Theorem 1arith 13093

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 12838	. . . . . 6 ⊢ ℙ ∈ V
2	1	mptex 5917	. . . . 5 ⊢ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ V
3		1arith.1	. . . . 5 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
4	2, 3	fnmpti 5492	. . . 4 ⊢ 𝑀 Fn ℕ
5	3	1arithlem3 13091	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (𝑀‘𝑥):ℙ⟶ℕ0)
6		nn0ex 9522	. . . . . . . 8 ⊢ ℕ0 ∈ V
7	6, 1	elmap 6924	. . . . . . 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 9624	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 1 ∈ ℤ)
10		nnz 9616	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
11	9, 10	fzfigd 10820	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1...𝑥) ∈ Fin)
12		ffn 5513	. . . . . . . . . 10 ⊢ ((𝑀‘𝑥):ℙ⟶ℕ0 → (𝑀‘𝑥) Fn ℙ)
13		elpreima 5802	. . . . . . . . . 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 13090	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
16	15	eleq1d 2303	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) ∈ ℕ ↔ (𝑞 pCnt 𝑥) ∈ ℕ))
17		prmz 12836	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
18		id 19	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
19		dvdsle 12558	. . . . . . . . . . . . 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 13047	. . . . . . . . . . . . 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 12835	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
24		nnuz 9911	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ≥‘1)
25	23, 24	eleqtrdi 2327	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ (ℤ≥‘1))
26		elfz5 10373	. . . . . . . . . . . . 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 3248	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (◡(𝑀‘𝑥) “ ℕ) ⊆ (1...𝑥))
33		elfznn 10412	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑥) → 𝑗 ∈ ℕ)
34	33	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → 𝑗 ∈ ℕ)
35		prmdc 12855	. . . . . . . . . . . . 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 5818	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → ((𝑀‘𝑥)‘𝑗) ∈ ℕ0)
41	40	nn0zd 9719	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → ((𝑀‘𝑥)‘𝑗) ∈ ℤ)
42		elnndc 9965	. . . . . . . . . . . 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 943	. . . . . . . . . . 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 940	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ¬ (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ))
48	47	olcd 742	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ((𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ) ∨ ¬ (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
49		df-dc 843	. . . . . . . . . . 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 844	. . . . . . . . . . 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 806	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ))
54		elpreima 5802	. . . . . . . . . . . 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 846	. . . . . . . . . 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 2617	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → ∀𝑗 ∈ (1...𝑥)DECID 𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ))
60		ssfidc 7211	. . . . . . 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 1274	. . . . . 6 ⊢ (𝑥 ∈ ℕ → (◡(𝑀‘𝑥) “ ℕ) ∈ Fin)
62		cnveq 4934	. . . . . . . . 9 ⊢ (𝑒 = (𝑀‘𝑥) → ◡𝑒 = ◡(𝑀‘𝑥))
63	62	imaeq1d 5105	. . . . . . . 8 ⊢ (𝑒 = (𝑀‘𝑥) → (◡𝑒 “ ℕ) = (◡(𝑀‘𝑥) “ ℕ))
64	63	eleq1d 2303	. . . . . . 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 2979	. . . . . 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 2597	. . . 4 ⊢ ∀𝑥 ∈ ℕ (𝑀‘𝑥) ∈ 𝑅
69		ffnfv 5840	. . . 4 ⊢ (𝑀:ℕ⟶𝑅 ↔ (𝑀 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝑀‘𝑥) ∈ 𝑅))
70	4, 68, 69	mpbir2an 951	. . 3 ⊢ 𝑀:ℕ⟶𝑅
71	15	adantlr 477	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
72	3	1arithlem2 13090	. . . . . . . . 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 2249	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞) ↔ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
75	74	ralbidva 2540	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑞 ∈ ℙ ((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
76	3	1arithlem3 13091	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → (𝑀‘𝑦):ℙ⟶ℕ0)
77		ffn 5513	. . . . . . . 8 ⊢ ((𝑀‘𝑦):ℙ⟶ℕ0 → (𝑀‘𝑦) Fn ℙ)
78		eqfnfv 5780	. . . . . . . 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 9523	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
82		nnnn0 9523	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
