Theorem sqff1o 25753

Description: There is a bijection from the squarefree divisors of a number 𝑁 to the powerset of the prime divisors of 𝑁. Among other things, this implies that a number has 2↑𝑘 squarefree divisors where 𝑘 is the number of prime divisors, and a squarefree number has 2↑𝑘 divisors (because all divisors of a squarefree number are squarefree). The inverse function to 𝐹 takes the product of all the primes in some subset of prime divisors of 𝑁. (Contributed by Mario Carneiro, 1-Jul-2015.)

Hypotheses

Ref	Expression
sqff1o.1	⊢ 𝑆 = {𝑥 ∈ ℕ ∣ ((μ‘𝑥) ≠ 0 ∧ 𝑥 ∥ 𝑁)}
sqff1o.2	⊢ 𝐹 = (𝑛 ∈ 𝑆 ↦ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛})
sqff1o.3	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))

Assertion

Ref	Expression
sqff1o	⊢ (𝑁 ∈ ℕ → 𝐹:𝑆–1-1-onto→𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})

Distinct variable groups:   𝑛,𝑝,𝑥,𝐺   𝑛,𝑁,𝑝,𝑥   𝑆,𝑛,𝑝

Allowed substitution hints:   𝑆(𝑥)   𝐹(𝑥,𝑛,𝑝)

Proof of Theorem sqff1o

Dummy variables 𝑘 𝑞 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sqff1o.2	. 2 ⊢ 𝐹 = (𝑛 ∈ 𝑆 ↦ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛})
2		fveq2 6665	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → (μ‘𝑥) = (μ‘𝑛))
3	2	neeq1d 3075	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → ((μ‘𝑥) ≠ 0 ↔ (μ‘𝑛) ≠ 0))
4		breq1 5062	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (𝑥 ∥ 𝑁 ↔ 𝑛 ∥ 𝑁))
5	3, 4	anbi12d 632	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (((μ‘𝑥) ≠ 0 ∧ 𝑥 ∥ 𝑁) ↔ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)))
6		sqff1o.1	. . . . . . . . 9 ⊢ 𝑆 = {𝑥 ∈ ℕ ∣ ((μ‘𝑥) ≠ 0 ∧ 𝑥 ∥ 𝑁)}
7	5, 6	elrab2 3683	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑆 ↔ (𝑛 ∈ ℕ ∧ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)))
8	7	simprbi 499	. . . . . . 7 ⊢ (𝑛 ∈ 𝑆 → ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁))
9	8	simprd 498	. . . . . 6 ⊢ (𝑛 ∈ 𝑆 → 𝑛 ∥ 𝑁)
10	9	ad2antlr 725	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → 𝑛 ∥ 𝑁)
11		prmz 16013	. . . . . . 7 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
12	11	adantl 484	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ)
13		simplr 767	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → 𝑛 ∈ 𝑆)
14	13, 7	sylib 220	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → (𝑛 ∈ ℕ ∧ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)))
15	14	simpld 497	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → 𝑛 ∈ ℕ)
16	15	nnzd 12080	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → 𝑛 ∈ ℤ)
17		nnz 11998	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
18	17	ad2antrr 724	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → 𝑁 ∈ ℤ)
19		dvdstr 15640	. . . . . 6 ⊢ ((𝑝 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑝 ∥ 𝑛 ∧ 𝑛 ∥ 𝑁) → 𝑝 ∥ 𝑁))
20	12, 16, 18, 19	syl3anc 1367	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ 𝑛 ∧ 𝑛 ∥ 𝑁) → 𝑝 ∥ 𝑁))
21	10, 20	mpan2d 692	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ 𝑛 → 𝑝 ∥ 𝑁))
22	21	ss2rabdv 4052	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛} ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
23		prmex 16015	. . . . 5 ⊢ ℙ ∈ V
24	23	rabex 5228	. . . 4 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛} ∈ V
25	24	elpw 4546	. . 3 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛} ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛} ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
26	22, 25	sylibr 236	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛} ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
27		cnveq 5739	. . . . . . 7 ⊢ (𝑦 = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) → ◡𝑦 = ◡(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))
28	27	imaeq1d 5923	. . . . . 6 ⊢ (𝑦 = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) → (◡𝑦 “ ℕ) = (◡(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) “ ℕ))
29	28	eleq1d 2897	. . . . 5 ⊢ (𝑦 = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) → ((◡𝑦 “ ℕ) ∈ Fin ↔ (◡(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) “ ℕ) ∈ Fin))
