2sqlem6 - Metamath Proof Explorer

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Theorem 2sqlem6 27467

Description: Lemma for 2sq 27474. 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.)

**Hypotheses**
Ref	Expression
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**
Ref	Expression
2sqlem6	⊢ (𝜑 → 𝐴 ∈ 𝑆)

Distinct variable groups: 𝑤,𝑝 𝜑,𝑝 𝐵,𝑝 𝑆,𝑝 Allowed substitution hints: 𝜑(𝑤) 𝐴(𝑤,𝑝) 𝐵(𝑤) 𝑆(𝑤)

Proof of Theorem 2sqlem6 Dummy variables 𝑛 𝑥 𝑦 𝑧 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2sqlem6.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℕ)
2		2sqlem6.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℕ)
3		2sqlem6.3	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ ℙ (𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆))
4		breq2 5147	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 1))
5	4	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 1 → ((𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) ↔ (𝑝 ∥ 1 → 𝑝 ∈ 𝑆)))
6	5	ralbidv 3178	. . . . 5 ⊢ (𝑥 = 1 → (∀𝑝 ∈ ℙ (𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) ↔ ∀𝑝 ∈ ℙ (𝑝 ∥ 1 → 𝑝 ∈ 𝑆)))
7		oveq2 7439	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝑚 · 𝑥) = (𝑚 · 1))
8	7	eleq1d 2826	. . . . . . 7 ⊢ (𝑥 = 1 → ((𝑚 · 𝑥) ∈ 𝑆 ↔ (𝑚 · 1) ∈ 𝑆))
9	8	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 1 → (((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ((𝑚 · 1) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
10	9	ralbidv 3178	. . . . 5 ⊢ (𝑥 = 1 → (∀𝑚 ∈ ℕ ((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ∀𝑚 ∈ ℕ ((𝑚 · 1) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
11	6, 10	imbi12d 344	. . . 4 ⊢ (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))))
12		breq2 5147	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝑦))
13	12	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) ↔ (𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆)))
14	13	ralbidv 3178	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑝 ∈ ℙ (𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) ↔ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆)))
15		oveq2 7439	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑚 · 𝑥) = (𝑚 · 𝑦))
16	15	eleq1d 2826	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑚 · 𝑥) ∈ 𝑆 ↔ (𝑚 · 𝑦) ∈ 𝑆))
17	16	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ((𝑚 · 𝑦) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
18	17	ralbidv 3178	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑚 ∈ ℕ ((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ∀𝑚 ∈ ℕ ((𝑚 · 𝑦) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
19	14, 18	imbi12d 344	. . . 4 ⊢ (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))))
20		breq2 5147	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝑧))
21	20	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) ↔ (𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆)))
22	21	ralbidv 3178	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑝 ∈ ℙ (𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) ↔ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆)))
23		oveq2 7439	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑚 · 𝑥) = (𝑚 · 𝑧))
