Theorem 1arith 12319

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 12067	. . . . . 6 ⊢ ℙ ∈ V
2	1	mptex 5722	. . . . 5 ⊢ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ V
3		1arith.1	. . . . 5 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
4	2, 3	fnmpti 5326	. . . 4 ⊢ 𝑀 Fn ℕ
5	3	1arithlem3 12317	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (𝑀‘𝑥):ℙ⟶ℕ0)
6		nn0ex 9141	. . . . . . . 8 ⊢ ℕ0 ∈ V
7	6, 1	elmap 6655	. . . . . . 7 ⊢ ((𝑀‘𝑥) ∈ (ℕ0 ↑𝑚 ℙ) ↔ (𝑀‘𝑥):ℙ⟶ℕ0)
8	5, 7	sylibr 133	. . . . . 6 ⊢ (𝑥 ∈ ℕ → (𝑀‘𝑥) ∈ (ℕ0 ↑𝑚 ℙ))
9		1zzd 9239	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 1 ∈ ℤ)
10		nnz 9231	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
11	9, 10	fzfigd 10387	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1...𝑥) ∈ Fin)
12		ffn 5347	. . . . . . . . . 10 ⊢ ((𝑀‘𝑥):ℙ⟶ℕ0 → (𝑀‘𝑥) Fn ℙ)
13		elpreima 5615	. . . . . . . . . 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 12316	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
16	15	eleq1d 2239	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) ∈ ℕ ↔ (𝑞 pCnt 𝑥) ∈ ℕ))
17		prmz 12065	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
18		id 19	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
19		dvdsle 11804	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℤ ∧ 𝑥 ∈ ℕ) → (𝑞 ∥ 𝑥 → 𝑞 ≤ 𝑥))
20	17, 18, 19	syl2anr 288	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∥ 𝑥 → 𝑞 ≤ 𝑥))
21		pcelnn 12274	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℙ ∧ 𝑥 ∈ ℕ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞 ∥ 𝑥))
22	21	ancoms 266	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞 ∥ 𝑥))
23		prmnn 12064	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
24		nnuz 9522	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ≥‘1)
25	23, 24	eleqtrdi 2263	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ (ℤ≥‘1))
26		elfz5 9973	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ (ℤ≥‘1) ∧ 𝑥 ∈ ℤ) → (𝑞 ∈ (1...𝑥) ↔ 𝑞 ≤ 𝑥))
27	25, 10, 26	syl2anr 288	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∈ (1...𝑥) ↔ 𝑞 ≤ 𝑥))
28	20, 22, 27	3imtr4d 202	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
29	16, 28	sylbid 149	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
30	29	expimpd 361	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → ((𝑞 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑞) ∈ ℕ) → 𝑞 ∈ (1...𝑥)))
31	14, 30	sylbid 149	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (𝑞 ∈ (◡(𝑀‘𝑥) “ ℕ) → 𝑞 ∈ (1...𝑥)))
32	31	ssrdv 3153	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (◡(𝑀‘𝑥) “ ℕ) ⊆ (1...𝑥))
33		elfznn 10010	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑥) → 𝑗 ∈ ℕ)
34	33	adantl 275	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → 𝑗 ∈ ℕ)
35		prmdc 12084	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → DECID 𝑗 ∈ ℙ)
36	34, 35	syl 14	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → DECID 𝑗 ∈ ℙ)
37	36	adantr 274	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → DECID 𝑗 ∈ ℙ)
38	5	ad2antrr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → (𝑀‘𝑥):ℙ⟶ℕ0)
39		simpr 109	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → 𝑗 ∈ ℙ)
40	38, 39	ffvelrnd 5632	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → ((𝑀‘𝑥)‘𝑗) ∈ ℕ0)
41	40	nn0zd 9332	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → ((𝑀‘𝑥)‘𝑗) ∈ ℤ)
42		elnndc 9571	. . . . . . . . . . . 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 929	. . . . . . . . . . 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 109	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ¬ 𝑗 ∈ ℙ)
47	46	intnanrd 927	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ¬ (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ))
48	47	olcd 729	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ((𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ) ∨ ¬ (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
49		df-dc 830	. . . . . . . . . . 11 ⊢ (DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ) ↔ ((𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ) ∨ ¬ (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
50	48, 49	sylibr 133	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ))
51		exmiddc 831	. . . . . . . . . . 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 793	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ))