30		1nn0 11907	. . . . . . . . . 10 ⊢ 1 ∈ ℕ0
31		0nn0 11906	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
32	30, 31	ifcli 4513	. . . . . . . . 9 ⊢ if(𝑘 ∈ 𝑧, 1, 0) ∈ ℕ0
33	32	rgenw 3150	. . . . . . . 8 ⊢ ∀𝑘 ∈ ℙ if(𝑘 ∈ 𝑧, 1, 0) ∈ ℕ0
34		eqid 2821	. . . . . . . . 9 ⊢ (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))
35	34	fmpt 6869	. . . . . . . 8 ⊢ (∀𝑘 ∈ ℙ if(𝑘 ∈ 𝑧, 1, 0) ∈ ℕ0 ↔ (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)):ℙ⟶ℕ0)
36	33, 35	mpbi 232	. . . . . . 7 ⊢ (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)):ℙ⟶ℕ0
37	36	a1i 11	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)):ℙ⟶ℕ0)
38		nn0ex 11897	. . . . . . 7 ⊢ ℕ0 ∈ V
39	38, 23	elmap 8429	. . . . . 6 ⊢ ((𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ∈ (ℕ0 ↑m ℙ) ↔ (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)):ℙ⟶ℕ0)
40	37, 39	sylibr 236	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ∈ (ℕ0 ↑m ℙ))
41		fzfi 13334	. . . . . 6 ⊢ (1...𝑁) ∈ Fin
42		ffn 6509	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)):ℙ⟶ℕ0 → (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) Fn ℙ)
43		elpreima 6823	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) Fn ℙ → (𝑥 ∈ (◡(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) “ ℕ) ↔ (𝑥 ∈ ℙ ∧ ((𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))‘𝑥) ∈ ℕ)))
44	36, 42, 43	mp2b 10	. . . . . . . . . 10 ⊢ (𝑥 ∈ (◡(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) “ ℕ) ↔ (𝑥 ∈ ℙ ∧ ((𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))‘𝑥) ∈ ℕ))
45		elequ1 2117	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑥 → (𝑘 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
46	45	ifbid 4489	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑥 → if(𝑘 ∈ 𝑧, 1, 0) = if(𝑥 ∈ 𝑧, 1, 0))
47	30, 31	ifcli 4513	. . . . . . . . . . . . . 14 ⊢ if(𝑥 ∈ 𝑧, 1, 0) ∈ ℕ0
48	47	elexi 3514	. . . . . . . . . . . . 13 ⊢ if(𝑥 ∈ 𝑧, 1, 0) ∈ V
49	46, 34, 48	fvmpt 6763	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℙ → ((𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))‘𝑥) = if(𝑥 ∈ 𝑧, 1, 0))
50	49	eleq1d 2897	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℙ → (((𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))‘𝑥) ∈ ℕ ↔ if(𝑥 ∈ 𝑧, 1, 0) ∈ ℕ))
51	50	biimpa 479	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℙ ∧ ((𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))‘𝑥) ∈ ℕ) → if(𝑥 ∈ 𝑧, 1, 0) ∈ ℕ)
52	44, 51	sylbi 219	. . . . . . . . 9 ⊢ (𝑥 ∈ (◡(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) “ ℕ) → if(𝑥 ∈ 𝑧, 1, 0) ∈ ℕ)
53		0nnn 11667	. . . . . . . . . . 11 ⊢ ¬ 0 ∈ ℕ
54		iffalse 4476	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ 𝑧 → if(𝑥 ∈ 𝑧, 1, 0) = 0)
55	54	eleq1d 2897	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝑧 → (if(𝑥 ∈ 𝑧, 1, 0) ∈ ℕ ↔ 0 ∈ ℕ))
56	53, 55	mtbiri 329	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝑧 → ¬ if(𝑥 ∈ 𝑧, 1, 0) ∈ ℕ)
57	56	con4i 114	. . . . . . . . 9 ⊢ (if(𝑥 ∈ 𝑧, 1, 0) ∈ ℕ → 𝑥 ∈ 𝑧)
58	52, 57	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (◡(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) “ ℕ) → 𝑥 ∈ 𝑧)
59	58	ssriv 3971	. . . . . . 7 ⊢ (◡(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) “ ℕ) ⊆ 𝑧
60		elpwi 4551	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} → 𝑧 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
61	60	adantl 484	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑧 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
62		prmssnn 16014	. . . . . . . . . 10 ⊢ ℙ ⊆ ℕ
63		rabss2 4054	. . . . . . . . . 10 ⊢ (ℙ ⊆ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ⊆ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑁})
64	62, 63	ax-mp 5	. . . . . . . . 9 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ⊆ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑁}
65		dvdsssfz1 15662	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑁} ⊆ (1...𝑁))
