → ( 𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝐵 ) )
39	38	imbi1d	⊢ ( 𝑥 = 𝐵 → ( ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ( 𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆 ) ) )
40	39	ralbidv	⊢ ( 𝑥 = 𝐵 → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆 ) ) )
41		oveq2	⊢ ( 𝑥 = 𝐵 → ( 𝑚 · 𝑥 ) = ( 𝑚 · 𝐵 ) )
42	41	eleq1d	⊢ ( 𝑥 = 𝐵 → ( ( 𝑚 · 𝑥 ) ∈ 𝑆 ↔ ( 𝑚 · 𝐵 ) ∈ 𝑆 ) )
43	42	imbi1d	⊢ ( 𝑥 = 𝐵 → ( ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
44	43	ralbidv	⊢ ( 𝑥 = 𝐵 → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
45	40 44	imbi12d	⊢ ( 𝑥 = 𝐵 → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) )
46		nncn	⊢ ( 𝑚 ∈ ℕ → 𝑚 ∈ ℂ )
47	46	mulridd	⊢ ( 𝑚 ∈ ℕ → ( 𝑚 · 1 ) = 𝑚 )
48	47	eleq1d	⊢ ( 𝑚 ∈ ℕ → ( ( 𝑚 · 1 ) ∈ 𝑆 ↔ 𝑚 ∈ 𝑆 ) )
49	48	biimpd	⊢ ( 𝑚 ∈ ℕ → ( ( 𝑚 · 1 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) )
50	49	rgen	⊢ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 1 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 )
51	50	a1i	⊢ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 1 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 1 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) )
52		breq1	⊢ ( 𝑝 = 𝑥 → ( 𝑝 ∥ 𝑥 ↔ 𝑥 ∥ 𝑥 ) )
53		eleq1	⊢ ( 𝑝 = 𝑥 → ( 𝑝 ∈ 𝑆 ↔ 𝑥 ∈ 𝑆 ) )
54	52 53	imbi12d	⊢ ( 𝑝 = 𝑥 → ( ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ( 𝑥 ∥ 𝑥 → 𝑥 ∈ 𝑆 ) ) )
55	54	rspcv	⊢ ( 𝑥 ∈ ℙ → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ( 𝑥 ∥ 𝑥 → 𝑥 ∈ 𝑆 ) ) )
56		prmz	⊢ ( 𝑥 ∈ ℙ → 𝑥 ∈ ℤ )
57		iddvds	⊢ ( 𝑥 ∈ ℤ → 𝑥 ∥ 𝑥 )
58	56 57	syl	⊢ ( 𝑥 ∈ ℙ → 𝑥 ∥ 𝑥 )
59		simprl	⊢ ( ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) ∧ ( 𝑚 ∈ ℕ ∧ ( 𝑚 · 𝑥 ) ∈ 𝑆 ) ) → 𝑚 ∈ ℕ )
60		simpll	⊢ ( ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) ∧ ( 𝑚 ∈ ℕ ∧ ( 𝑚 · 𝑥 ) ∈ 𝑆 ) ) → 𝑥 ∈ ℙ )
61		simprr	⊢ ( ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) ∧ ( 𝑚 ∈ ℕ ∧ ( 𝑚 · 𝑥 ) ∈ 𝑆 ) ) → ( 𝑚 · 𝑥 ) ∈ 𝑆 )
62		simplr	⊢ ( ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) ∧ ( 𝑚 ∈ ℕ ∧ ( 𝑚 · 𝑥 ) ∈ 𝑆 ) ) → 𝑥 ∈ 𝑆 )
63	1 59 60 61 62	2sqlem5	⊢ ( ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) ∧ ( 𝑚 ∈ ℕ ∧ ( 𝑚 · 𝑥 ) ∈ 𝑆 ) ) → 𝑚 ∈ 𝑆 )
64	63	expr	⊢ ( ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) )
65	64	ralrimiva	⊢ ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) )
66	65	ex	⊢ ( 𝑥 ∈ ℙ → ( 𝑥 ∈ 𝑆 → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
67	58 66	embantd	⊢ ( 𝑥 ∈ ℙ → ( ( 𝑥 ∥ 𝑥 → 𝑥 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
68	55 67	syld	⊢ ( 𝑥 ∈ ℙ → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
69		anim12	⊢ ( ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ∧ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) )
70		simpr	⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ 𝑝 ∈ ℙ ) → 𝑝 ∈ ℙ )
