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Theorem 1arith 15678
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 11424 . . . . . . 7 ℤ ∈ V
2 prmz 15436 . . . . . . . 8 (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
32ssriv 3640 . . . . . . 7 ℙ ⊆ ℤ
41, 3ssexi 4836 . . . . . 6 ℙ ∈ V
54mptex 6527 . . . . 5 (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ V
6 1arith.1 . . . . 5 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
75, 6fnmpti 6060 . . . 4 𝑀 Fn ℕ
861arithlem3 15676 . . . . . . 7 (𝑥 ∈ ℕ → (𝑀𝑥):ℙ⟶ℕ0)
9 nn0ex 11336 . . . . . . . 8 0 ∈ V
109, 4elmap 7928 . . . . . . 7 ((𝑀𝑥) ∈ (ℕ0𝑚 ℙ) ↔ (𝑀𝑥):ℙ⟶ℕ0)
118, 10sylibr 224 . . . . . 6 (𝑥 ∈ ℕ → (𝑀𝑥) ∈ (ℕ0𝑚 ℙ))
12 fzfi 12811 . . . . . . 7 (1...𝑥) ∈ Fin
13 ffn 6083 . . . . . . . . . 10 ((𝑀𝑥):ℙ⟶ℕ0 → (𝑀𝑥) Fn ℙ)
14 elpreima 6377 . . . . . . . . . 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 15675 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
1716eleq1d 2715 . . . . . . . . . . 11 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) ∈ ℕ ↔ (𝑞 pCnt 𝑥) ∈ ℕ))
18 id 22 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
19 dvdsle 15079 . . . . . . . . . . . . 13 ((𝑞 ∈ ℤ ∧ 𝑥 ∈ ℕ) → (𝑞𝑥𝑞𝑥))
202, 18, 19syl2anr 494 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞𝑥𝑞𝑥))
21 pcelnn 15621 . . . . . . . . . . . . 13 ((𝑞 ∈ ℙ ∧ 𝑥 ∈ ℕ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞𝑥))
2221ancoms 468 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞𝑥))
23 prmnn 15435 . . . . . . . . . . . . . 14 (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
24 nnuz 11761 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
2523, 24syl6eleq 2740 . . . . . . . . . . . . 13 (𝑞 ∈ ℙ → 𝑞 ∈ (ℤ‘1))
26 nnz 11437 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
27 elfz5 12372 . . . . . . . . . . . . 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 283 . . . . . . . . . . 11 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
3017, 29sylbid 230 . . . . . . . . . 10 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
3130expimpd 628 . . . . . . . . 9 (𝑥 ∈ ℕ → ((𝑞 ∈ ℙ ∧ ((𝑀𝑥)‘𝑞) ∈ ℕ) → 𝑞 ∈ (1...𝑥)))
3215, 31sylbid 230 . . . . . . . 8 (𝑥 ∈ ℕ → (𝑞 ∈ ((𝑀𝑥) “ ℕ) → 𝑞 ∈ (1...𝑥)))
3332ssrdv 3642 . . . . . . 7 (𝑥 ∈ ℕ → ((𝑀𝑥) “ ℕ) ⊆ (1...𝑥))
34 ssfi 8221 . . . . . . 7 (((1...𝑥) ∈ Fin ∧ ((𝑀𝑥) “ ℕ) ⊆ (1...𝑥)) → ((𝑀𝑥) “ ℕ) ∈ Fin)
3512, 33, 34sylancr 696 . . . . . 6 (𝑥 ∈ ℕ → ((𝑀𝑥) “ ℕ) ∈ Fin)
36 cnveq 5328 . . . . . . . . 9 (𝑒 = (𝑀𝑥) → 𝑒 = (𝑀𝑥))
3736imaeq1d 5500 . . . . . . . 8 (𝑒 = (𝑀𝑥) → (𝑒 “ ℕ) = ((𝑀𝑥) “ ℕ))
3837eleq1d 2715 . . . . . . 7 (𝑒 = (𝑀𝑥) → ((𝑒 “ ℕ) ∈ Fin ↔ ((𝑀𝑥) “ ℕ) ∈ Fin))
39 1arith.2 . . . . . . 7 𝑅 = {𝑒 ∈ (ℕ0𝑚 ℙ) ∣ (𝑒 “ ℕ) ∈ Fin}
4038, 39elrab2 3399 . . . . . 6 ((𝑀𝑥) ∈ 𝑅 ↔ ((𝑀𝑥) ∈ (ℕ0𝑚 ℙ) ∧ ((𝑀𝑥) “ ℕ) ∈ Fin))
4111, 35, 40sylanbrc 699 . . . . 5 (𝑥 ∈ ℕ → (𝑀𝑥) ∈ 𝑅)
4241rgen 2951 . . . 4 𝑥 ∈ ℕ (𝑀𝑥) ∈ 𝑅
43 ffnfv 6428 . . . 4 (𝑀:ℕ⟶𝑅 ↔ (𝑀 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝑀𝑥) ∈ 𝑅))
447, 42, 43mpbir2an 975 . . 3 𝑀:ℕ⟶𝑅
4516adantlr 751 . . . . . . . 8 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
4661arithlem2 15675 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
4746adantll 750 . . . . . . . 8 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
4845, 47eqeq12d 2666 . . . . . . 7 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞) ↔ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
4948ralbidva 3014 . . . . . 6 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
5061arithlem3 15676 . . . . . . 7 (𝑦 ∈ ℕ → (𝑀𝑦):ℙ⟶ℕ0)
51 ffn 6083 . . . . . . . 8 ((𝑀𝑦):ℙ⟶ℕ0 → (𝑀𝑦) Fn ℙ)
52 eqfnfv 6351 . . . . . . . 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 11337 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
56 nnnn0 11337 . . . . . . 7 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
57 pc11 15631 . . . . . . 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 300 . . . . 5 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀𝑥) = (𝑀𝑦) ↔ 𝑥 = 𝑦))
6059biimpd 219 . . . 4 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦))
6160rgen2a 3006 . . 3 𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦)
62 dff13 6552 . . 3 (𝑀:ℕ–1-1𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦)))
6344, 61, 62mpbir2an 975 . 2 𝑀:ℕ–1-1𝑅
64 eqid 2651 . . . . . 6 (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓𝑔)), 1)) = (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓𝑔)), 1))
65 cnveq 5328 . . . . . . . . . . . 12 (𝑒 = 𝑓𝑒 = 𝑓)
6665imaeq1d 5500 . . . . . . . . . . 11 (𝑒 = 𝑓 → (𝑒 “ ℕ) = (𝑓 “ ℕ))
6766eleq1d 2715 . . . . . . . . . 10 (𝑒 = 𝑓 → ((𝑒 “ ℕ) ∈ Fin ↔ (𝑓 “ ℕ) ∈ Fin))
6867, 39elrab2 3399 . . . . . . . . 9 (𝑓𝑅 ↔ (𝑓 ∈ (ℕ0𝑚 ℙ) ∧ (𝑓 “ ℕ) ∈ Fin))
6968simplbi 475 . . . . . . . 8 (𝑓𝑅𝑓 ∈ (ℕ0𝑚 ℙ))
709, 4elmap 7928 . . . . . . . 8 (𝑓 ∈ (ℕ0𝑚 ℙ) ↔ 𝑓:ℙ⟶ℕ0)
7169, 70sylib 208 . . . . . . 7 (𝑓𝑅𝑓:ℙ⟶ℕ0)
7271ad2antrr 762 . . . . . 6 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → 𝑓:ℙ⟶ℕ0)
73 simplr 807 . . . . . . . . 9 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → 𝑦 ∈ ℝ)
74 0re 10078 . . . . . . . . 9 0 ∈ ℝ
75 ifcl 4163 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
7673, 74, 75sylancl 695 . . . . . . . 8 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
77 max1 12054 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
7874, 73, 77sylancr 696 . . . . . . . 8 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
79 flge0nn0 12661 . . . . . . . 8 ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → (⌊‘if(0 ≤ 𝑦, 𝑦, 0)) ∈ ℕ0)
8076, 78, 79syl2anc 694 . . . . . . 7 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → (⌊‘if(0 ≤ 𝑦, 𝑦, 0)) ∈ ℕ0)
81 nn0p1nn 11370 . . . . . . 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 11073 . . . . . . . . . 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 11422 . . . . . . . . . . . 12 ℤ ⊆ ℝ
873, 86sstri 3645 . . . . . . . . . . 11 ℙ ⊆ ℝ
88 simprl 809 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑞 ∈ ℙ)
8987, 88sseldi 3634 . . . . . . . . . 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 12056 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
