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Theorem 1arith 15469
 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 ∈ (ℕ0𝑚 ℙ) ∣ (𝑒 “ ℕ) ∈ Fin}
Assertion
Ref Expression
1arith 𝑀:ℕ–1-1-onto𝑅
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.
StepHypRef Expression
1 zex 11263 . . . . . . 7 ℤ ∈ V
2 prmz 15227 . . . . . . . 8 (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
32ssriv 3572 . . . . . . 7 ℙ ⊆ ℤ
41, 3ssexi 4731 . . . . . 6 ℙ ∈ V
54mptex 6390 . . . . 5 (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ V
6 1arith.1 . . . . 5 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
75, 6fnmpti 5935 . . . 4 𝑀 Fn ℕ
861arithlem3 15467 . . . . . . 7 (𝑥 ∈ ℕ → (𝑀𝑥):ℙ⟶ℕ0)
9 nn0ex 11175 . . . . . . . 8 0 ∈ V
109, 4elmap 7772 . . . . . . 7 ((𝑀𝑥) ∈ (ℕ0𝑚 ℙ) ↔ (𝑀𝑥):ℙ⟶ℕ0)
118, 10sylibr 223 . . . . . 6 (𝑥 ∈ ℕ → (𝑀𝑥) ∈ (ℕ0𝑚 ℙ))
12 fzfi 12633 . . . . . . 7 (1...𝑥) ∈ Fin
13 ffn 5958 . . . . . . . . . 10 ((𝑀𝑥):ℙ⟶ℕ0 → (𝑀𝑥) Fn ℙ)
14 elpreima 6245 . . . . . . . . . 10 ((𝑀𝑥) Fn ℙ → (𝑞 ∈ ((𝑀𝑥) “ ℕ) ↔ (𝑞 ∈ ℙ ∧ ((𝑀𝑥)‘𝑞) ∈ ℕ)))
158, 13, 143syl 18 . . . . . . . . 9 (𝑥 ∈ ℕ → (𝑞 ∈ ((𝑀𝑥) “ ℕ) ↔ (𝑞 ∈ ℙ ∧ ((𝑀𝑥)‘𝑞) ∈ ℕ)))
1661arithlem2 15466 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
1716eleq1d 2672 . . . . . . . . . . 11 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) ∈ ℕ ↔ (𝑞 pCnt 𝑥) ∈ ℕ))
18 id 22 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
19 dvdsle 14870 . . . . . . . . . . . . 13 ((𝑞 ∈ ℤ ∧ 𝑥 ∈ ℕ) → (𝑞𝑥𝑞𝑥))
202, 18, 19syl2anr 494 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞𝑥𝑞𝑥))
21 pcelnn 15412 . . . . . . . . . . . . 13 ((𝑞 ∈ ℙ ∧ 𝑥 ∈ ℕ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞𝑥))
2221ancoms 468 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞𝑥))
23 prmnn 15226 . . . . . . . . . . . . . 14 (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
24 nnuz 11599 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
2523, 24syl6eleq 2698 . . . . . . . . . . . . 13 (𝑞 ∈ ℙ → 𝑞 ∈ (ℤ‘1))
26 nnz 11276 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
27 elfz5 12205 . . . . . . . . . . . . 13 ((𝑞 ∈ (ℤ‘1) ∧ 𝑥 ∈ ℤ) → (𝑞 ∈ (1...𝑥) ↔ 𝑞𝑥))
2825, 26, 27syl2anr 494 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∈ (1...𝑥) ↔ 𝑞𝑥))
2920, 22, 283imtr4d 282 . . . . . . . . . . 11 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
3017, 29sylbid 229 . . . . . . . . . 10 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
3130expimpd 627 . . . . . . . . 9 (𝑥 ∈ ℕ → ((𝑞 ∈ ℙ ∧ ((𝑀𝑥)‘𝑞) ∈ ℕ) → 𝑞 ∈ (1...𝑥)))
3215, 31sylbid 229 . . . . . . . 8 (𝑥 ∈ ℕ → (𝑞 ∈ ((𝑀𝑥) “ ℕ) → 𝑞 ∈ (1...𝑥)))
3332ssrdv 3574 . . . . . . 7 (𝑥 ∈ ℕ → ((𝑀𝑥) “ ℕ) ⊆ (1...𝑥))
34 ssfi 8065 . . . . . . 7 (((1...𝑥) ∈ Fin ∧ ((𝑀𝑥) “ ℕ) ⊆ (1...𝑥)) → ((𝑀𝑥) “ ℕ) ∈ Fin)
3512, 33, 34sylancr 694 . . . . . 6 (𝑥 ∈ ℕ → ((𝑀𝑥) “ ℕ) ∈ Fin)
36 cnveq 5218 . . . . . . . . 9 (𝑒 = (𝑀𝑥) → 𝑒 = (𝑀𝑥))