54		elpreima 5615	. . . . . . . . . . . 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 833	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → (DECID 𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ) ↔ DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
57	56	adantr 274	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → (DECID 𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ) ↔ DECID (𝑗 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑗) ∈ ℕ)))
58	53, 57	mpbird 166	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → DECID 𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ))
59	58	ralrimiva 2543	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → ∀𝑗 ∈ (1...𝑥)DECID 𝑗 ∈ (◡(𝑀‘𝑥) “ ℕ))
60		ssfidc 6912	. . . . . . 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 1233	. . . . . 6 ⊢ (𝑥 ∈ ℕ → (◡(𝑀‘𝑥) “ ℕ) ∈ Fin)
62		cnveq 4785	. . . . . . . . 9 ⊢ (𝑒 = (𝑀‘𝑥) → ◡𝑒 = ◡(𝑀‘𝑥))
63	62	imaeq1d 4952	. . . . . . . 8 ⊢ (𝑒 = (𝑀‘𝑥) → (◡𝑒 “ ℕ) = (◡(𝑀‘𝑥) “ ℕ))
64	63	eleq1d 2239	. . . . . . 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 2889	. . . . . 6 ⊢ ((𝑀‘𝑥) ∈ 𝑅 ↔ ((𝑀‘𝑥) ∈ (ℕ0 ↑𝑚 ℙ) ∧ (◡(𝑀‘𝑥) “ ℕ) ∈ Fin))
67	8, 61, 66	sylanbrc 415	. . . . 5 ⊢ (𝑥 ∈ ℕ → (𝑀‘𝑥) ∈ 𝑅)
68	67	rgen 2523	. . . 4 ⊢ ∀𝑥 ∈ ℕ (𝑀‘𝑥) ∈ 𝑅
69		ffnfv 5654	. . . 4 ⊢ (𝑀:ℕ⟶𝑅 ↔ (𝑀 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝑀‘𝑥) ∈ 𝑅))
70	4, 68, 69	mpbir2an 937	. . 3 ⊢ 𝑀:ℕ⟶𝑅
71	15	adantlr 474	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
72	3	1arithlem2 12316	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
73	72	adantll 473	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
74	71, 73	eqeq12d 2185	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞) ↔ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
75	74	ralbidva 2466	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑞 ∈ ℙ ((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
76	3	1arithlem3 12317	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → (𝑀‘𝑦):ℙ⟶ℕ0)
77		ffn 5347	. . . . . . . 8 ⊢ ((𝑀‘𝑦):ℙ⟶ℕ0 → (𝑀‘𝑦) Fn ℙ)
78		eqfnfv 5593	. . . . . . . 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 287	. . . . . . 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 287	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀‘𝑥) = (𝑀‘𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞)))
81		nnnn0 9142	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
82		nnnn0 9142	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
83		pc11 12284	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 = 𝑦 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
84	81, 82, 83	syl2an 287	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 = 𝑦 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
85	75, 80, 84	3bitr4d 219	. . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀‘𝑥) = (𝑀‘𝑦) ↔ 𝑥 = 𝑦))
86	85	biimpd 143	. . . 4 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦))
87	86	rgen2 2556	. . 3 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦)
88		dff13 5747	. . 3 ⊢ (𝑀:ℕ–1-1→𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦)))
89	70, 87, 88	mpbir2an 937	. 2 ⊢ 𝑀:ℕ–1-1→𝑅
90		eqid 2170	. . . . . 6 ⊢ (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓‘𝑔)), 1)) = (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓‘𝑔)), 1))
91		cnveq 4785	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑓 → ◡𝑒 = ◡𝑓)
92	91	imaeq1d 4952	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑓 → (◡𝑒 “ ℕ) = (◡𝑓 “ ℕ))
93	92	eleq1d 2239	. . . . . . . . . 10 ⊢ (𝑒 = 𝑓 → ((◡𝑒 “ ℕ) ∈ Fin ↔ (◡𝑓 “ ℕ) ∈ Fin))
94	93, 65	elrab2 2889	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑅 ↔ (𝑓 ∈ (ℕ0 ↑𝑚 ℙ) ∧ (◡𝑓 “ ℕ) ∈ Fin))
95	94	simplbi 272	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑅 → 𝑓 ∈ (ℕ0 ↑𝑚 ℙ))
96	6, 1	elmap 6655	. . . . . . . 8 ⊢ (𝑓 ∈ (ℕ0 ↑𝑚 ℙ) ↔ 𝑓:ℙ⟶ℕ0)
97	95, 96	sylib 121	. . . . . . 7 ⊢ (𝑓 ∈ 𝑅 → 𝑓:ℙ⟶ℕ0)
98	97	ad2antrr 485	. . . . . 6 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → 𝑓:ℙ⟶ℕ0)
99		simplr 525	. . . . . . 7 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → 𝑦 ∈ ℕ)
100	99	peano2nnd 8893	. . . . . 6 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → (𝑦 + 1) ∈ ℕ)
101	99	adantr 274	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 ∈ ℕ)
102	101	nnred 8891	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 ∈ ℝ)