66	65	adantr 483	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑁} ⊆ (1...𝑁))
67	64, 66	sstrid 3978	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ⊆ (1...𝑁))
68	61, 67	sstrd 3977	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑧 ⊆ (1...𝑁))
69	59, 68	sstrid 3978	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (◡(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) “ ℕ) ⊆ (1...𝑁))
70		ssfi 8732	. . . . . 6 ⊢ (((1...𝑁) ∈ Fin ∧ (◡(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) “ ℕ) ⊆ (1...𝑁)) → (◡(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) “ ℕ) ∈ Fin)
71	41, 69, 70	sylancr 589	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (◡(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) “ ℕ) ∈ Fin)
72	29, 40, 71	elrabd 3682	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ∈ {𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin})
73		sqff1o.3	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
74		eqid 2821	. . . . . . 7 ⊢ {𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin} = {𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin}
75	73, 74	1arith 16257	. . . . . 6 ⊢ 𝐺:ℕ–1-1-onto→{𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin}
76		f1ocnv 6622	. . . . . 6 ⊢ (𝐺:ℕ–1-1-onto→{𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin} → ◡𝐺:{𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin}–1-1-onto→ℕ)
77		f1of 6610	. . . . . 6 ⊢ (◡𝐺:{𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin}–1-1-onto→ℕ → ◡𝐺:{𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin}⟶ℕ)
78	75, 76, 77	mp2b 10	. . . . 5 ⊢ ◡𝐺:{𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin}⟶ℕ
79	78	ffvelrni 6845	. . . 4 ⊢ ((𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ∈ {𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin} → (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∈ ℕ)
80	72, 79	syl 17	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∈ ℕ)
81		f1ocnvfv2 7028	. . . . . . . . . . . 12 ⊢ ((𝐺:ℕ–1-1-onto→{𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin} ∧ (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ∈ {𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin}) → (𝐺‘(◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))
82	75, 72, 81	sylancr 589	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (𝐺‘(◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))
83	73	1arithlem1 16253	. . . . . . . . . . . 12 ⊢ ((◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∈ ℕ → (𝐺‘(◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))))))
84	80, 83	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (𝐺‘(◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))))))
85	82, 84	eqtr3d 2858	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))))))
86	85	fveq1d 6667	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ((𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))‘𝑞) = ((𝑝 ∈ ℙ ↦ (𝑝 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))))‘𝑞))
87		elequ1 2117	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑞 → (𝑘 ∈ 𝑧 ↔ 𝑞 ∈ 𝑧))
88	87	ifbid 4489	. . . . . . . . . 10 ⊢ (𝑘 = 𝑞 → if(𝑘 ∈ 𝑧, 1, 0) = if(𝑞 ∈ 𝑧, 1, 0))
89	30, 31	ifcli 4513	. . . . . . . . . . 11 ⊢ if(𝑞 ∈ 𝑧, 1, 0) ∈ ℕ0
90	89	elexi 3514	. . . . . . . . . 10 ⊢ if(𝑞 ∈ 𝑧, 1, 0) ∈ V
91	88, 34, 90	fvmpt 6763	. . . . . . . . 9 ⊢ (𝑞 ∈ ℙ → ((𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))‘𝑞) = if(𝑞 ∈ 𝑧, 1, 0))
92	86, 91	sylan9req 2877	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ ℙ) → ((𝑝 ∈ ℙ ↦ (𝑝 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))))‘𝑞) = if(𝑞 ∈ 𝑧, 1, 0))
93		oveq1 7157	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → (𝑝 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) = (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))))
94		eqid 2821	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ ↦ (𝑝 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))))) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))))
