Theorem 1arith 16556

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmex 16310	. . . . . 6 ⊢ ℙ ∈ V
2	1	mptex 7081	. . . . 5 ⊢ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ V
3		1arith.1	. . . . 5 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
4	2, 3	fnmpti 6560	. . . 4 ⊢ 𝑀 Fn ℕ
5	3	1arithlem3 16554	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (𝑀‘𝑥):ℙ⟶ℕ0)
6		nn0ex 12169	. . . . . . . 8 ⊢ ℕ0 ∈ V
7	6, 1	elmap 8617	. . . . . . 7 ⊢ ((𝑀‘𝑥) ∈ (ℕ0 ↑m ℙ) ↔ (𝑀‘𝑥):ℙ⟶ℕ0)
8	5, 7	sylibr 233	. . . . . 6 ⊢ (𝑥 ∈ ℕ → (𝑀‘𝑥) ∈ (ℕ0 ↑m ℙ))
9		fzfi 13620	. . . . . . 7 ⊢ (1...𝑥) ∈ Fin
10		ffn 6584	. . . . . . . . . 10 ⊢ ((𝑀‘𝑥):ℙ⟶ℕ0 → (𝑀‘𝑥) Fn ℙ)
11		elpreima 6917	. . . . . . . . . 10 ⊢ ((𝑀‘𝑥) Fn ℙ → (𝑞 ∈ (◡(𝑀‘𝑥) “ ℕ) ↔ (𝑞 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑞) ∈ ℕ)))
12	5, 10, 11	3syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → (𝑞 ∈ (◡(𝑀‘𝑥) “ ℕ) ↔ (𝑞 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑞) ∈ ℕ)))
13	3	1arithlem2 16553	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
14	13	eleq1d 2823	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) ∈ ℕ ↔ (𝑞 pCnt 𝑥) ∈ ℕ))
15		prmz 16308	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
16		id 22	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
17		dvdsle 15947	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℤ ∧ 𝑥 ∈ ℕ) → (𝑞 ∥ 𝑥 → 𝑞 ≤ 𝑥))
18	15, 16, 17	syl2anr 596	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∥ 𝑥 → 𝑞 ≤ 𝑥))
19		pcelnn 16499	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℙ ∧ 𝑥 ∈ ℕ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞 ∥ 𝑥))
20	19	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞 ∥ 𝑥))
21		prmnn 16307	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
22		nnuz 12550	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ≥‘1)
23	21, 22	eleqtrdi 2849	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ (ℤ≥‘1))
24		nnz 12272	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
25		elfz5 13177	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ (ℤ≥‘1) ∧ 𝑥 ∈ ℤ) → (𝑞 ∈ (1...𝑥) ↔ 𝑞 ≤ 𝑥))
26	23, 24, 25	syl2anr 596	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∈ (1...𝑥) ↔ 𝑞 ≤ 𝑥))
27	18, 20, 26	3imtr4d 293	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
28	14, 27	sylbid 239	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
29	28	expimpd 453	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → ((𝑞 ∈ ℙ ∧ ((𝑀‘𝑥)‘𝑞) ∈ ℕ) → 𝑞 ∈ (1...𝑥)))
30	12, 29	sylbid 239	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (𝑞 ∈ (◡(𝑀‘𝑥) “ ℕ) → 𝑞 ∈ (1...𝑥)))
31	30	ssrdv 3923	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (◡(𝑀‘𝑥) “ ℕ) ⊆ (1...𝑥))
32		ssfi 8918	. . . . . . 7 ⊢ (((1...𝑥) ∈ Fin ∧ (◡(𝑀‘𝑥) “ ℕ) ⊆ (1...𝑥)) → (◡(𝑀‘𝑥) “ ℕ) ∈ Fin)
33	9, 31, 32	sylancr 586	. . . . . 6 ⊢ (𝑥 ∈ ℕ → (◡(𝑀‘𝑥) “ ℕ) ∈ Fin)
34		cnveq 5771	. . . . . . . . 9 ⊢ (𝑒 = (𝑀‘𝑥) → ◡𝑒 = ◡(𝑀‘𝑥))
35	34	imaeq1d 5957	. . . . . . . 8 ⊢ (𝑒 = (𝑀‘𝑥) → (◡𝑒 “ ℕ) = (◡(𝑀‘𝑥) “ ℕ))
36	35	eleq1d 2823	. . . . . . 7 ⊢ (𝑒 = (𝑀‘𝑥) → ((◡𝑒 “ ℕ) ∈ Fin ↔ (◡(𝑀‘𝑥) “ ℕ) ∈ Fin))
37		1arith.2	. . . . . . 7 ⊢ 𝑅 = {𝑒 ∈ (ℕ0 ↑m ℙ) ∣ (◡𝑒 “ ℕ) ∈ Fin}
