Theorem 1arith 12330

Description: Fundamental theorem of arithmetic, where a prime factorization is represented as a sequence of prime exponents, for which only finitely many primes have nonzero exponent. The function 𝑀 maps the set of positive integers one-to-one onto the set of prime factorizations 𝑅. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 30-May-2014.)

Hypotheses

Ref	Expression
1arith.1	⊢ 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
1arith.2	⊢ 𝑅 = {𝑒 ∈ (ℕ0 ↑𝑚 ℙ) ∣ (◡𝑒 “ ℕ) ∈ Fin}

Assertion

Ref	Expression
1arith	⊢ 𝑀:ℕ–1-1-onto→𝑅

Distinct variable groups:   𝑒,𝑛,𝑝   𝑒,𝑀   𝑅,𝑛

Allowed substitution hints:   𝑅(𝑒,𝑝)   𝑀(𝑛,𝑝)

Proof of Theorem 1arith

Dummy variables 𝑓 𝑔 𝑘 𝑞 𝑥 𝑦 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmex 12078	. . . . . 6 ⊢ ℙ ∈ V
2	1	mptex 5734	. . . . 5 ⊢ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ V
3		1arith.1	. . . . 5 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
4	2, 3	fnmpti 5336	. . . 4 ⊢ 𝑀 Fn ℕ
5	3	1arithlem3 12328	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (𝑀‘𝑥):ℙ⟶ℕ0)
6		nn0ex 9153	. . . . . . . 8 ⊢ ℕ0 ∈ V
7	6, 1	elmap 6667	. . . . . . 7 ⊢ ((𝑀‘𝑥) ∈ (ℕ0 ↑𝑚 ℙ) ↔ (𝑀‘𝑥):ℙ⟶ℕ0)
8	5, 7	sylibr 134	. . . . . 6 ⊢ (𝑥 ∈ ℕ → (𝑀‘𝑥) ∈ (ℕ0 ↑𝑚 ℙ))
9		1zzd 9251	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 1 ∈ ℤ)
10		nnz 9243	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
11	9, 10	fzfigd 10399	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1...𝑥) ∈ Fin)
12		ffn 5357	. . . . . . . . . 10 ⊢ ((𝑀‘𝑥):ℙ⟶ℕ0 → (𝑀‘𝑥) Fn ℙ)
13		elpreima 5627	. . . . . . . . . 10 ⊢ ((𝑀‘𝑥) Fn ℙ → (𝑞 ∈ (◡(𝑀‘𝑥) “ ℕ) ↔ (𝑞 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑞) ∈ ℕ)))
14	5, 12, 13	3syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → (𝑞 ∈ (◡(𝑀‘𝑥) “ ℕ) ↔ (𝑞 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑞) ∈ ℕ)))
15	3	1arithlem2 12327	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
16	15	eleq1d 2244	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) ∈ ℕ ↔ (𝑞 pCnt 𝑥) ∈ ℕ))
17		prmz 12076	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
18		id 19	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
19		dvdsle 11815	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℤ ∧ 𝑥 ∈ ℕ) → (𝑞 ∥ 𝑥 → 𝑞 ≤ 𝑥))
20	17, 18, 19	syl2anr 290	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∥ 𝑥 → 𝑞 ≤ 𝑥))
21		pcelnn 12285	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℙ ∧ 𝑥 ∈ ℕ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞 ∥ 𝑥))
22	21	ancoms 268	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞 ∥ 𝑥))
23		prmnn 12075	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
24		nnuz 9534	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ≥‘1)
25	23, 24	eleqtrdi 2268	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ (ℤ≥‘1))
26		elfz5 9985	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ (ℤ≥‘1) ∧ 𝑥 ∈ ℤ) → (𝑞 ∈ (1...𝑥) ↔ 𝑞 ≤ 𝑥))
27	25, 10, 26	syl2anr 290	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∈ (1...𝑥) ↔ 𝑞 ≤ 𝑥))
28	20, 22, 27	3imtr4d 203	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