24	23	eleq1d 2826	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑚 · 𝑥) ∈ 𝑆 ↔ (𝑚 · 𝑧) ∈ 𝑆))
25	24	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ((𝑚 · 𝑧) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
26	25	ralbidv 3178	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑚 ∈ ℕ ((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ∀𝑚 ∈ ℕ ((𝑚 · 𝑧) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
27	22, 26	imbi12d 344	. . . 4 ⊢ (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))))
28		breq2 5147	. . . . . . 7 ⊢ (𝑥 = (𝑦 · 𝑧) → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ (𝑦 · 𝑧)))
29	28	imbi1d 341	. . . . . 6 ⊢ (𝑥 = (𝑦 · 𝑧) → ((𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) ↔ (𝑝 ∥ (𝑦 · 𝑧) → 𝑝 ∈ 𝑆)))
30	29	ralbidv 3178	. . . . 5 ⊢ (𝑥 = (𝑦 · 𝑧) → (∀𝑝 ∈ ℙ (𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) ↔ ∀𝑝 ∈ ℙ (𝑝 ∥ (𝑦 · 𝑧) → 𝑝 ∈ 𝑆)))
31		oveq2 7439	. . . . . . . 8 ⊢ (𝑥 = (𝑦 · 𝑧) → (𝑚 · 𝑥) = (𝑚 · (𝑦 · 𝑧)))
32	31	eleq1d 2826	. . . . . . 7 ⊢ (𝑥 = (𝑦 · 𝑧) → ((𝑚 · 𝑥) ∈ 𝑆 ↔ (𝑚 · (𝑦 · 𝑧)) ∈ 𝑆))
33	32	imbi1d 341	. . . . . 6 ⊢ (𝑥 = (𝑦 · 𝑧) → (((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ((𝑚 · (𝑦 · 𝑧)) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
34	33	ralbidv 3178	. . . . 5 ⊢ (𝑥 = (𝑦 · 𝑧) → (∀𝑚 ∈ ℕ ((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ∀𝑚 ∈ ℕ ((𝑚 · (𝑦 · 𝑧)) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
35	30, 34	imbi12d 344	. . . 4 ⊢ (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))))
36		breq2 5147	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝐵))
37	36	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) ↔ (𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆)))
38	37	ralbidv 3178	. . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑝 ∈ ℙ (𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) ↔ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆)))
39		oveq2 7439	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑚 · 𝑥) = (𝑚 · 𝐵))
40	39	eleq1d 2826	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝑚 · 𝑥) ∈ 𝑆 ↔ (𝑚 · 𝐵) ∈ 𝑆))
41	40	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 𝐵 → (((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ((𝑚 · 𝐵) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
42	41	ralbidv 3178	. . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑚 ∈ ℕ ((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ∀𝑚 ∈ ℕ ((𝑚 · 𝐵) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
43	38, 42	imbi12d 344	. . . 4 ⊢ (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))))
44		nncn 12274	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
45	44	mulridd 11278	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → (𝑚 · 1) = 𝑚)
46	45	eleq1d 2826	. . . . . . 7 ⊢ (𝑚 ∈ ℕ → ((𝑚 · 1) ∈ 𝑆 ↔ 𝑚 ∈ 𝑆))
47	46	biimpd 229	. . . . . 6 ⊢ (𝑚 ∈ ℕ → ((𝑚 · 1) ∈ 𝑆 → 𝑚 ∈ 𝑆))
48	47	rgen 3063	. . . . 5 ⊢ ∀𝑚 ∈ ℕ ((𝑚 · 1) ∈ 𝑆 → 𝑚 ∈ 𝑆)
49	48	a1i 11	. . . 4 ⊢ (∀𝑝 ∈ ℙ (𝑝 ∥ 1 → 𝑝 ∈ 𝑆) → ∀𝑚 ∈ ℕ ((𝑚 · 1) ∈ 𝑆 → 𝑚 ∈ 𝑆))
50		breq1 5146	. . . . . . 7 ⊢ (𝑝 = 𝑥 → (𝑝 ∥ 𝑥 ↔ 𝑥 ∥ 𝑥))
51		eleq1 2829	. . . . . . 7 ⊢ (𝑝 = 𝑥 → (𝑝 ∈ 𝑆 ↔ 𝑥 ∈ 𝑆))