103		peano2re 8055	. . . . . . . . . . 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 487	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℕ)
106	105	nnred 8891	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℝ)
107	102	ltp1d 8846	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 < (𝑦 + 1))
108		simprr 527	. . . . . . . . . 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 8342	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 < 𝑞)
110	101	nnzd 9333	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 ∈ ℤ)
111	17	ad2antrl 487	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℤ)
112		zltnle 9258	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ 𝑞 ∈ ℤ) → (𝑦 < 𝑞 ↔ ¬ 𝑞 ≤ 𝑦))
113	110, 111, 112	syl2anc 409	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑦 < 𝑞 ↔ ¬ 𝑞 ≤ 𝑦))
114	109, 113	mpbid 146	. . . . . . . 8 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ¬ 𝑞 ≤ 𝑦)
115		simprl 526	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℙ)
116	115	biantrurd 303	. . . . . . . . . 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 489	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑓:ℙ⟶ℕ0)
118		ffn 5347	. . . . . . . . . . 11 ⊢ (𝑓:ℙ⟶ℕ0 → 𝑓 Fn ℙ)
119		elpreima 5615	. . . . . . . . . . 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 190	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ((𝑓‘𝑞) ∈ ℕ ↔ 𝑞 ∈ (◡𝑓 “ ℕ)))
122		breq1 3992	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑞 → (𝑘 ≤ 𝑦 ↔ 𝑞 ≤ 𝑦))
123	122	rspccv 2831	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦 → (𝑞 ∈ (◡𝑓 “ ℕ) → 𝑞 ≤ 𝑦))
124	123	ad2antlr 486	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑞 ∈ (◡𝑓 “ ℕ) → 𝑞 ≤ 𝑦))
125	121, 124	sylbid 149	. . . . . . . 8 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ((𝑓‘𝑞) ∈ ℕ → 𝑞 ≤ 𝑦))
126	114, 125	mtod 658	. . . . . . 7 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ¬ (𝑓‘𝑞) ∈ ℕ)
127	117, 115	ffvelrnd 5632	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑓‘𝑞) ∈ ℕ0)
128		elnn0 9137	. . . . . . . . 9 ⊢ ((𝑓‘𝑞) ∈ ℕ0 ↔ ((𝑓‘𝑞) ∈ ℕ ∨ (𝑓‘𝑞) = 0))
129	127, 128	sylib 121	. . . . . . . 8 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ((𝑓‘𝑞) ∈ ℕ ∨ (𝑓‘𝑞) = 0))
130	129	ord 719	. . . . . . 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 12318	. . . . 5 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥))
133		cnvimass 4974	. . . . . . 7 ⊢ (◡𝑓 “ ℕ) ⊆ dom 𝑓
134	97	fdmd 5354	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑅 → dom 𝑓 = ℙ)
135		prmssnn 12066	. . . . . . . 8 ⊢ ℙ ⊆ ℕ
136	134, 135	eqsstrdi 3199	. . . . . . 7 ⊢ (𝑓 ∈ 𝑅 → dom 𝑓 ⊆ ℕ)
137	133, 136	sstrid 3158	. . . . . 6 ⊢ (𝑓 ∈ 𝑅 → (◡𝑓 “ ℕ) ⊆ ℕ)
138	94	simprbi 273	. . . . . 6 ⊢ (𝑓 ∈ 𝑅 → (◡𝑓 “ ℕ) ∈ Fin)
139		fiubnn 10765	. . . . . 6 ⊢ (((◡𝑓 “ ℕ) ⊆ ℕ ∧ (◡𝑓 “ ℕ) ∈ Fin) → ∃𝑦 ∈ ℕ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦)
140	137, 138, 139	syl2anc 409	. . . . 5 ⊢ (𝑓 ∈ 𝑅 → ∃𝑦 ∈ ℕ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦)
141	132, 140	r19.29a 2613	. . . 4 ⊢ (𝑓 ∈ 𝑅 → ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥))
142	141	rgen 2523	. . 3 ⊢ ∀𝑓 ∈ 𝑅 ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥)
143		dffo3 5643	. . 3 ⊢ (𝑀:ℕ–onto→𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑓 ∈ 𝑅 ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥)))
144	70, 142, 143	mpbir2an 937	. 2 ⊢ 𝑀:ℕ–onto→𝑅
145		df-f1o 5205	. 2 ⊢ (𝑀:ℕ–1-1-onto→𝑅 ↔ (𝑀:ℕ–1-1→𝑅 ∧ 𝑀:ℕ–onto→𝑅))
146	89, 144, 145	mpbir2an 937	1 ⊢ 𝑀:ℕ–1-1-onto→𝑅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  DECID wdc 829   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  {crab 2452   ⊆ wss 3121  ifcif 3526   class class class wbr 3989   ↦ cmpt 4050  ^◡ccnv 4610  dom cdm 4611   “ cima 4614   Fn wfn 5193  ⟶wf 5194  –1-1→wf1 5195  –onto→wfo 5196  –1-1-onto→wf1o 5197  ‘cfv 5198  (class class class)co 5853   ↑_𝑚 cmap 6626  Fincfn 6718  ℝcr 7773  0cc0 7774  1c1 7775   + caddc 7777   < clt 7954   ≤ cle 7955  ℕcn 8878  ℕ₀cn0 9135  ℤcz 9212  ℤ_≥cuz 9487  ...cfz 9965  ↑cexp 10475   ∥ cdvds 11749  ℙcprime 12061   pCnt cpc 12238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-2o 6396  df-er 6513  df-map 6628  df-en 6719  df-fin 6721  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-xnn0 9199  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898  df-prm 12062  df-pc 12239

This theorem is referenced by:  1arith2  12320