29	16, 28	sylbid 150	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
30	29	expimpd 363	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → ((𝑞 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑞) ∈ ℕ) → 𝑞 ∈ (1...𝑥)))
31	14, 30	sylbid 150	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (𝑞 ∈ (◡(𝑀‘𝑥) “ ℕ) → 𝑞 ∈ (1...𝑥)))
32	31	ssrdv 3159	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (◡(𝑀‘𝑥) “ ℕ) ⊆ (1...𝑥))
33		elfznn 10022	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑥) → 𝑗 ∈ ℕ)
34	33	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → 𝑗 ∈ ℕ)
35		prmdc 12095	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → DECID 𝑗 ∈ ℙ)
36	34, 35	syl 14	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → DECID 𝑗 ∈ ℙ)
37	36	adantr 276	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → DECID 𝑗 ∈ ℙ)
38	5	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → (𝑀‘𝑥):ℙ⟶ℕ0)
39		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → 𝑗 ∈ ℙ)
40	38, 39	ffvelcdmd 5644	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → ((𝑀‘𝑥)‘𝑗) ∈ ℕ0)
41	40	nn0zd 9344	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → ((𝑀‘𝑥)‘𝑗) ∈ ℤ)
42		elnndc 9583	. . . . . . . . . . . 12 ⊢ (((𝑀‘𝑥)‘𝑗) ∈ ℤ → DECID ((𝑀‘𝑥)‘𝑗) ∈ ℕ)
43	41, 42	syl 14	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → DECID ((𝑀‘𝑥)‘𝑗) ∈ ℕ)
44		dcan2 934	. . . . . . . . . . 11 ⊢ (DECID 𝑗 ∈ ℙ → (DECID ((𝑀‘𝑥)‘𝑗) ∈ ℕ → DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
45	37, 43, 44	sylc 62	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ))
46		simpr 110	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ¬ 𝑗 ∈ ℙ)
47	46	intnanrd 932	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ¬ (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ))
48	47	olcd 734	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ((𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ) ∨ ¬ (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
49		df-dc 835	. . . . . . . . . . 11 ⊢ (DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ) ↔ ((𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ) ∨ ¬ (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
50	48, 49	sylibr 134	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ))
51		exmiddc 836	. . . . . . . . . . 11 ⊢ (DECID 𝑗 ∈ ℙ → (𝑗 ∈ ℙ ∨ ¬ 𝑗 ∈ ℙ))
52	36, 51	syl 14	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → (𝑗 ∈ ℙ ∨ ¬ 𝑗 ∈ ℙ))
53	45, 50, 52	mpjaodan 798	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ))
54		elpreima 5627	. . . . . . . . . . . 12 ⊢ ((𝑀‘𝑥) Fn ℙ → (𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ) ↔ (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
55	5, 12, 54	3syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → (𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ) ↔ (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
56	55	dcbid 838	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → (DECID 𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ) ↔ DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