52	50, 51	imbi12d 344	. . . . . 6 ⊢ (𝑝 = 𝑥 → ((𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) ↔ (𝑥 ∥ 𝑥 → 𝑥 ∈ 𝑆)))
53	52	rspcv 3618	. . . . 5 ⊢ (𝑥 ∈ ℙ → (∀𝑝 ∈ ℙ (𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) → (𝑥 ∥ 𝑥 → 𝑥 ∈ 𝑆)))
54		prmz 16712	. . . . . . 7 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℤ)
55		iddvds 16307	. . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∥ 𝑥)
56	54, 55	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℙ → 𝑥 ∥ 𝑥)
57		2sq.1	. . . . . . . . . 10 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
58		simprl 771	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆) ∧ (𝑚 ∈ ℕ ∧ (𝑚 · 𝑥) ∈ 𝑆)) → 𝑚 ∈ ℕ)
59		simpll 767	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆) ∧ (𝑚 ∈ ℕ ∧ (𝑚 · 𝑥) ∈ 𝑆)) → 𝑥 ∈ ℙ)
60		simprr 773	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆) ∧ (𝑚 ∈ ℕ ∧ (𝑚 · 𝑥) ∈ 𝑆)) → (𝑚 · 𝑥) ∈ 𝑆)
61		simplr 769	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆) ∧ (𝑚 ∈ ℕ ∧ (𝑚 · 𝑥) ∈ 𝑆)) → 𝑥 ∈ 𝑆)
62	57, 58, 59, 60, 61	2sqlem5 27466	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆) ∧ (𝑚 ∈ ℕ ∧ (𝑚 · 𝑥) ∈ 𝑆)) → 𝑚 ∈ 𝑆)
63	62	expr 456	. . . . . . . 8 ⊢ (((𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆) ∧ 𝑚 ∈ ℕ) → ((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆))
64	63	ralrimiva 3146	. . . . . . 7 ⊢ ((𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆) → ∀𝑚 ∈ ℕ ((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆))
65	64	ex 412	. . . . . 6 ⊢ (𝑥 ∈ ℙ → (𝑥 ∈ 𝑆 → ∀𝑚 ∈ ℕ ((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
66	56, 65	embantd 59	. . . . 5 ⊢ (𝑥 ∈ ℙ → ((𝑥 ∥ 𝑥 → 𝑥 ∈ 𝑆) → ∀𝑚 ∈ ℕ ((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
67	53, 66	syld 47	. . . 4 ⊢ (𝑥 ∈ ℙ → (∀𝑝 ∈ ℙ (𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆) → ∀𝑚 ∈ ℕ ((𝑚 · 𝑥) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
68		anim12 809	. . . . 5 ⊢ (((∀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))))
69		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
70		eluzelz 12888	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (ℤ_≥‘2) → 𝑦 ∈ ℤ)
71	70	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ 𝑝 ∈ ℙ) → 𝑦 ∈ ℤ)
72		eluzelz 12888	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℤ)
73	72	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ 𝑝 ∈ ℙ) → 𝑧 ∈ ℤ)
74		euclemma 16750	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ ℙ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (𝑝 ∥ (𝑦 · 𝑧) ↔ (𝑝 ∥ 𝑦 ∨ 𝑝 ∥ 𝑧)))
75	69, 71, 73, 74	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝑦 · 𝑧) ↔ (𝑝 ∥ 𝑦 ∨ 𝑝 ∥ 𝑧)))
76	75	imbi1d 341	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ (𝑦 · 𝑧) → 𝑝 ∈ 𝑆) ↔ ((𝑝 ∥ 𝑦 ∨ 𝑝 ∥ 𝑧) → 𝑝 ∈ 𝑆)))
77		jaob 964	. . . . . . . . . . . 12 ⊢ (((𝑝 ∥ 𝑦 ∨ 𝑝 ∥ 𝑧) → 𝑝 ∈ 𝑆) ↔ ((𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆) ∧ (𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆)))
78	76, 77	bitrdi 287	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ (𝑦 · 𝑧) → 𝑝 ∈ 𝑆) ↔ ((𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆) ∧ (𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆))))