95		ovex 7183	. . . . . . . . . 10 ⊢ (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ∈ V
96	93, 94, 95	fvmpt 6763	. . . . . . . . 9 ⊢ (𝑞 ∈ ℙ → ((𝑝 ∈ ℙ ↦ (𝑝 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))))‘𝑞) = (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))))
97	96	adantl 484	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ ℙ) → ((𝑝 ∈ ℙ ↦ (𝑝 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))))‘𝑞) = (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))))
98	92, 97	eqtr3d 2858	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ ℙ) → if(𝑞 ∈ 𝑧, 1, 0) = (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))))
99		breq1 5062	. . . . . . . 8 ⊢ (1 = if(𝑞 ∈ 𝑧, 1, 0) → (1 ≤ 1 ↔ if(𝑞 ∈ 𝑧, 1, 0) ≤ 1))
100		breq1 5062	. . . . . . . 8 ⊢ (0 = if(𝑞 ∈ 𝑧, 1, 0) → (0 ≤ 1 ↔ if(𝑞 ∈ 𝑧, 1, 0) ≤ 1))
101		1le1 11262	. . . . . . . 8 ⊢ 1 ≤ 1
102		0le1 11157	. . . . . . . 8 ⊢ 0 ≤ 1
103	99, 100, 101, 102	keephyp 4536	. . . . . . 7 ⊢ if(𝑞 ∈ 𝑧, 1, 0) ≤ 1
104	98, 103	eqbrtrrdi 5099	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≤ 1)
105	104	ralrimiva 3182	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ∀𝑞 ∈ ℙ (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≤ 1)
106		issqf 25707	. . . . . 6 ⊢ ((◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∈ ℕ → ((μ‘(◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≠ 0 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≤ 1))
107	80, 106	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ((μ‘(◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≠ 0 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≤ 1))
108	105, 107	mpbird 259	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (μ‘(◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≠ 0)
109		iftrue 4473	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝑧 → if(𝑞 ∈ 𝑧, 1, 0) = 1)
110	109	adantl 484	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ 𝑧) → if(𝑞 ∈ 𝑧, 1, 0) = 1)
111	61	sselda 3967	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ 𝑧) → 𝑞 ∈ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
112		breq1 5062	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = 𝑞 → (𝑝 ∥ 𝑁 ↔ 𝑞 ∥ 𝑁))
113	112	elrab 3680	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ↔ (𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑁))
114	111, 113	sylib 220	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ 𝑧) → (𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑁))
115	114	simprd 498	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ 𝑧) → 𝑞 ∥ 𝑁)
116	114	simpld 497	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ 𝑧) → 𝑞 ∈ ℙ)
117		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ 𝑧) → 𝑁 ∈ ℕ)
118		pcelnn 16200	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ((𝑞 pCnt 𝑁) ∈ ℕ ↔ 𝑞 ∥ 𝑁))
119	116, 117, 118	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ 𝑧) → ((𝑞 pCnt 𝑁) ∈ ℕ ↔ 𝑞 ∥ 𝑁))
120	115, 119	mpbird 259	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ 𝑧) → (𝑞 pCnt 𝑁) ∈ ℕ)
121	120	nnge1d 11679	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ 𝑧) → 1 ≤ (𝑞 pCnt 𝑁))
122	110, 121	eqbrtrd 5081	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ 𝑧) → if(𝑞 ∈ 𝑧, 1, 0) ≤ (𝑞 pCnt 𝑁))
123	122	ex 415	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (𝑞 ∈ 𝑧 → if(𝑞 ∈ 𝑧, 1, 0) ≤ (𝑞 pCnt 𝑁)))
124	123	adantr 483	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ ℙ) → (𝑞 ∈ 𝑧 → if(𝑞 ∈ 𝑧, 1, 0) ≤ (𝑞 pCnt 𝑁)))
125		simpr 487	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ ℙ) → 𝑞 ∈ ℙ)
126	17	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ ℙ) → 𝑁 ∈ ℤ)
127		pcge0 16192	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℙ ∧ 𝑁 ∈ ℤ) → 0 ≤ (𝑞 pCnt 𝑁))
128	125, 126, 127	syl2anc 586	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ ℙ) → 0 ≤ (𝑞 pCnt 𝑁))
129		iffalse 4476	. . . . . . . . . 10 ⊢ (¬ 𝑞 ∈ 𝑧 → if(𝑞 ∈ 𝑧, 1, 0) = 0)
130	129	breq1d 5069	. . . . . . . . 9 ⊢ (¬ 𝑞 ∈ 𝑧 → (if(𝑞 ∈ 𝑧, 1, 0) ≤ (𝑞 pCnt 𝑁) ↔ 0 ≤ (𝑞 pCnt 𝑁)))