57	56	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → (DECID 𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ) ↔ DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
58	53, 57	mpbird 167	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → DECID 𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ))
59	58	ralrimiva 2548	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → ∀𝑗 ∈ (1...𝑥)DECID 𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ))
60		ssfidc 6924	. . . . . . 7 ⊢ (((1...𝑥) ∈ Fin ∧ (◡(𝑀‘𝑥) “ ℕ) ⊆ (1...𝑥) ∧ ∀𝑗 ∈ (1...𝑥)DECID 𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ)) → (◡(𝑀‘𝑥) “ ℕ) ∈ Fin)
61	11, 32, 59, 60	syl3anc 1238	. . . . . 6 ⊢ (𝑥 ∈ ℕ → (◡(𝑀‘𝑥) “ ℕ) ∈ Fin)
62		cnveq 4794	. . . . . . . . 9 ⊢ (𝑒 = (𝑀‘𝑥) → ◡𝑒 = ◡(𝑀‘𝑥))
63	62	imaeq1d 4962	. . . . . . . 8 ⊢ (𝑒 = (𝑀‘𝑥) → (◡𝑒 “ ℕ) = (◡(𝑀‘𝑥) “ ℕ))
64	63	eleq1d 2244	. . . . . . 7 ⊢ (𝑒 = (𝑀‘𝑥) → ((◡𝑒 “ ℕ) ∈ Fin ↔ (◡(𝑀‘𝑥) “ ℕ) ∈ Fin))
65		1arith.2	. . . . . . 7 ⊢ 𝑅 = {𝑒 ∈ (ℕ0 ↑𝑚 ℙ) ∣ (◡𝑒 “ ℕ) ∈ Fin}
66	64, 65	elrab2 2894	. . . . . 6 ⊢ ((𝑀‘𝑥) ∈ 𝑅 ↔ ((𝑀‘𝑥) ∈ (ℕ0 ↑𝑚 ℙ) ∧ (◡(𝑀‘𝑥) “ ℕ) ∈ Fin))
67	8, 61, 66	sylanbrc 417	. . . . 5 ⊢ (𝑥 ∈ ℕ → (𝑀‘𝑥) ∈ 𝑅)
68	67	rgen 2528	. . . 4 ⊢ ∀𝑥 ∈ ℕ (𝑀‘𝑥) ∈ 𝑅
69		ffnfv 5666	. . . 4 ⊢ (𝑀:ℕ⟶𝑅 ↔ (𝑀 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝑀‘𝑥) ∈ 𝑅))
70	4, 68, 69	mpbir2an 942	. . 3 ⊢ 𝑀:ℕ⟶𝑅
71	15	adantlr 477	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
72	3	1arithlem2 12327	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
73	72	adantll 476	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
74	71, 73	eqeq12d 2190	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞) ↔ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
75	74	ralbidva 2471	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑞 ∈ ℙ ((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
76	3	1arithlem3 12328	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → (𝑀‘𝑦):ℙ⟶ℕ0)
77		ffn 5357	. . . . . . . 8 ⊢ ((𝑀‘𝑦):ℙ⟶ℕ0 → (𝑀‘𝑦) Fn ℙ)
78		eqfnfv 5605	. . . . . . . 8 ⊢ (((𝑀‘𝑥) Fn ℙ ∧ (𝑀‘𝑦) Fn ℙ) → ((𝑀‘𝑥) = (𝑀‘𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞)))
79	12, 77, 78	syl2an 289	. . . . . . 7 ⊢ (((𝑀‘𝑥):ℙ⟶ℕ0 ∧ (𝑀‘𝑦):ℙ⟶ℕ0) → ((𝑀‘𝑥) = (𝑀‘𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞)))
80	5, 76, 79	syl2an 289	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀‘𝑥) = (𝑀‘𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞)))
81		nnnn0 9154	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
82		nnnn0 9154	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
83		pc11 12295	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 = 𝑦 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
84	81, 82, 83	syl2an 289	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 = 𝑦 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
85	75, 80, 84	3bitr4d 220	. . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀‘𝑥) = (𝑀‘𝑦) ↔ 𝑥 = 𝑦))
86	85	biimpd 144	. . . 4 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦))
87	86	rgen2 2561	. . 3 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦)