79	78	ralbidva 3176	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) → (∀𝑝 ∈ ℙ (𝑝 ∥ (𝑦 · 𝑧) → 𝑝 ∈ 𝑆) ↔ ∀𝑝 ∈ ℙ ((𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆) ∧ (𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆))))
80		r19.26 3111	. . . . . . . . . 10 ⊢ (∀𝑝 ∈ ℙ ((𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆) ∧ (𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆)) ↔ (∀𝑝 ∈ ℙ (𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆) ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆)))
81	79, 80	bitrdi 287	. . . . . . . . 9 ⊢ ((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))))
82	81	biimpa 476	. . . . . . . 8 ⊢ (((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		oveq1 7438	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑚 · 𝑦) = (𝑛 · 𝑦))
84	83	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑚 · 𝑦) ∈ 𝑆 ↔ (𝑛 · 𝑦) ∈ 𝑆))
85		eleq1 2829	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ 𝑆 ↔ 𝑛 ∈ 𝑆))
86	84, 85	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((𝑚 · 𝑦) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ((𝑛 · 𝑦) ∈ 𝑆 → 𝑛 ∈ 𝑆)))
87	86	cbvralvw 3237	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ ℕ ((𝑚 · 𝑦) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ∀𝑛 ∈ ℕ ((𝑛 · 𝑦) ∈ 𝑆 → 𝑛 ∈ 𝑆))
88	44	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ (𝑦 · 𝑧) → 𝑝 ∈ 𝑆)) ∧ ∀𝑛 ∈ ℕ ((𝑛 · 𝑦) ∈ 𝑆 → 𝑛 ∈ 𝑆)) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
89		uzssz 12899	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ_≥‘2) ⊆ ℤ
90		zsscn 12621	. . . . . . . . . . . . . . . . 17 ⊢ ℤ ⊆ ℂ
91	89, 90	sstri 3993	. . . . . . . . . . . . . . . 16 ⊢ (ℤ_≥‘2) ⊆ ℂ
92		simpll 767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ (𝑦 · 𝑧) → 𝑝 ∈ 𝑆)) → 𝑦 ∈ (ℤ_≥‘2))
93	92	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((((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))
94	91, 93	sselid 3981	. . . . . . . . . . . . . . 15 ⊢ (((((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ (𝑦 · 𝑧) → 𝑝 ∈ 𝑆)) ∧ ∀𝑛 ∈ ℕ ((𝑛 · 𝑦) ∈ 𝑆 → 𝑛 ∈ 𝑆)) ∧ 𝑚 ∈ ℕ) → 𝑦 ∈ ℂ)
95		simplr 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ (𝑦 · 𝑧) → 𝑝 ∈ 𝑆)) → 𝑧 ∈ (ℤ_≥‘2))
96	95	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((((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))
97	91, 96	sselid 3981	. . . . . . . . . . . . . . 15 ⊢ (((((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ (𝑦 · 𝑧) → 𝑝 ∈ 𝑆)) ∧ ∀𝑛 ∈ ℕ ((𝑛 · 𝑦) ∈ 𝑆 → 𝑛 ∈ 𝑆)) ∧ 𝑚 ∈ ℕ) → 𝑧 ∈ ℂ)
98		mul32 11427	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑚 · 𝑦) · 𝑧) = ((𝑚 · 𝑧) · 𝑦))
99		mulass 11243	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑚 · 𝑦) · 𝑧) = (𝑚 · (𝑦 · 𝑧)))
100	98, 99	eqtr3d 2779	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑚 · 𝑧) · 𝑦) = (𝑚 · (𝑦 · 𝑧)))
101	88, 94, 97, 100	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((((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)))
102	101	eleq1d 2826	. . . . . . . . . . . . 13 ⊢ (((((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))
103		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ (𝑦 · 𝑧) → 𝑝 ∈ 𝑆)) ∧ ∀𝑛 ∈ ℕ ((𝑛 · 𝑦) ∈ 𝑆 → 𝑛 ∈ 𝑆)) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
104		eluz2nn 12924	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℕ)