71		eluzelz	⊢ ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) → 𝑦 ∈ ℤ )
72	71	ad2antrr	⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ 𝑝 ∈ ℙ ) → 𝑦 ∈ ℤ )
73		eluzelz	⊢ ( 𝑧 ∈ ( ℤ≥ ‘ 2 ) → 𝑧 ∈ ℤ )
74	73	ad2antlr	⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ 𝑝 ∈ ℙ ) → 𝑧 ∈ ℤ )
75		euclemma	⊢ ( ( 𝑝 ∈ ℙ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ ) → ( 𝑝 ∥ ( 𝑦 · 𝑧 ) ↔ ( 𝑝 ∥ 𝑦 ∨ 𝑝 ∥ 𝑧 ) ) )
76	70 72 74 75	syl3anc	⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( 𝑦 · 𝑧 ) ↔ ( 𝑝 ∥ 𝑦 ∨ 𝑝 ∥ 𝑧 ) ) )
77	76	imbi1d	⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ↔ ( ( 𝑝 ∥ 𝑦 ∨ 𝑝 ∥ 𝑧 ) → 𝑝 ∈ 𝑆 ) ) )
78		jaob	⊢ ( ( ( 𝑝 ∥ 𝑦 ∨ 𝑝 ∥ 𝑧 ) → 𝑝 ∈ 𝑆 ) ↔ ( ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) )
79	77 78	bitrdi	⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ↔ ( ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ) )
80	79	ralbidva	⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ↔ ∀ 𝑝 ∈ ℙ ( ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ) )
81		r19.26	⊢ ( ∀ 𝑝 ∈ ℙ ( ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) )
82	80 81	bitrdi	⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ) )
83	82	biimpa	⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) )
84		oveq1	⊢ ( 𝑚 = 𝑛 → ( 𝑚 · 𝑦 ) = ( 𝑛 · 𝑦 ) )
85	84	eleq1d	⊢ ( 𝑚 = 𝑛 → ( ( 𝑚 · 𝑦 ) ∈ 𝑆 ↔ ( 𝑛 · 𝑦 ) ∈ 𝑆 ) )
86		eleq1	⊢ ( 𝑚 = 𝑛 → ( 𝑚 ∈ 𝑆 ↔ 𝑛 ∈ 𝑆 ) )
87	85 86	imbi12d	⊢ ( 𝑚 = 𝑛 → ( ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) )
88	87	cbvralvw	⊢ ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) )
89	46	adantl	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑚 ∈ ℂ )
90		uzssz	⊢ ( ℤ≥ ‘ 2 ) ⊆ ℤ
91		zsscn	⊢ ℤ ⊆ ℂ
92	90 91	sstri	⊢ ( ℤ≥ ‘ 2 ) ⊆ ℂ
93		simpll	⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) → 𝑦 ∈ ( ℤ≥ ‘ 2 ) )
94	93	ad2antrr	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑦 ∈ ( ℤ≥ ‘ 2 ) )
95	92 94	sselid	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑦 ∈ ℂ )
96		simplr	⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) → 𝑧 ∈ ( ℤ≥ ‘ 2 ) )
97	96	ad2antrr	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑧 ∈ ( ℤ≥ ‘ 2 ) )
98	92 97	sselid	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑧 ∈ ℂ )
99		mul32	⊢ ( ( 𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → ( ( 𝑚 · 𝑦 ) · 𝑧 ) = ( ( 𝑚 · 𝑧 ) · 𝑦 ) )
100		mulass	⊢ ( ( 𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → ( ( 𝑚 · 𝑦 ) · 𝑧 ) = ( 𝑚 · ( 𝑦 · 𝑧 ) ) )
101	99 100	eqtr3d	⊢ ( ( 𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → ( ( 𝑚 · 𝑧 ) · 𝑦 ) = ( 𝑚 · ( 𝑦 · 𝑧 ) ) )
102	89 95 98 101	syl3anc	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑚 · 𝑧 ) · 𝑦 ) = ( 𝑚 · ( 𝑦 · 𝑧 ) ) )
103	102	eleq1d	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑚 · 𝑧 ) · 𝑦 ) ∈ 𝑆 ↔ ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 ) )