38	36, 37	elrab2 3620	. . . . . 6 ⊢ ((𝑀‘𝑥) ∈ 𝑅 ↔ ((𝑀‘𝑥) ∈ (ℕ0 ↑m ℙ) ∧ (◡(𝑀‘𝑥) “ ℕ) ∈ Fin))
39	8, 33, 38	sylanbrc 582	. . . . 5 ⊢ (𝑥 ∈ ℕ → (𝑀‘𝑥) ∈ 𝑅)
40	39	rgen 3073	. . . 4 ⊢ ∀𝑥 ∈ ℕ (𝑀‘𝑥) ∈ 𝑅
41		ffnfv 6974	. . . 4 ⊢ (𝑀:ℕ⟶𝑅 ↔ (𝑀 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝑀‘𝑥) ∈ 𝑅))
42	4, 40, 41	mpbir2an 707	. . 3 ⊢ 𝑀:ℕ⟶𝑅
43	13	adantlr 711	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
44	3	1arithlem2 16553	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
45	44	adantll 710	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀‘𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
46	43, 45	eqeq12d 2754	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → (((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞) ↔ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
47	46	ralbidva 3119	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑞 ∈ ℙ ((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
48	3	1arithlem3 16554	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → (𝑀‘𝑦):ℙ⟶ℕ0)
49		ffn 6584	. . . . . . . 8 ⊢ ((𝑀‘𝑦):ℙ⟶ℕ0 → (𝑀‘𝑦) Fn ℙ)
50		eqfnfv 6891	. . . . . . . 8 ⊢ (((𝑀‘𝑥) Fn ℙ ∧ (𝑀‘𝑦) Fn ℙ) → ((𝑀‘𝑥) = (𝑀‘𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞)))
51	10, 49, 50	syl2an 595	. . . . . . 7 ⊢ (((𝑀‘𝑥):ℙ⟶ℕ0 ∧ (𝑀‘𝑦):ℙ⟶ℕ0) → ((𝑀‘𝑥) = (𝑀‘𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞)))
52	5, 48, 51	syl2an 595	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀‘𝑥) = (𝑀‘𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀‘𝑥)‘𝑞) = ((𝑀‘𝑦)‘𝑞)))
53		nnnn0 12170	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
54		nnnn0 12170	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
55		pc11 16509	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 = 𝑦 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
56	53, 54, 55	syl2an 595	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 = 𝑦 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
57	47, 52, 56	3bitr4d 310	. . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀‘𝑥) = (𝑀‘𝑦) ↔ 𝑥 = 𝑦))
58	57	biimpd 228	. . . 4 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦))
59	58	rgen2 3126	. . 3 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦)
60		dff13 7109	. . 3 ⊢ (𝑀:ℕ–1-1→𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀‘𝑥) = (𝑀‘𝑦) → 𝑥 = 𝑦)))
61	42, 59, 60	mpbir2an 707	. 2 ⊢ 𝑀:ℕ–1-1→𝑅
62		eqid 2738	. . . . . 6 ⊢ (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓‘𝑔)), 1)) = (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓‘𝑔)), 1))
63		cnveq 5771	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑓 → ◡𝑒 = ◡𝑓)
64	63	imaeq1d 5957	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑓 → (◡𝑒 “ ℕ) = (◡𝑓 “ ℕ))
65	64	eleq1d 2823	. . . . . . . . . 10 ⊢ (𝑒 = 𝑓 → ((◡𝑒 “ ℕ) ∈ Fin ↔ (◡𝑓 “ ℕ) ∈ Fin))
66	65, 37	elrab2 3620	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑅 ↔ (𝑓 ∈ (ℕ0 ↑m ℙ) ∧ (◡𝑓 “ ℕ) ∈ Fin))
67	66	simplbi 497	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑅 → 𝑓 ∈ (ℕ0 ↑m ℙ))
68	6, 1	elmap 8617	. . . . . . . 8 ⊢ (𝑓 ∈ (ℕ0 ↑m ℙ) ↔ 𝑓:ℙ⟶ℕ0)
69	67, 68	sylib 217	. . . . . . 7 ⊢ (𝑓 ∈ 𝑅 → 𝑓:ℙ⟶ℕ0)
