Theorem 1arith 12402

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 12148	. . . . . 6 ⊢ ℙ ∈ V
2	1	mptex 5763	. . . . 5 ⊢ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ V
3		1arith.1	. . . . 5 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
4	2, 3	fnmpti 5363	. . . 4 ⊢ 𝑀 Fn ℕ
5	3	1arithlem3 12400	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (𝑀‘𝑥):ℙ⟶ℕ0)
6		nn0ex 9213	. . . . . . . 8 ⊢ ℕ0 ∈ V
7	6, 1	elmap 6704	. . . . . . 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 9311	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 1 ∈ ℤ)
10		nnz 9303	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
11	9, 10	fzfigd 10464	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1...𝑥) ∈ Fin)
12		ffn 5384	. . . . . . . . . 10 ⊢ ((𝑀‘𝑥):ℙ⟶ℕ0 → (𝑀‘𝑥) Fn ℙ)
13		elpreima 5656	. . . . . . . . . 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 12399	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
16	15	eleq1d 2258	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) ∈ ℕ ↔ (𝑞 pCnt 𝑥) ∈ ℕ))
17		prmz 12146	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
18		id 19	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
19		dvdsle 11885	. . . . . . . . . . . . 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 12356	. . . . . . . . . . . . 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 12145	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
24		nnuz 9595	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ≥‘1)
25	23, 24	eleqtrdi 2282	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ (ℤ≥‘1))
26		elfz5 10049	. . . . . . . . . . . . 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 3176	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (◡(𝑀‘𝑥) “ ℕ) ⊆ (1...𝑥))
33		elfznn 10086	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑥) → 𝑗 ∈ ℕ)
34	33	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → 𝑗 ∈ ℕ)
35		prmdc 12165	. . . . . . . . . . . . 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 5673	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → ((𝑀‘𝑥)‘𝑗) ∈ ℕ0)
41	40	nn0zd 9404	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → ((𝑀‘𝑥)‘𝑗) ∈ ℤ)
42		elnndc 9644	. . . . . . . . . . . 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 936	. . . . . . . . . . 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 933	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ¬ (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ))
48	47	olcd 735	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ((𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ) ∨ ¬ (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
49		df-dc 836	. . . . . . . . . . 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 837	. . . . . . . . . . 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 799	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ))
54		elpreima 5656	. . . . . . . . . . . 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 839	. . . . . . . . . 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 2563	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → ∀𝑗 ∈ (1...𝑥)DECID 𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ))
60		ssfidc 6964	. . . . . . 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 1249	. . . . . 6 ⊢ (𝑥 ∈ ℕ → (◡(𝑀‘𝑥) “ ℕ) ∈ Fin)
62		cnveq 4819	. . . . . . . . 9 ⊢ (𝑒 = (𝑀‘𝑥) → ◡𝑒 = ◡(𝑀‘𝑥))
63	62	imaeq1d 4987	. . . . . . . 8 ⊢ (𝑒 = (𝑀‘𝑥) → (◡𝑒 “ ℕ) = (◡(𝑀‘𝑥) “ ℕ))
64	63	eleq1d 2258	. . . . . . 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 2911	. . . . . 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 2543	. . . 4 ⊢ ∀𝑥 ∈ ℕ (𝑀‘𝑥) ∈ 𝑅
69		ffnfv 5695	. . . 4 ⊢ (𝑀:ℕ⟶𝑅 ↔ (𝑀 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝑀‘𝑥) ∈ 𝑅))
70	4, 68, 69	mpbir2an 944	. . 3 ⊢ 𝑀:ℕ⟶𝑅
71	15	adantlr 477	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
72	3	1arithlem2 12399	. . . . . . . . 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 2204	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞) ↔ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
75	74	ralbidva 2486	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑞 ∈ ℙ ((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
76	3	1arithlem3 12400	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → (𝑀‘𝑦):ℙ⟶ℕ0)
77		ffn 5384	. . . . . . . 8 ⊢ ((𝑀‘𝑦):ℙ⟶ℕ0 → (𝑀‘𝑦) Fn ℙ)