9274, 83, 91sylancr 696 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
93 flltp1 12641 . . . . . . . . . . . 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 10233 . . . . . . . . . 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 811 . . . . . . . . . 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 10235 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 < 𝑞)
9883, 89ltnled 10222 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑦 < 𝑞 ↔ ¬ 𝑞𝑦))
9997, 98mpbid 222 . . . . . . . 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 6083 . . . . . . . . . . 11 (𝑓:ℙ⟶ℕ0𝑓 Fn ℙ)
103 elpreima 6377 . . . . . . . . . . 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 271 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ ↔ 𝑞 ∈ (𝑓 “ ℕ)))
106 simplr 807 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦)
107 breq1 4688 . . . . . . . . . . 11 (𝑘 = 𝑞 → (𝑘𝑦𝑞𝑦))
108107rspccv 3337 . . . . . . . . . 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 230 . . . . . . . 8 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ → 𝑞𝑦))
11199, 110mtod 189 . . . . . . 7 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ¬ (𝑓𝑞) ∈ ℕ)
112101, 88ffvelrnd 6400 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑓𝑞) ∈ ℕ0)
113 elnn0 11332 . . . . . . . . 9 ((𝑓𝑞) ∈ ℕ0 ↔ ((𝑓𝑞) ∈ ℕ ∨ (𝑓𝑞) = 0))
114112, 113sylib 208 . . . . . . . 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 15677 . . . . 5 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → ∃𝑥 ∈ ℕ 𝑓 = (𝑀𝑥))
118 cnvimass 5520 . . . . . . 7 (𝑓 “ ℕ) ⊆ dom 𝑓
119 fdm 6089 . . . . . . . . 9 (𝑓:ℙ⟶ℕ0 → dom 𝑓 = ℙ)
12071, 119syl 17 . . . . . . . 8 (𝑓𝑅 → dom 𝑓 = ℙ)
121120, 87syl6eqss 3688 . . . . . . 7 (𝑓𝑅 → dom 𝑓 ⊆ ℝ)
122118, 121syl5ss 3647 . . . . . 6 (𝑓𝑅 → (𝑓 “ ℕ) ⊆ ℝ)
12368simprbi 479 . . . . . 6 (𝑓𝑅 → (𝑓 “ ℕ) ∈ Fin)
124 fimaxre2 11007 . . . . . 6 (((𝑓 “ ℕ) ⊆ ℝ ∧ (𝑓 “ ℕ) ∈ Fin) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦)
125122, 123, 124syl2anc 694 . . . . 5 (𝑓𝑅 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦)
126117, 125r19.29a 3107 . . . 4 (𝑓𝑅 → ∃𝑥 ∈ ℕ 𝑓 = (𝑀𝑥))
127126rgen 2951 . . 3 𝑓𝑅𝑥 ∈ ℕ 𝑓 = (𝑀𝑥)
128 dffo3 6414 . . 3 (𝑀:ℕ–onto𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑓𝑅𝑥 ∈ ℕ 𝑓 = (𝑀𝑥)))
12944, 127, 128mpbir2an 975 . 2 𝑀:ℕ–onto𝑅
130 df-f1o 5933 . 2 (𝑀:ℕ–1-1-onto𝑅 ↔ (𝑀:ℕ–1-1𝑅𝑀:ℕ–onto𝑅))
13163, 129, 130mpbir2an 975 1 𝑀:ℕ–1-1-onto𝑅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  wss 3607  ifcif 4119   class class class wbr 4685  cmpt 4762  ccnv 5142  dom cdm 5143  cima 5146   Fn wfn 5921  wf 5922  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Fincfn 7997  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cn 11058  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  cfl 12631  cexp 12900  cdvds 15027  cprime 15432   pCnt cpc 15588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-fz 12365  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-dvds 15028  df-gcd 15264  df-prm 15433  df-pc 15589
This theorem is referenced by:  1arith2  15679  sqff1o  24953
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