88		dff13 5759	. . 3 ⊢ (𝑀:ℕ–1-1→𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦)))
89	70, 87, 88	mpbir2an 942	. 2 ⊢ 𝑀:ℕ–1-1→𝑅
90		eqid 2175	. . . . . 6 ⊢ (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓‘𝑔)), 1)) = (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓‘𝑔)), 1))
91		cnveq 4794	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑓 → ◡𝑒 = ◡𝑓)
92	91	imaeq1d 4962	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑓 → (◡𝑒 “ ℕ) = (◡𝑓 “ ℕ))
93	92	eleq1d 2244	. . . . . . . . . 10 ⊢ (𝑒 = 𝑓 → ((◡𝑒 “ ℕ) ∈ Fin ↔ (◡𝑓 “ ℕ) ∈ Fin))
94	93, 65	elrab2 2894	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑅 ↔ (𝑓 ∈ (ℕ0 ↑𝑚 ℙ) ∧ (◡𝑓 “ ℕ) ∈ Fin))
95	94	simplbi 274	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑅 → 𝑓 ∈ (ℕ0 ↑𝑚 ℙ))
96	6, 1	elmap 6667	. . . . . . . 8 ⊢ (𝑓 ∈ (ℕ0 ↑𝑚 ℙ) ↔ 𝑓:ℙ⟶ℕ0)
97	95, 96	sylib 122	. . . . . . 7 ⊢ (𝑓 ∈ 𝑅 → 𝑓:ℙ⟶ℕ0)
98	97	ad2antrr 488	. . . . . 6 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → 𝑓:ℙ⟶ℕ0)
99		simplr 528	. . . . . . 7 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → 𝑦 ∈ ℕ)
100	99	peano2nnd 8905	. . . . . 6 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → (𝑦 + 1) ∈ ℕ)
101	99	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 ∈ ℕ)
102	101	nnred 8903	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 ∈ ℝ)
103		peano2re 8067	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ)
104	102, 103	syl 14	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑦 + 1) ∈ ℝ)
105	23	ad2antrl 490	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℕ)
106	105	nnred 8903	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℝ)
107	102	ltp1d 8858	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 < (𝑦 + 1))
108		simprr 531	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑦 + 1) ≤ 𝑞)
109	102, 104, 106, 107, 108	ltletrd 8354	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 < 𝑞)
110	101	nnzd 9345	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 ∈ ℤ)
111	17	ad2antrl 490	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℤ)
112		zltnle 9270	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ 𝑞 ∈ ℤ) → (𝑦 < 𝑞 ↔ ¬ 𝑞 ≤ 𝑦))
113	110, 111, 112	syl2anc 411	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑦 < 𝑞 ↔ ¬ 𝑞 ≤ 𝑦))
114	109, 113	mpbid 147	. . . . . . . 8 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ¬ 𝑞 ≤ 𝑦)
115		simprl 529	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℙ)
116	115	biantrurd 305	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ((𝑓‘𝑞) ∈ ℕ ↔ (𝑞 ∈ ℙ ∧ (𝑓‘𝑞) ∈ ℕ)))
117	97	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑓:ℙ⟶ℕ0)
118		ffn 5357	. . . . . . . . . . 11 ⊢ (𝑓:ℙ⟶ℕ0 → 𝑓 Fn ℙ)
119		elpreima 5627	. . . . . . . . . . 11 ⊢ (𝑓 Fn ℙ → (𝑞 ∈ (◡𝑓 “ ℕ) ↔ (𝑞 ∈ ℙ ∧ (𝑓‘𝑞) ∈ ℕ)))
120	117, 118, 119	3syl 17	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑞 ∈ (◡𝑓 “ ℕ) ↔ (𝑞 ∈ ℙ ∧ (𝑓‘𝑞) ∈ ℕ)))
121	116, 120	bitr4d 191	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ((𝑓‘𝑞) ∈ ℕ ↔ 𝑞 ∈ (◡𝑓 “ ℕ)))
122		breq1 4001	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑞 → (𝑘 ≤ 𝑦 ↔ 𝑞 ≤ 𝑦))