3736imaeq1d 5384 . . . . . . . 8 (𝑒 = (𝑀𝑥) → (𝑒 “ ℕ) = ((𝑀𝑥) “ ℕ))
3837eleq1d 2672 . . . . . . 7 (𝑒 = (𝑀𝑥) → ((𝑒 “ ℕ) ∈ Fin ↔ ((𝑀𝑥) “ ℕ) ∈ Fin))
39 1arith.2 . . . . . . 7 𝑅 = {𝑒 ∈ (ℕ0𝑚 ℙ) ∣ (𝑒 “ ℕ) ∈ Fin}
4038, 39elrab2 3333 . . . . . 6 ((𝑀𝑥) ∈ 𝑅 ↔ ((𝑀𝑥) ∈ (ℕ0𝑚 ℙ) ∧ ((𝑀𝑥) “ ℕ) ∈ Fin))
4111, 35, 40sylanbrc 695 . . . . 5 (𝑥 ∈ ℕ → (𝑀𝑥) ∈ 𝑅)
4241rgen 2906 . . . 4 𝑥 ∈ ℕ (𝑀𝑥) ∈ 𝑅
43 ffnfv 6295 . . . 4 (𝑀:ℕ⟶𝑅 ↔ (𝑀 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝑀𝑥) ∈ 𝑅))
447, 42, 43mpbir2an 957 . . 3 𝑀:ℕ⟶𝑅
4516adantlr 747 . . . . . . . 8 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
4661arithlem2 15466 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
4746adantll 746 . . . . . . . 8 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
4845, 47eqeq12d 2625 . . . . . . 7 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞) ↔ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
4948ralbidva 2968 . . . . . 6 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
5061arithlem3 15467 . . . . . . 7 (𝑦 ∈ ℕ → (𝑀𝑦):ℙ⟶ℕ0)
51 ffn 5958 . . . . . . . 8 ((𝑀𝑦):ℙ⟶ℕ0 → (𝑀𝑦) Fn ℙ)
52 eqfnfv 6219 . . . . . . . 8 (((𝑀𝑥) Fn ℙ ∧ (𝑀𝑦) Fn ℙ) → ((𝑀𝑥) = (𝑀𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞)))
5313, 51, 52syl2an 493 . . . . . . 7 (((𝑀𝑥):ℙ⟶ℕ0 ∧ (𝑀𝑦):ℙ⟶ℕ0) → ((𝑀𝑥) = (𝑀𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞)))
548, 50, 53syl2an 493 . . . . . 6 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀𝑥) = (𝑀𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞)))
55 nnnn0 11176 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
56 nnnn0 11176 . . . . . . 7 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
57 pc11 15422 . . . . . . 7 ((𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑥 = 𝑦 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
5855, 56, 57syl2an 493 . . . . . 6 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 = 𝑦 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
5949, 54, 583bitr4d 299 . . . . 5 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀𝑥) = (𝑀𝑦) ↔ 𝑥 = 𝑦))
6059biimpd 218 . . . 4 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦))
6160rgen2a 2960 . . 3 𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦)
62 dff13 6416 . . 3 (𝑀:ℕ–1-1𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦)))
6344, 61, 62mpbir2an 957 . 2 𝑀:ℕ–1-1𝑅
64 eqid 2610 . . . . . 6 (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓𝑔)), 1)) = (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓𝑔)), 1))
65 cnveq 5218 . . . . . . . . . . . 12 (𝑒 = 𝑓𝑒 = 𝑓)
6665imaeq1d 5384 . . . . . . . . . . 11 (𝑒 = 𝑓 → (𝑒 “ ℕ) = (𝑓 “ ℕ))
6766eleq1d 2672 . . . . . . . . . 10 (𝑒 = 𝑓 → ((𝑒 “ ℕ) ∈ Fin ↔ (𝑓 “ ℕ) ∈ Fin))
6867, 39elrab2 3333 . . . . . . . . 9 (𝑓𝑅 ↔ (𝑓 ∈ (ℕ0𝑚 ℙ) ∧ (𝑓 “ ℕ) ∈ Fin))