131	128, 130	syl5ibrcom 249	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ ℙ) → (¬ 𝑞 ∈ 𝑧 → if(𝑞 ∈ 𝑧, 1, 0) ≤ (𝑞 pCnt 𝑁)))
132	124, 131	pm2.61d 181	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ ℙ) → if(𝑞 ∈ 𝑧, 1, 0) ≤ (𝑞 pCnt 𝑁))
133	98, 132	eqbrtrrd 5083	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≤ (𝑞 pCnt 𝑁))
134	133	ralrimiva 3182	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ∀𝑞 ∈ ℙ (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≤ (𝑞 pCnt 𝑁))
135	80	nnzd 12080	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∈ ℤ)
136	17	adantr 483	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑁 ∈ ℤ)
137		pc2dvds 16209	. . . . . 6 ⊢ (((◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∥ 𝑁 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≤ (𝑞 pCnt 𝑁)))
138	135, 136, 137	syl2anc 586	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ((◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∥ 𝑁 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≤ (𝑞 pCnt 𝑁)))
139	134, 138	mpbird 259	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∥ 𝑁)
140	108, 139	jca 514	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ((μ‘(◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≠ 0 ∧ (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∥ 𝑁))
141		fveq2 6665	. . . . . 6 ⊢ (𝑥 = (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) → (μ‘𝑥) = (μ‘(◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))))
142	141	neeq1d 3075	. . . . 5 ⊢ (𝑥 = (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) → ((μ‘𝑥) ≠ 0 ↔ (μ‘(◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≠ 0))
143		breq1 5062	. . . . 5 ⊢ (𝑥 = (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) → (𝑥 ∥ 𝑁 ↔ (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∥ 𝑁))
144	142, 143	anbi12d 632	. . . 4 ⊢ (𝑥 = (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) → (((μ‘𝑥) ≠ 0 ∧ 𝑥 ∥ 𝑁) ↔ ((μ‘(◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≠ 0 ∧ (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∥ 𝑁)))
145	144, 6	elrab2 3683	. . 3 ⊢ ((◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∈ 𝑆 ↔ ((◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∈ ℕ ∧ ((μ‘(◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))) ≠ 0 ∧ (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∥ 𝑁)))
146	80, 140, 145	sylanbrc 585	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ∈ 𝑆)
147		eqcom 2828	. . 3 ⊢ (𝑛 = (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ↔ (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) = 𝑛)
148	7	simplbi 500	. . . . . . 7 ⊢ (𝑛 ∈ 𝑆 → 𝑛 ∈ ℕ)
149	148	ad2antrl 726	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 𝑛 ∈ ℕ)
150	23	mptex 6980	. . . . . 6 ⊢ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ V
151	73	fvmpt2 6774	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ V) → (𝐺‘𝑛) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
152	149, 150, 151	sylancl 588	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → (𝐺‘𝑛) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
153	152	eqeq1d 2823	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → ((𝐺‘𝑛) = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ↔ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))))
154	75	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 𝐺:ℕ–1-1-onto→{𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin})
155	72	adantrl 714	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ∈ {𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin})
156		f1ocnvfvb 7030	. . . . 5 ⊢ ((𝐺:ℕ–1-1-onto→{𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin} ∧ 𝑛 ∈ ℕ ∧ (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ∈ {𝑦 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑦 “ ℕ) ∈ Fin}) → ((𝐺‘𝑛) = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ↔ (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) = 𝑛))