105	96, 104	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝑦 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ (𝑦 · 𝑧) → 𝑝 ∈ 𝑆)) ∧ ∀𝑛 ∈ ℕ ((𝑛 · 𝑦) ∈ 𝑆 → 𝑛 ∈ 𝑆)) ∧ 𝑚 ∈ ℕ) → 𝑧 ∈ ℕ)
106	103, 105	nnmulcld 12319	. . . . . . . . . . . . . 14 ⊢ (((((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)
107		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((((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))
108		oveq1 7438	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑚 · 𝑧) → (𝑛 · 𝑦) = ((𝑚 · 𝑧) · 𝑦))
109	108	eleq1d 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑚 · 𝑧) → ((𝑛 · 𝑦) ∈ 𝑆 ↔ ((𝑚 · 𝑧) · 𝑦) ∈ 𝑆))
110		eleq1 2829	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑚 · 𝑧) → (𝑛 ∈ 𝑆 ↔ (𝑚 · 𝑧) ∈ 𝑆))
111	109, 110	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑚 · 𝑧) → (((𝑛 · 𝑦) ∈ 𝑆 → 𝑛 ∈ 𝑆) ↔ (((𝑚 · 𝑧) · 𝑦) ∈ 𝑆 → (𝑚 · 𝑧) ∈ 𝑆)))
112	111	rspcv 3618	. . . . . . . . . . . . . 14 ⊢ ((𝑚 · 𝑧) ∈ ℕ → (∀𝑛 ∈ ℕ ((𝑛 · 𝑦) ∈ 𝑆 → 𝑛 ∈ 𝑆) → (((𝑚 · 𝑧) · 𝑦) ∈ 𝑆 → (𝑚 · 𝑧) ∈ 𝑆)))
113	106, 107, 112	sylc 65	. . . . . . . . . . . . 13 ⊢ (((((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))
114	102, 113	sylbird 260	. . . . . . . . . . . 12 ⊢ (((((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))
115	114	imim1d 82	. . . . . . . . . . 11 ⊢ (((((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)))
116	115	ralimdva 3167	. . . . . . . . . 10 ⊢ ((((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)))
117	87, 116	sylan2b 594	. . . . . . . . 9 ⊢ ((((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)))
118	117	expimpd 453	. . . . . . . 8 ⊢ (((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	82, 118	embantd 59	. . . . . . 7 ⊢ (((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)))
120	119	ex 412	. . . . . 6 ⊢ ((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	com23 86	. . . . 5 ⊢ ((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))))
122	68, 121	syl5 34	. . . 4 ⊢ ((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))))
123	11, 19, 27, 35, 43, 49, 67, 122	prmind 16723	. . 3 ⊢ (𝐵 ∈ ℕ → (∀𝑝 ∈ ℙ (𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆) → ∀𝑚 ∈ ℕ ((𝑚 · 𝐵) ∈ 𝑆 → 𝑚 ∈ 𝑆)))
124	2, 3, 123	sylc 65	. 2 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ((𝑚 · 𝐵) ∈ 𝑆 → 𝑚 ∈ 𝑆))
125		2sqlem6.4	. 2 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ 𝑆)
126		oveq1 7438	. . . . 5 ⊢ (𝑚 = 𝐴 → (𝑚 · 𝐵) = (𝐴 · 𝐵))
127	126	eleq1d 2826	. . . 4 ⊢ (𝑚 = 𝐴 → ((𝑚 · 𝐵) ∈ 𝑆 ↔ (𝐴 · 𝐵) ∈ 𝑆))
128		eleq1 2829	. . . 4 ⊢ (𝑚 = 𝐴 → (𝑚 ∈ 𝑆 ↔ 𝐴 ∈ 𝑆))
129	127, 128	imbi12d 344	. . 3 ⊢ (𝑚 = 𝐴 → (((𝑚 · 𝐵) ∈ 𝑆 → 𝑚 ∈ 𝑆) ↔ ((𝐴 · 𝐵) ∈ 𝑆 → 𝐴 ∈ 𝑆)))
130	129	rspcv 3618	. 2 ⊢ (𝐴 ∈ ℕ → (∀𝑚 ∈ ℕ ((𝑚 · 𝐵) ∈ 𝑆 → 𝑚 ∈ 𝑆) → ((𝐴 · 𝐵) ∈ 𝑆 → 𝐴 ∈ 𝑆)))
131	1, 124, 125, 130	syl3c 66	1 ⊢ (𝜑 → 𝐴 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 1c1 11156 · cmul 11160 ℕcn 12266 2c2 12321 ℤcz 12613 ℤ_≥cuz 12878 ↑cexp 14102 abscabs 15273 ∥ cdvds 16290 ℙcprime 16708 ℤ[i]cgz 16967

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-prm 16709 df-gz 16968

This theorem is referenced by: 2sqlem8 27470

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