6968simplbi 475 . . . . . . . 8 (𝑓𝑅𝑓 ∈ (ℕ0𝑚 ℙ))
709, 4elmap 7772 . . . . . . . 8 (𝑓 ∈ (ℕ0𝑚 ℙ) ↔ 𝑓:ℙ⟶ℕ0)
7169, 70sylib 207 . . . . . . 7 (𝑓𝑅𝑓:ℙ⟶ℕ0)
7271ad2antrr 758 . . . . . 6 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → 𝑓:ℙ⟶ℕ0)
73 simplr 788 . . . . . . . . 9 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → 𝑦 ∈ ℝ)
74 0re 9919 . . . . . . . . 9 0 ∈ ℝ
75 ifcl 4080 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
7673, 74, 75sylancl 693 . . . . . . . 8 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
77 max1 11890 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
7874, 73, 77sylancr 694 . . . . . . . 8 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
79 flge0nn0 12483 . . . . . . . 8 ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → (⌊‘if(0 ≤ 𝑦, 𝑦, 0)) ∈ ℕ0)
8076, 78, 79syl2anc 691 . . . . . . 7 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → (⌊‘if(0 ≤ 𝑦, 𝑦, 0)) ∈ ℕ0)
81 nn0p1nn 11209 . . . . . . 7 ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) ∈ ℕ0 → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ∈ ℕ)
8280, 81syl 17 . . . . . 6 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ∈ ℕ)
8373adantr 480 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 ∈ ℝ)
8482adantr 480 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ∈ ℕ)
8584nnred 10912 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ∈ ℝ)
86 zssre 11261 . . . . . . . . . . . 12 ℤ ⊆ ℝ
873, 86sstri 3577 . . . . . . . . . . 11 ℙ ⊆ ℝ
88 simprl 790 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑞 ∈ ℙ)
8987, 88sseldi 3566 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑞 ∈ ℝ)
9076adantr 480 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
91 max2 11892 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
9274, 83, 91sylancr 694 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
93 flltp1 12463 . . . . . . . . . . . 12 (if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ → if(0 ≤ 𝑦, 𝑦, 0) < ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1))
9490, 93syl 17 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → if(0 ≤ 𝑦, 𝑦, 0) < ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1))
9583, 90, 85, 92, 94lelttrd 10074 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 < ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1))
96 simprr 792 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)
9783, 85, 89, 95, 96ltletrd 10076 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 < 𝑞)
9883, 89ltnled 10063 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑦 < 𝑞 ↔ ¬ 𝑞𝑦))
9997, 98mpbid 221 . . . . . . . 8 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ¬ 𝑞𝑦)
10088biantrurd 528 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ ↔ (𝑞 ∈ ℙ ∧ (𝑓𝑞) ∈ ℕ)))
10172adantr 480 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑓:ℙ⟶ℕ0)