104		simpr	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑚 ∈ ℕ )
105		eluz2nn	⊢ ( 𝑧 ∈ ( ℤ≥ ‘ 2 ) → 𝑧 ∈ ℕ )
106	97 105	syl	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑧 ∈ ℕ )
107	104 106	nnmulcld	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝑚 · 𝑧 ) ∈ ℕ )
108		simplr	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) )
109		oveq1	⊢ ( 𝑛 = ( 𝑚 · 𝑧 ) → ( 𝑛 · 𝑦 ) = ( ( 𝑚 · 𝑧 ) · 𝑦 ) )
110	109	eleq1d	⊢ ( 𝑛 = ( 𝑚 · 𝑧 ) → ( ( 𝑛 · 𝑦 ) ∈ 𝑆 ↔ ( ( 𝑚 · 𝑧 ) · 𝑦 ) ∈ 𝑆 ) )
111		eleq1	⊢ ( 𝑛 = ( 𝑚 · 𝑧 ) → ( 𝑛 ∈ 𝑆 ↔ ( 𝑚 · 𝑧 ) ∈ 𝑆 ) )
112	110 111	imbi12d	⊢ ( 𝑛 = ( 𝑚 · 𝑧 ) → ( ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ↔ ( ( ( 𝑚 · 𝑧 ) · 𝑦 ) ∈ 𝑆 → ( 𝑚 · 𝑧 ) ∈ 𝑆 ) ) )
113	112	rspcv	⊢ ( ( 𝑚 · 𝑧 ) ∈ ℕ → ( ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) → ( ( ( 𝑚 · 𝑧 ) · 𝑦 ) ∈ 𝑆 → ( 𝑚 · 𝑧 ) ∈ 𝑆 ) ) )
114	107 108 113	sylc	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑚 · 𝑧 ) · 𝑦 ) ∈ 𝑆 → ( 𝑚 · 𝑧 ) ∈ 𝑆 ) )
115	103 114	sylbird	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → ( 𝑚 · 𝑧 ) ∈ 𝑆 ) )
116	115	imim1d	⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) → ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
117	116	ralimdva	⊢ ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
118	88 117	sylan2b	⊢ ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
119	118	expimpd	⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) → ( ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
120	83 119	embantd	⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) → ( ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
121	120	ex	⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) → ( ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) )
122	121	com23	⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) → ( ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) )
123	69 122	syl5	⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) → ( ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ∧ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) )
124	13 21 29 37 45 51 68 123	prmind	⊢ ( 𝐵 ∈ ℕ → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) )
125	3 4 124	sylc	⊢ ( 𝜑 → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) )
126		oveq1	⊢ ( 𝑚 = 𝐴 → ( 𝑚 · 𝐵 ) = ( 𝐴 · 𝐵 ) )
127	126	eleq1d	⊢ ( 𝑚 = 𝐴 → ( ( 𝑚 · 𝐵 ) ∈ 𝑆 ↔ ( 𝐴 · 𝐵 ) ∈ 𝑆 ) )
128		eleq1	⊢ ( 𝑚 = 𝐴 → ( 𝑚 ∈ 𝑆 ↔ 𝐴 ∈ 𝑆 ) )
129	127 128	imbi12d	⊢ ( 𝑚 = 𝐴 → ( ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝐴 · 𝐵 ) ∈ 𝑆 → 𝐴 ∈ 𝑆 ) ) )
130	129	rspcv	⊢ ( 𝐴 ∈ ℕ → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) → ( ( 𝐴 · 𝐵 ) ∈ 𝑆 → 𝐴 ∈ 𝑆 ) ) )
131	2 125 5 130	syl3c	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )