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Theorem 1arith 16899
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 prmex 16651 . . . . . 6 ℙ ∈ V
21mptex 7235 . . . . 5 (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ V
3 1arith.1 . . . . 5 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
42, 3fnmpti 6699 . . . 4 𝑀 Fn ℕ
531arithlem3 16897 . . . . . . 7 (𝑥 ∈ ℕ → (𝑀𝑥):ℙ⟶ℕ0)
6 nn0ex 12511 . . . . . . . 8 0 ∈ V
76, 1elmap 8890 . . . . . . 7 ((𝑀𝑥) ∈ (ℕ0m ℙ) ↔ (𝑀𝑥):ℙ⟶ℕ0)
85, 7sylibr 233 . . . . . 6 (𝑥 ∈ ℕ → (𝑀𝑥) ∈ (ℕ0m ℙ))
9 fzfi 13973 . . . . . . 7 (1...𝑥) ∈ Fin
10 ffn 6723 . . . . . . . . . 10 ((𝑀𝑥):ℙ⟶ℕ0 → (𝑀𝑥) Fn ℙ)
11 elpreima 7066 . . . . . . . . . 10 ((𝑀𝑥) Fn ℙ → (𝑞 ∈ ((𝑀𝑥) “ ℕ) ↔ (𝑞 ∈ ℙ ∧ ((𝑀𝑥)‘𝑞) ∈ ℕ)))
125, 10, 113syl 18 . . . . . . . . 9 (𝑥 ∈ ℕ → (𝑞 ∈ ((𝑀𝑥) “ ℕ) ↔ (𝑞 ∈ ℙ ∧ ((𝑀𝑥)‘𝑞) ∈ ℕ)))
1331arithlem2 16896 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
1413eleq1d 2810 . . . . . . . . . . 11 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) ∈ ℕ ↔ (𝑞 pCnt 𝑥) ∈ ℕ))
15 prmz 16649 . . . . . . . . . . . . 13 (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
16 id 22 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
17 dvdsle 16290 . . . . . . . . . . . . 13 ((𝑞 ∈ ℤ ∧ 𝑥 ∈ ℕ) → (𝑞𝑥𝑞𝑥))
1815, 16, 17syl2anr 595 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞𝑥𝑞𝑥))
19 pcelnn 16842 . . . . . . . . . . . . 13 ((𝑞 ∈ ℙ ∧ 𝑥 ∈ ℕ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞𝑥))
2019ancoms 457 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞𝑥))
21 prmnn 16648 . . . . . . . . . . . . . 14 (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
22 nnuz 12898 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
2321, 22eleqtrdi 2835 . . . . . . . . . . . . 13 (𝑞 ∈ ℙ → 𝑞 ∈ (ℤ‘1))
24 nnz 12612 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
25 elfz5 13528 . . . . . . . . . . . . 13 ((𝑞 ∈ (ℤ‘1) ∧ 𝑥 ∈ ℤ) → (𝑞 ∈ (1...𝑥) ↔ 𝑞𝑥))
2623, 24, 25syl2anr 595 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∈ (1...𝑥) ↔ 𝑞𝑥))
2718, 20, 263imtr4d 293 . . . . . . . . . . 11 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
2814, 27sylbid 239 . . . . . . . . . 10 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
2928expimpd 452 . . . . . . . . 9 (𝑥 ∈ ℕ → ((𝑞 ∈ ℙ ∧ ((𝑀𝑥)‘𝑞) ∈ ℕ) → 𝑞 ∈ (1...𝑥)))
3012, 29sylbid 239 . . . . . . . 8 (𝑥 ∈ ℕ → (𝑞 ∈ ((𝑀𝑥) “ ℕ) → 𝑞 ∈ (1...𝑥)))
3130ssrdv 3982 . . . . . . 7 (𝑥 ∈ ℕ → ((𝑀𝑥) “ ℕ) ⊆ (1...𝑥))
32 ssfi 9198 . . . . . . 7 (((1...𝑥) ∈ Fin ∧ ((𝑀𝑥) “ ℕ) ⊆ (1...𝑥)) → ((𝑀𝑥) “ ℕ) ∈ Fin)
339, 31, 32sylancr 585 . . . . . 6 (𝑥 ∈ ℕ → ((𝑀𝑥) “ ℕ) ∈ Fin)
34 cnveq 5876 . . . . . . . . 9 (𝑒 = (𝑀𝑥) → 𝑒 = (𝑀𝑥))
3534imaeq1d 6063 . . . . . . . 8 (𝑒 = (𝑀𝑥) → (𝑒 “ ℕ) = ((𝑀𝑥) “ ℕ))
3635eleq1d 2810 . . . . . . 7 (𝑒 = (𝑀𝑥) → ((𝑒 “ ℕ) ∈ Fin ↔ ((𝑀𝑥) “ ℕ) ∈ Fin))
37 1arith.2 . . . . . . 7 𝑅 = {𝑒 ∈ (ℕ0m ℙ) ∣ (𝑒 “ ℕ) ∈ Fin}
3836, 37elrab2 3682 . . . . . 6 ((𝑀𝑥) ∈ 𝑅 ↔ ((𝑀𝑥) ∈ (ℕ0m ℙ) ∧ ((𝑀𝑥) “ ℕ) ∈ Fin))
398, 33, 38sylanbrc 581 . . . . 5 (𝑥 ∈ ℕ → (𝑀𝑥) ∈ 𝑅)
4039rgen 3052 . . . 4 𝑥 ∈ ℕ (𝑀𝑥) ∈ 𝑅
41 ffnfv 7128 . . . 4 (𝑀:ℕ⟶𝑅 ↔ (𝑀 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝑀𝑥) ∈ 𝑅))
424, 40, 41mpbir2an 709 . . 3 𝑀:ℕ⟶𝑅
4313adantlr 713 . . . . . . . 8 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
4431arithlem2 16896 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
4544adantll 712 . . . . . . . 8 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
4643, 45eqeq12d 2741 . . . . . . 7 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞) ↔ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
4746ralbidva 3165 . . . . . 6 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
4831arithlem3 16897 . . . . . . 7 (𝑦 ∈ ℕ → (𝑀𝑦):ℙ⟶ℕ0)
49 ffn 6723 . . . . . . . 8 ((𝑀𝑦):ℙ⟶ℕ0 → (𝑀𝑦) Fn ℙ)
50 eqfnfv 7039 . . . . . . . 8 (((𝑀𝑥) Fn ℙ ∧ (𝑀𝑦) Fn ℙ) → ((𝑀𝑥) = (𝑀𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞)))
5110, 49, 50syl2an 594 . . . . . . 7 (((𝑀𝑥):ℙ⟶ℕ0 ∧ (𝑀𝑦):ℙ⟶ℕ0) → ((𝑀𝑥) = (𝑀𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞)))
525, 48, 51syl2an 594 . . . . . 6 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀𝑥) = (𝑀𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞)))
53 nnnn0 12512 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
54 nnnn0 12512 . . . . . . 7 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
55 pc11 16852 . . . . . . 7 ((𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑥 = 𝑦 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
5653, 54, 55syl2an 594 . . . . . 6 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 = 𝑦 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
5747, 52, 563bitr4d 310 . . . . 5 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀𝑥) = (𝑀𝑦) ↔ 𝑥 = 𝑦))
5857biimpd 228 . . . 4 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦))
5958rgen2 3187 . . 3 𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦)
60 dff13 7265 . . 3 (𝑀:ℕ–1-1𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦)))
6142, 59, 60mpbir2an 709 . 2 𝑀:ℕ–1-1𝑅
62 eqid 2725 . . . . . 6 (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓𝑔)), 1)) = (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓𝑔)), 1))
63 cnveq 5876 . . . . . . . . . . . 12 (𝑒 = 𝑓𝑒 = 𝑓)
6463imaeq1d 6063 . . . . . . . . . . 11 (𝑒 = 𝑓 → (𝑒 “ ℕ) = (𝑓 “ ℕ))
6564eleq1d 2810 . . . . . . . . . 10 (𝑒 = 𝑓 → ((𝑒 “ ℕ) ∈ Fin ↔ (𝑓 “ ℕ) ∈ Fin))
6665, 37elrab2 3682 . . . . . . . . 9 (𝑓𝑅 ↔ (𝑓 ∈ (ℕ0m ℙ) ∧ (𝑓 “ ℕ) ∈ Fin))
6766simplbi 496 . . . . . . . 8 (𝑓𝑅𝑓 ∈ (ℕ0m ℙ))
686, 1elmap 8890 . . . . . . . 8 (𝑓 ∈ (ℕ0m ℙ) ↔ 𝑓:ℙ⟶ℕ0)
6967, 68sylib 217 . . . . . . 7 (𝑓𝑅𝑓:ℙ⟶ℕ0)
7069ad2antrr 724 . . . . . 6 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → 𝑓:ℙ⟶ℕ0)
71 simplr 767 . . . . . . . . 9 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → 𝑦 ∈ ℝ)
72 0re 11248 . . . . . . . . 9 0 ∈ ℝ
73 ifcl 4575 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
7471, 72, 73sylancl 584 . . . . . . . 8 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
75 max1 13199 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
7672, 71, 75sylancr 585 . . . . . . . 8 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
77 flge0nn0 13821 . . . . . . . 8 ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → (⌊‘if(0 ≤ 𝑦, 𝑦, 0)) ∈ ℕ0)
7874, 76, 77syl2anc 582 . . . . . . 7 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → (⌊‘if(0 ≤ 𝑦, 𝑦, 0)) ∈ ℕ0)
79 nn0p1nn 12544 . . . . . . 7 ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) ∈ ℕ0 → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ∈ ℕ)
8078, 79syl 17 . . . . . 6 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ∈ ℕ)
8171adantr 479 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 ∈ ℝ)
8280adantr 479 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ∈ ℕ)
8382nnred 12260 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ∈ ℝ)
8415ssriv 3980 . . . . . . . . . . . 12 ℙ ⊆ ℤ
85 zssre 12598 . . . . . . . . . . . 12 ℤ ⊆ ℝ
8684, 85sstri 3986 . . . . . . . . . . 11 ℙ ⊆ ℝ
87 simprl 769 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑞 ∈ ℙ)