123	122	rspccv 2836	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦 → (𝑞 ∈ (◡𝑓 “ ℕ) → 𝑞 ≤ 𝑦))
124	123	ad2antlr 489	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑞 ∈ (◡𝑓 “ ℕ) → 𝑞 ≤ 𝑦))
125	121, 124	sylbid 150	. . . . . . . 8 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ((𝑓‘𝑞) ∈ ℕ → 𝑞 ≤ 𝑦))
126	114, 125	mtod 663	. . . . . . 7 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ¬ (𝑓‘𝑞) ∈ ℕ)
127	117, 115	ffvelcdmd 5644	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑓‘𝑞) ∈ ℕ0)
128		elnn0 9149	. . . . . . . . 9 ⊢ ((𝑓‘𝑞) ∈ ℕ0 ↔ ((𝑓‘𝑞) ∈ ℕ ∨ (𝑓‘𝑞) = 0))
129	127, 128	sylib 122	. . . . . . . 8 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ((𝑓‘𝑞) ∈ ℕ ∨ (𝑓‘𝑞) = 0))
130	129	ord 724	. . . . . . 7 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (¬ (𝑓‘𝑞) ∈ ℕ → (𝑓‘𝑞) = 0))
131	126, 130	mpd 13	. . . . . 6 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑓‘𝑞) = 0)
132	3, 90, 98, 100, 131	1arithlem4 12329	. . . . 5 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥))
133		cnvimass 4984	. . . . . . 7 ⊢ (◡𝑓 “ ℕ) ⊆ dom 𝑓
134	97	fdmd 5364	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑅 → dom 𝑓 = ℙ)
135		prmssnn 12077	. . . . . . . 8 ⊢ ℙ ⊆ ℕ
136	134, 135	eqsstrdi 3205	. . . . . . 7 ⊢ (𝑓 ∈ 𝑅 → dom 𝑓 ⊆ ℕ)
137	133, 136	sstrid 3164	. . . . . 6 ⊢ (𝑓 ∈ 𝑅 → (◡𝑓 “ ℕ) ⊆ ℕ)
138	94	simprbi 275	. . . . . 6 ⊢ (𝑓 ∈ 𝑅 → (◡𝑓 “ ℕ) ∈ Fin)
139		fiubnn 10776	. . . . . 6 ⊢ (((◡𝑓 “ ℕ) ⊆ ℕ ∧ (◡𝑓 “ ℕ) ∈ Fin) → ∃𝑦 ∈ ℕ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦)
140	137, 138, 139	syl2anc 411	. . . . 5 ⊢ (𝑓 ∈ 𝑅 → ∃𝑦 ∈ ℕ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦)
141	132, 140	r19.29a 2618	. . . 4 ⊢ (𝑓 ∈ 𝑅 → ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥))
142	141	rgen 2528	. . 3 ⊢ ∀𝑓 ∈ 𝑅 ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥)
143		dffo3 5655	. . 3 ⊢ (𝑀:ℕ–onto→𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑓 ∈ 𝑅 ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥)))
144	70, 142, 143	mpbir2an 942	. 2 ⊢ 𝑀:ℕ–onto→𝑅
145		df-f1o 5215	. 2 ⊢ (𝑀:ℕ–1-1-onto→𝑅 ↔ (𝑀:ℕ–1-1→𝑅 ∧ 𝑀:ℕ–onto→𝑅))
146	89, 144, 145	mpbir2an 942	1 ⊢ 𝑀:ℕ–1-1-onto→𝑅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834   = wceq 1353   ∈ wcel 2146  ∀wral 2453  ∃wrex 2454  {crab 2457   ⊆ wss 3127  ifcif 3532   class class class wbr 3998   ↦ cmpt 4059  ^◡ccnv 4619  dom cdm 4620   “ cima 4623   Fn wfn 5203  ⟶wf 5204  –1-1→wf1 5205  –onto→wfo 5206  –1-1-onto→wf1o 5207  ‘cfv 5208  (class class class)co 5865   ↑_𝑚 cmap 6638  Fincfn 6730  ℝcr 7785  0cc0 7786  1c1 7787   + caddc 7789   < clt 7966   ≤ cle 7967  ℕcn 8890  ℕ₀cn0 9147  ℤcz 9224  ℤ_≥cuz 9499  ...cfz 9977  ↑cexp 10487   ∥ cdvds 11760  ℙcprime 12072   pCnt cpc 12249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-1o 6407  df-2o 6408  df-er 6525  df-map 6640  df-en 6731  df-fin 6733  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-xnn0 9211  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-fl 10238  df-mod 10291  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-dvds 11761  df-gcd 11909  df-prm 12073  df-pc 12250

This theorem is referenced by:  1arith2  12331