70	69	ad2antrr 722	. . . . . 6 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → 𝑓:ℙ⟶ℕ0)
71		simplr 765	. . . . . . . . 9 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → 𝑦 ∈ ℝ)
72		0re 10908	. . . . . . . . 9 ⊢ 0 ∈ ℝ
73		ifcl 4501	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
74	71, 72, 73	sylancl 585	. . . . . . . 8 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
75		max1 12848	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
76	72, 71, 75	sylancr 586	. . . . . . . 8 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
77		flge0nn0 13468	. . . . . . . 8 ⊢ ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → (⌊‘if(0 ≤ 𝑦, 𝑦, 0)) ∈ ℕ0)
78	74, 76, 77	syl2anc 583	. . . . . . 7 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → (⌊‘if(0 ≤ 𝑦, 𝑦, 0)) ∈ ℕ0)
79		nn0p1nn 12202	. . . . . . 7 ⊢ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) ∈ ℕ0 → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ∈ ℕ)
80	78, 79	syl 17	. . . . . 6 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ∈ ℕ)
81	71	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 ∈ ℝ)
82	80	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ∈ ℕ)
83	82	nnred 11918	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ∈ ℝ)
84	15	ssriv 3921	. . . . . . . . . . . 12 ⊢ ℙ ⊆ ℤ
85		zssre 12256	. . . . . . . . . . . 12 ⊢ ℤ ⊆ ℝ
86	84, 85	sstri 3926	. . . . . . . . . . 11 ⊢ ℙ ⊆ ℝ
87		simprl 767	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑞 ∈ ℙ)
88	86, 87	sselid 3915	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑞 ∈ ℝ)
89	74	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
90		max2 12850	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
91	72, 81, 90	sylancr 586	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
92		flltp1 13448	. . . . . . . . . . . 12 ⊢ (if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ → if(0 ≤ 𝑦, 𝑦, 0) < ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1))
93	89, 92	syl 17	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → if(0 ≤ 𝑦, 𝑦, 0) < ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1))
94	81, 89, 83, 91, 93	lelttrd 11063	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 < ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1))
95		simprr 769	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)
96	81, 83, 88, 94, 95	ltletrd 11065	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 < 𝑞)
97	81, 88	ltnled 11052	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑦 < 𝑞 ↔ ¬ 𝑞 ≤ 𝑦))
98	96, 97	mpbid 231	. . . . . . . 8 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ¬ 𝑞 ≤ 𝑦)
99	87	biantrurd 532	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓‘𝑞) ∈ ℕ ↔ (𝑞 ∈ ℙ ∧ (𝑓‘𝑞) ∈ ℕ)))
100	70	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑓:ℙ⟶ℕ0)
101		ffn 6584	. . . . . . . . . . 11 ⊢ (𝑓:ℙ⟶ℕ0 → 𝑓 Fn ℙ)
102		elpreima 6917	. . . . . . . . . . 11 ⊢ (𝑓 Fn ℙ → (𝑞 ∈ (◡𝑓 “ ℕ) ↔ (𝑞 ∈ ℙ ∧ (𝑓‘𝑞) ∈ ℕ)))
103	100, 101, 102	3syl 18	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑞 ∈ (◡𝑓 “ ℕ) ↔ (𝑞 ∈ ℙ ∧ (𝑓‘𝑞) ∈ ℕ)))
104	99, 103	bitr4d 281	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓‘𝑞) ∈ ℕ ↔ 𝑞 ∈ (◡𝑓 “ ℕ)))