78		eqfnfv 5634	. . . . . . . 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 9214	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
82		nnnn0 9214	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
83		pc11 12366	. . . . . . 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 2576	. . 3 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦)
88		dff13 5790	. . 3 ⊢ (𝑀:ℕ–1-1→𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦)))
89	70, 87, 88	mpbir2an 944	. 2 ⊢ 𝑀:ℕ–1-1→𝑅
90		eqid 2189	. . . . . 6 ⊢ (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓‘𝑔)), 1)) = (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓‘𝑔)), 1))
91		cnveq 4819	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑓 → ◡𝑒 = ◡𝑓)
92	91	imaeq1d 4987	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑓 → (◡𝑒 “ ℕ) = (◡𝑓 “ ℕ))
93	92	eleq1d 2258	. . . . . . . . . 10 ⊢ (𝑒 = 𝑓 → ((◡𝑒 “ ℕ) ∈ Fin ↔ (◡𝑓 “ ℕ) ∈ Fin))
94	93, 65	elrab2 2911	. . . . . . . . 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 6704	. . . . . . . 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 8965	. . . . . 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 8963	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 ∈ ℝ)
103		peano2re 8124	. . . . . . . . . . 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 8963	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℝ)
107	102	ltp1d 8918	. . . . . . . . . 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 8411	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 < 𝑞)
110	101	nnzd 9405	. . . . . . . . . 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 9330	. . . . . . . . . 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 5384	. . . . . . . . . . 11 ⊢ (𝑓:ℙ⟶ℕ0 → 𝑓 Fn ℙ)
119		elpreima 5656	. . . . . . . . . . 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 4021	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑞 → (𝑘 ≤ 𝑦 ↔ 𝑞 ≤ 𝑦))
123	122	rspccv 2853	. . . . . . . . . 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 664	. . . . . . 7 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ¬ (𝑓‘𝑞) ∈ ℕ)
127	117, 115	ffvelcdmd 5673	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑓‘𝑞) ∈ ℕ0)
128		elnn0 9209	. . . . . . . . 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 725	. . . . . . 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 12401	. . . . 5 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥))
133		cnvimass 5009	. . . . . . 7 ⊢ (◡𝑓 “ ℕ) ⊆ dom 𝑓
134	97	fdmd 5391	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑅 → dom 𝑓 = ℙ)
135		prmssnn 12147	. . . . . . . 8 ⊢ ℙ ⊆ ℕ
136	134, 135	eqsstrdi 3222	. . . . . . 7 ⊢ (𝑓 ∈ 𝑅 → dom 𝑓 ⊆ ℕ)
137	133, 136	sstrid 3181	. . . . . 6 ⊢ (𝑓 ∈ 𝑅 → (◡𝑓 “ ℕ) ⊆ ℕ)
138	94	simprbi 275	. . . . . 6 ⊢ (𝑓 ∈ 𝑅 → (◡𝑓 “ ℕ) ∈ Fin)
139		fiubnn 10845	. . . . . 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 2633	. . . 4 ⊢ (𝑓 ∈ 𝑅 → ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥))
142	141	rgen 2543	. . 3 ⊢ ∀𝑓 ∈ 𝑅 ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥)
143		dffo3 5684	. . 3 ⊢ (𝑀:ℕ–onto→𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑓 ∈ 𝑅 ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥)))
144	70, 142, 143	mpbir2an 944	. 2 ⊢ 𝑀:ℕ–onto→𝑅
145		df-f1o 5242	. 2 ⊢ (𝑀:ℕ–1-1-onto→𝑅 ↔ (𝑀:ℕ–1-1→𝑅 ∧ 𝑀:ℕ–onto→𝑅))
146	89, 144, 145	mpbir2an 944	1 ⊢ 𝑀:ℕ–1-1-onto→𝑅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2160  ∀wral 2468  ∃wrex 2469  {crab 2472   ⊆ wss 3144  ifcif 3549   class class class wbr 4018   ↦ cmpt 4079  ^◡ccnv 4643  dom cdm 4644   “ cima 4647   Fn wfn 5230  ⟶wf 5231  –1-1→wf1 5232  –onto→wfo 5233  –1-1-onto→wf1o 5234  ‘cfv 5235  (class class class)co 5897   ↑_𝑚 cmap 6675  Fincfn 6767  ℝcr 7841  0cc0 7842  1c1 7843   + caddc 7845   < clt 8023   ≤ cle 8024  ℕcn 8950  ℕ₀cn0 9207  ℤcz 9284  ℤ_≥cuz 9559  ...cfz 10040  ↑cexp 10553   ∥ cdvds 11829  ℙcprime 12142   pCnt cpc 12319

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-1o 6442  df-2o 6443  df-er 6560  df-map 6677  df-en 6768  df-fin 6770  df-sup 7014  df-inf 7015  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-xnn0 9271  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-fz 10041  df-fzo 10175  df-fl 10303  df-mod 10356  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-dvds 11830  df-gcd 11979  df-prm 12143  df-pc 12320

This theorem is referenced by:  1arith2  12403