102 ffn 5958 . . . . . . . . . . 11 (𝑓:ℙ⟶ℕ0𝑓 Fn ℙ)
103 elpreima 6245 . . . . . . . . . . 11 (𝑓 Fn ℙ → (𝑞 ∈ (𝑓 “ ℕ) ↔ (𝑞 ∈ ℙ ∧ (𝑓𝑞) ∈ ℕ)))
104101, 102, 1033syl 18 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑞 ∈ (𝑓 “ ℕ) ↔ (𝑞 ∈ ℙ ∧ (𝑓𝑞) ∈ ℕ)))
105100, 104bitr4d 270 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ ↔ 𝑞 ∈ (𝑓 “ ℕ)))
106 simplr 788 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦)
107 breq1 4586 . . . . . . . . . . 11 (𝑘 = 𝑞 → (𝑘𝑦𝑞𝑦))
108107rspccv 3279 . . . . . . . . . 10 (∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦 → (𝑞 ∈ (𝑓 “ ℕ) → 𝑞𝑦))
109106, 108syl 17 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑞 ∈ (𝑓 “ ℕ) → 𝑞𝑦))
110105, 109sylbid 229 . . . . . . . 8 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ → 𝑞𝑦))
11199, 110mtod 188 . . . . . . 7 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ¬ (𝑓𝑞) ∈ ℕ)
112101, 88ffvelrnd 6268 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑓𝑞) ∈ ℕ0)
113 elnn0 11171 . . . . . . . . 9 ((𝑓𝑞) ∈ ℕ0 ↔ ((𝑓𝑞) ∈ ℕ ∨ (𝑓𝑞) = 0))
114112, 113sylib 207 . . . . . . . 8 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ ∨ (𝑓𝑞) = 0))
115114ord 391 . . . . . . 7 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (¬ (𝑓𝑞) ∈ ℕ → (𝑓𝑞) = 0))
116111, 115mpd 15 . . . . . 6 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑓𝑞) = 0)
1176, 64, 72, 82, 1161arithlem4 15468 . . . . 5 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → ∃𝑥 ∈ ℕ 𝑓 = (𝑀𝑥))
118 cnvimass 5404 . . . . . . 7 (𝑓 “ ℕ) ⊆ dom 𝑓
119 fdm 5964 . . . . . . . . 9 (𝑓:ℙ⟶ℕ0 → dom 𝑓 = ℙ)
12071, 119syl 17 . . . . . . . 8 (𝑓𝑅 → dom 𝑓 = ℙ)
121120, 87syl6eqss 3618 . . . . . . 7 (𝑓𝑅 → dom 𝑓 ⊆ ℝ)
122118, 121syl5ss 3579 . . . . . 6 (𝑓𝑅 → (𝑓 “ ℕ) ⊆ ℝ)
12368simprbi 479 . . . . . 6 (𝑓𝑅 → (𝑓 “ ℕ) ∈ Fin)
124 fimaxre2 10848 . . . . . 6 (((𝑓 “ ℕ) ⊆ ℝ ∧ (𝑓 “ ℕ) ∈ Fin) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦)
125122, 123, 124syl2anc 691 . . . . 5 (𝑓𝑅 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦)
126117, 125r19.29a 3060 . . . 4 (𝑓𝑅 → ∃𝑥 ∈ ℕ 𝑓 = (𝑀𝑥))
127126rgen 2906 . . 3 𝑓𝑅𝑥 ∈ ℕ 𝑓 = (𝑀𝑥)
128 dffo3 6282 . . 3 (𝑀:ℕ–onto𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑓𝑅𝑥 ∈ ℕ 𝑓 = (𝑀𝑥)))
12944, 127, 128mpbir2an 957 . 2 𝑀:ℕ–onto𝑅
130 df-f1o 5811 . 2 (𝑀:ℕ–1-1-onto𝑅 ↔ (𝑀:ℕ–1-1𝑅𝑀:ℕ–onto𝑅))
13163, 129, 130mpbir2an 957 1 𝑀:ℕ–1-1-onto𝑅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Fincfn 7841  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  ⌊cfl 12453  ↑cexp 12722   ∥ cdvds 14821  ℙcprime 15223   pCnt cpc 15379 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-fz 12198  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-gcd 15055  df-prm 15224  df-pc 15380 This theorem is referenced by:  1arith2  15470  sqff1o  24708
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