157	154, 149, 155, 156	syl3anc 1367	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → ((𝐺‘𝑛) = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ↔ (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) = 𝑛))
158	23	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → ℙ ∈ V)
159		0cnd 10628	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 0 ∈ ℂ)
160		1cnd 10630	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 1 ∈ ℂ)
161		0ne1 11702	. . . . . . . 8 ⊢ 0 ≠ 1
162	161	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 0 ≠ 1)
163	158, 159, 160, 162	pw2f1olem 8615	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → ((𝑧 ∈ 𝒫 ℙ ∧ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ↔ ((𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ ({0, 1} ↑m ℙ) ∧ 𝑧 = (◡(𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) “ {1}))))
164		ssrab2 4056	. . . . . . . . 9 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ⊆ ℙ
165		sspwb 5334	. . . . . . . . 9 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ⊆ ℙ ↔ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ⊆ 𝒫 ℙ)
166	164, 165	mpbi 232	. . . . . . . 8 ⊢ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ⊆ 𝒫 ℙ
167		simprr 771	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
168	166, 167	sseldi 3965	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 𝑧 ∈ 𝒫 ℙ)
169	168	biantrurd 535	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → ((𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ↔ (𝑧 ∈ 𝒫 ℙ ∧ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)))))
170		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℙ)
171	148	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) → 𝑛 ∈ ℕ)
172		pccl 16180	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ ℙ ∧ 𝑛 ∈ ℕ) → (𝑝 pCnt 𝑛) ∈ ℕ0)
173	170, 171, 172	syl2anr 598	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝑛) ∈ ℕ0)
174		elnn0 11893	. . . . . . . . . . . . . 14 ⊢ ((𝑝 pCnt 𝑛) ∈ ℕ0 ↔ ((𝑝 pCnt 𝑛) ∈ ℕ ∨ (𝑝 pCnt 𝑛) = 0))
175	173, 174	sylib 220	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝑛) ∈ ℕ ∨ (𝑝 pCnt 𝑛) = 0))
176	175	orcomd 867	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝑛) = 0 ∨ (𝑝 pCnt 𝑛) ∈ ℕ))
177	8	simpld 497	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑆 → (μ‘𝑛) ≠ 0)
178	177	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) → (μ‘𝑛) ≠ 0)
179		issqf 25707	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ((μ‘𝑛) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑛) ≤ 1))
180	171, 179	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) → ((μ‘𝑛) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑛) ≤ 1))
181	178, 180	mpbid 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) → ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑛) ≤ 1)
182	181	r19.21bi 3208	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝑛) ≤ 1)
183		nnle1eq1 11661	. . . . . . . . . . . . . 14 ⊢ ((𝑝 pCnt 𝑛) ∈ ℕ → ((𝑝 pCnt 𝑛) ≤ 1 ↔ (𝑝 pCnt 𝑛) = 1))
184	182, 183	syl5ibcom 247	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝑛) ∈ ℕ → (𝑝 pCnt 𝑛) = 1))
185	184	orim2d 963	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝑛) = 0 ∨ (𝑝 pCnt 𝑛) ∈ ℕ) → ((𝑝 pCnt 𝑛) = 0 ∨ (𝑝 pCnt 𝑛) = 1)))
186	176, 185	mpd 15	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝑛) = 0 ∨ (𝑝 pCnt 𝑛) = 1))
187		ovex 7183	. . . . . . . . . . . 12 ⊢ (𝑝 pCnt 𝑛) ∈ V
188	187	elpr 4584	. . . . . . . . . . 11 ⊢ ((𝑝 pCnt 𝑛) ∈ {0, 1} ↔ ((𝑝 pCnt 𝑛) = 0 ∨ (𝑝 pCnt 𝑛) = 1))
189	186, 188	sylibr 236	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝑛) ∈ {0, 1})
190	189	fmpttd 6874	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) → (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)):ℙ⟶{0, 1})
191	190	adantrr 715	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)):ℙ⟶{0, 1})
192		prex 5325	. . . . . . . . 9 ⊢ {0, 1} ∈ V
193	192, 23	elmap 8429	. . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ ({0, 1} ↑m ℙ) ↔ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)):ℙ⟶{0, 1})
194	191, 193	sylibr 236	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ ({0, 1} ↑m ℙ))