83		pc11 13057	. . . . . . 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 2630	. . 3 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦)
88		dff13 5947	. . 3 ⊢ (𝑀:ℕ–1-1→𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦)))
89	70, 87, 88	mpbir2an 951	. 2 ⊢ 𝑀:ℕ–1-1→𝑅
90		eqid 2234	. . . . . 6 ⊢ (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓‘𝑔)), 1)) = (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓‘𝑔)), 1))
91		cnveq 4934	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑓 → ◡𝑒 = ◡𝑓)
92	91	imaeq1d 5105	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑓 → (◡𝑒 “ ℕ) = (◡𝑓 “ ℕ))
93	92	eleq1d 2303	. . . . . . . . . 10 ⊢ (𝑒 = 𝑓 → ((◡𝑒 “ ℕ) ∈ Fin ↔ (◡𝑓 “ ℕ) ∈ Fin))
94	93, 65	elrab2 2979	. . . . . . . . 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 6924	. . . . . . . 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 529	. . . . . . 7 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → 𝑦 ∈ ℕ)
100	99	peano2nnd 9272	. . . . . 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 9270	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 ∈ ℝ)
103		peano2re 8426	. . . . . . . . . . 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 9270	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℝ)
107	102	ltp1d 9224	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 < (𝑦 + 1))
108		simprr 533	. . . . . . . . . 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 8715	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 < 𝑞)
110	101	nnzd 9720	. . . . . . . . . 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 9643	. . . . . . . . . 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 531	. . . . . . . . . . 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 5513	. . . . . . . . . . 11 ⊢ (𝑓:ℙ⟶ℕ0 → 𝑓 Fn ℙ)
119		elpreima 5802	. . . . . . . . . . 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 4117	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑞 → (𝑘 ≤ 𝑦 ↔ 𝑞 ≤ 𝑦))
123	122	rspccv 2920	. . . . . . . . . 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 669	. . . . . . 7 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ¬ (𝑓‘𝑞) ∈ ℕ)
127	117, 115	ffvelcdmd 5818	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑓‘𝑞) ∈ ℕ0)
128		elnn0 9518	. . . . . . . . 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 732	. . . . . . 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 13092	. . . . 5 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥))
133		cnvimass 5130	. . . . . . 7 ⊢ (◡𝑓 “ ℕ) ⊆ dom 𝑓
134	97	fdmd 5520	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑅 → dom 𝑓 = ℙ)
135		prmssnn 12837	. . . . . . . 8 ⊢ ℙ ⊆ ℕ
136	134, 135	eqsstrdi 3294	. . . . . . 7 ⊢ (𝑓 ∈ 𝑅 → dom 𝑓 ⊆ ℕ)
137	133, 136	sstrid 3253	. . . . . 6 ⊢ (𝑓 ∈ 𝑅 → (◡𝑓 “ ℕ) ⊆ ℕ)
138	94	simprbi 275	. . . . . 6 ⊢ (𝑓 ∈ 𝑅 → (◡𝑓 “ ℕ) ∈ Fin)
139		fiubnn 11225	. . . . . 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 2688	. . . 4 ⊢ (𝑓 ∈ 𝑅 → ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥))
142	141	rgen 2597	. . 3 ⊢ ∀𝑓 ∈ 𝑅 ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥)
143		dffo3 5829	. . 3 ⊢ (𝑀:ℕ–onto→𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑓 ∈ 𝑅 ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥)))
144	70, 142, 143	mpbir2an 951	. 2 ⊢ 𝑀:ℕ–onto→𝑅
145		df-f1o 5364	. 2 ⊢ (𝑀:ℕ–1-1-onto→𝑅 ↔ (𝑀:ℕ–1-1→𝑅 ∧ 𝑀:ℕ–onto→𝑅))
146	89, 144, 145	mpbir2an 951	1 ⊢ 𝑀:ℕ–1-1-onto→𝑅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  DECID wdc 842   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  {crab 2526   ⊆ wss 3214  ifcif 3624   class class class wbr 4114   ↦ cmpt 4176  ^◡ccnv 4753  dom cdm 4754   “ cima 4757   Fn wfn 5352  ⟶wf 5353  –1-1→wf1 5354  –onto→wfo 5355  –1-1-onto→wf1o 5356  ‘cfv 5357  (class class class)co 6058   ↑_𝑚 cmap 6895  Fincfn 6988  ℝcr 8142  0cc0 8143  1c1 8144   + caddc 8146   < clt 8324   ≤ cle 8325  ℕcn 9257  ℕ₀cn0 9516  ℤcz 9597  ℤ_≥cuz 9874  ...cfz 10364  ↑cexp 10927   ∥ cdvds 12501  ℙcprime 12832   pCnt cpc 13010

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-map 6897  df-en 6989  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-reap 8867  df-ap 8874  df-div 8967  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-n0 9517  df-xnn0 9584  df-z 9598  df-uz 9875  df-q 9973  df-rp 10008  df-fz 10365  df-fzo 10502  df-fl 10657  df-mod 10712  df-seqfrec 10837  df-exp 10928  df-cj 11555  df-re 11556  df-im 11557  df-rsqrt 11711  df-abs 11712  df-dvds 12502  df-gcd 12678  df-prm 12833  df-pc 13011

This theorem is referenced by:  1arith2  13094