105		simplr 765	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦)
106		breq1 5073	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑞 → (𝑘 ≤ 𝑦 ↔ 𝑞 ≤ 𝑦))
107	106	rspccv 3549	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦 → (𝑞 ∈ (◡𝑓 “ ℕ) → 𝑞 ≤ 𝑦))
108	105, 107	syl 17	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑞 ∈ (◡𝑓 “ ℕ) → 𝑞 ≤ 𝑦))
109	104, 108	sylbid 239	. . . . . . . 8 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓‘𝑞) ∈ ℕ → 𝑞 ≤ 𝑦))
110	98, 109	mtod 197	. . . . . . 7 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ¬ (𝑓‘𝑞) ∈ ℕ)
111	100, 87	ffvelrnd 6944	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑓‘𝑞) ∈ ℕ0)
112		elnn0 12165	. . . . . . . . 9 ⊢ ((𝑓‘𝑞) ∈ ℕ0 ↔ ((𝑓‘𝑞) ∈ ℕ ∨ (𝑓‘𝑞) = 0))
113	111, 112	sylib 217	. . . . . . . 8 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓‘𝑞) ∈ ℕ ∨ (𝑓‘𝑞) = 0))
114	113	ord 860	. . . . . . 7 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (¬ (𝑓‘𝑞) ∈ ℕ → (𝑓‘𝑞) = 0))
115	110, 114	mpd 15	. . . . . 6 ⊢ ((((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑓‘𝑞) = 0)
116	3, 62, 70, 80, 115	1arithlem4 16555	. . . . 5 ⊢ (((𝑓 ∈ 𝑅 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦) → ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥))
117		cnvimass 5978	. . . . . . 7 ⊢ (◡𝑓 “ ℕ) ⊆ dom 𝑓
118	69	fdmd 6595	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑅 → dom 𝑓 = ℙ)
119	118, 86	eqsstrdi 3971	. . . . . . 7 ⊢ (𝑓 ∈ 𝑅 → dom 𝑓 ⊆ ℝ)
120	117, 119	sstrid 3928	. . . . . 6 ⊢ (𝑓 ∈ 𝑅 → (◡𝑓 “ ℕ) ⊆ ℝ)
121	66	simprbi 496	. . . . . 6 ⊢ (𝑓 ∈ 𝑅 → (◡𝑓 “ ℕ) ∈ Fin)
122		fimaxre2 11850	. . . . . 6 ⊢ (((◡𝑓 “ ℕ) ⊆ ℝ ∧ (◡𝑓 “ ℕ) ∈ Fin) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦)
123	120, 121, 122	syl2anc 583	. . . . 5 ⊢ (𝑓 ∈ 𝑅 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (◡𝑓 “ ℕ)𝑘 ≤ 𝑦)
124	116, 123	r19.29a 3217	. . . 4 ⊢ (𝑓 ∈ 𝑅 → ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥))
125	124	rgen 3073	. . 3 ⊢ ∀𝑓 ∈ 𝑅 ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥)
126		dffo3 6960	. . 3 ⊢ (𝑀:ℕ–onto→𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑓 ∈ 𝑅 ∃𝑥 ∈ ℕ 𝑓 = (𝑀‘𝑥)))
127	42, 125, 126	mpbir2an 707	. 2 ⊢ 𝑀:ℕ–onto→𝑅
128		df-f1o 6425	. 2 ⊢ (𝑀:ℕ–1-1-onto→𝑅 ↔ (𝑀:ℕ–1-1→𝑅 ∧ 𝑀:ℕ–onto→𝑅))
129	61, 127, 128	mpbir2an 707	1 ⊢ 𝑀:ℕ–1-1-onto→𝑅

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  {crab 3067   ⊆ wss 3883  ifcif 4456   class class class wbr 5070   ↦ cmpt 5153  ^◡ccnv 5579  dom cdm 5580   “ cima 5583   Fn wfn 6413  ⟶wf 6414  –1-1→wf1 6415  –onto→wfo 6416  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  Fincfn 8691  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805   < clt 10940   ≤ cle 10941  ℕcn 11903  ℕ₀cn0 12163  ℤcz 12249  ℤ_≥cuz 12511  ...cfz 13168  ⌊cfl 13438  ↑cexp 13710   ∥ cdvds 15891  ℙcprime 16304   pCnt cpc 16465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-fz 13169  df-fl 13440  df-mod 13518  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-dvds 15892  df-gcd 16130  df-prm 16305  df-pc 16466

This theorem is referenced by:  1arith2  16557  sqff1o  26236