195	194	biantrurd 535	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → (𝑧 = (◡(𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) “ {1}) ↔ ((𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ ({0, 1} ↑m ℙ) ∧ 𝑧 = (◡(𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) “ {1}))))
196	163, 169, 195	3bitr4d 313	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → ((𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ↔ 𝑧 = (◡(𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) “ {1})))
197		eqid 2821	. . . . . . . . 9 ⊢ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛))
198	197	mptiniseg 6088	. . . . . . . 8 ⊢ (1 ∈ ℕ0 → (◡(𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) “ {1}) = {𝑝 ∈ ℙ ∣ (𝑝 pCnt 𝑛) = 1})
199	30, 198	ax-mp 5	. . . . . . 7 ⊢ (◡(𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) “ {1}) = {𝑝 ∈ ℙ ∣ (𝑝 pCnt 𝑛) = 1}
200		id 22	. . . . . . . . . . . 12 ⊢ ((𝑝 pCnt 𝑛) = 1 → (𝑝 pCnt 𝑛) = 1)
201		1nn 11643	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
202	200, 201	eqeltrdi 2921	. . . . . . . . . . 11 ⊢ ((𝑝 pCnt 𝑛) = 1 → (𝑝 pCnt 𝑛) ∈ ℕ)
203	202, 184	impbid2 228	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝑛) = 1 ↔ (𝑝 pCnt 𝑛) ∈ ℕ))
204		simpr 487	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
205		pcelnn 16200	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ 𝑛 ∈ ℕ) → ((𝑝 pCnt 𝑛) ∈ ℕ ↔ 𝑝 ∥ 𝑛))
206	204, 15, 205	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝑛) ∈ ℕ ↔ 𝑝 ∥ 𝑛))
207	203, 206	bitrd 281	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝑛) = 1 ↔ 𝑝 ∥ 𝑛))
208	207	rabbidva 3479	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ 𝑆) → {𝑝 ∈ ℙ ∣ (𝑝 pCnt 𝑛) = 1} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛})
209	208	adantrr 715	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → {𝑝 ∈ ℙ ∣ (𝑝 pCnt 𝑛) = 1} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛})
210	199, 209	syl5eq 2868	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → (◡(𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) “ {1}) = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛})
211	210	eqeq2d 2832	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → (𝑧 = (◡(𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) “ {1}) ↔ 𝑧 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛}))
212	196, 211	bitrd 281	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → ((𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) = (𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0)) ↔ 𝑧 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛}))
213	153, 157, 212	3bitr3d 311	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → ((◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) = 𝑛 ↔ 𝑧 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛}))
214	147, 213	syl5bb 285	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑛 ∈ 𝑆 ∧ 𝑧 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → (𝑛 = (◡𝐺‘(𝑘 ∈ ℙ ↦ if(𝑘 ∈ 𝑧, 1, 0))) ↔ 𝑧 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑛}))
215	1, 26, 146, 214	f1o2d 7393	1 ⊢ (𝑁 ∈ ℕ → 𝐹:𝑆–1-1-onto→𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  {crab 3142  Vcvv 3495   ⊆ wss 3936  ifcif 4467  𝒫 cpw 4539  {csn 4561  {cpr 4563   class class class wbr 5059   ↦ cmpt 5139  ^◡ccnv 5549   “ cima 5553   Fn wfn 6345  ⟶wf 6346  –1-1-onto→wf1o 6349  ‘cfv 6350  (class class class)co 7150   ↑_m cmap 8400  Fincfn 8503  ℂcc 10529  0cc0 10531  1c1 10532   ≤ cle 10670  ℕcn 11632  ℕ₀cn0 11891  ℤcz 11975  ...cfz 12886   ∥ cdvds 15601  ℙcprime 16009   pCnt cpc 16167  μcmu 25666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-inf 8901  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-q 12343  df-rp 12384  df-fz 12887  df-fl 13156  df-mod 13232  df-seq 13364  df-exp 13424  df-hash 13685  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-dvds 15602  df-gcd 15838  df-prm 16010  df-pc 16168  df-mu 25672

This theorem is referenced by:  musum  25762