8886, 87sselid 3974 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑞 ∈ ℝ)
8974adantr 479 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
90 max2 13201 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
9172, 81, 90sylancr 585 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
92 flltp1 13801 . . . . . . . . . . . 12 (if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ → if(0 ≤ 𝑦, 𝑦, 0) < ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1))
9389, 92syl 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))
9481, 89, 83, 91, 93lelttrd 11404 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 < ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1))
95 simprr 771 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)
9681, 83, 88, 94, 95ltletrd 11406 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑦 < 𝑞)
9781, 88ltnled 11393 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑦 < 𝑞 ↔ ¬ 𝑞𝑦))
9896, 97mpbid 231 . . . . . . . 8 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ¬ 𝑞𝑦)
9987biantrurd 531 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ ↔ (𝑞 ∈ ℙ ∧ (𝑓𝑞) ∈ ℕ)))
10070adantr 479 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → 𝑓:ℙ⟶ℕ0)
101 ffn 6723 . . . . . . . . . . 11 (𝑓:ℙ⟶ℕ0𝑓 Fn ℙ)
102 elpreima 7066 . . . . . . . . . . 11 (𝑓 Fn ℙ → (𝑞 ∈ (𝑓 “ ℕ) ↔ (𝑞 ∈ ℙ ∧ (𝑓𝑞) ∈ ℕ)))
103100, 101, 1023syl 18 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑞 ∈ (𝑓 “ ℕ) ↔ (𝑞 ∈ ℙ ∧ (𝑓𝑞) ∈ ℕ)))
10499, 103bitr4d 281 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ ↔ 𝑞 ∈ (𝑓 “ ℕ)))
105 simplr 767 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦)
106 breq1 5152 . . . . . . . . . . 11 (𝑘 = 𝑞 → (𝑘𝑦𝑞𝑦))
107106rspccv 3603 . . . . . . . . . 10 (∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦 → (𝑞 ∈ (𝑓 “ ℕ) → 𝑞𝑦))
108105, 107syl 17 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑞 ∈ (𝑓 “ ℕ) → 𝑞𝑦))
109104, 108sylbid 239 . . . . . . . 8 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ → 𝑞𝑦))
11098, 109mtod 197 . . . . . . 7 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ¬ (𝑓𝑞) ∈ ℕ)
111100, 87ffvelcdmd 7094 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑓𝑞) ∈ ℕ0)
112 elnn0 12507 . . . . . . . . 9 ((𝑓𝑞) ∈ ℕ0 ↔ ((𝑓𝑞) ∈ ℕ ∨ (𝑓𝑞) = 0))
113111, 112sylib 217 . . . . . . . 8 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ ∨ (𝑓𝑞) = 0))
114113ord 862 . . . . . . 7 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (¬ (𝑓𝑞) ∈ ℕ → (𝑓𝑞) = 0))
115110, 114mpd 15 . . . . . 6 ((((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ ((⌊‘if(0 ≤ 𝑦, 𝑦, 0)) + 1) ≤ 𝑞)) → (𝑓𝑞) = 0)
1163, 62, 70, 80, 1151arithlem4 16898 . . . . 5 (((𝑓𝑅𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → ∃𝑥 ∈ ℕ 𝑓 = (𝑀𝑥))
117 cnvimass 6086 . . . . . . 7 (𝑓 “ ℕ) ⊆ dom 𝑓
11869fdmd 6733 . . . . . . . 8 (𝑓𝑅 → dom 𝑓 = ℙ)
119118, 86eqsstrdi 4031 . . . . . . 7 (𝑓𝑅 → dom 𝑓 ⊆ ℝ)
120117, 119sstrid 3988 . . . . . 6 (𝑓𝑅 → (𝑓 “ ℕ) ⊆ ℝ)
12166simprbi 495 . . . . . 6 (𝑓𝑅 → (𝑓 “ ℕ) ∈ Fin)
122 fimaxre2 12192 . . . . . 6 (((𝑓 “ ℕ) ⊆ ℝ ∧ (𝑓 “ ℕ) ∈ Fin) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦)
123120, 121, 122syl2anc 582 . . . . 5 (𝑓𝑅 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦)
124116, 123r19.29a 3151 . . . 4 (𝑓𝑅 → ∃𝑥 ∈ ℕ 𝑓 = (𝑀𝑥))
125124rgen 3052 . . 3 𝑓𝑅𝑥 ∈ ℕ 𝑓 = (𝑀𝑥)
126 dffo3 7111 . . 3 (𝑀:ℕ–onto𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑓𝑅𝑥 ∈ ℕ 𝑓 = (𝑀𝑥)))
12742, 125, 126mpbir2an 709 . 2 𝑀:ℕ–onto𝑅
128 df-f1o 6556 . 2 (𝑀:ℕ–1-1-onto𝑅 ↔ (𝑀:ℕ–1-1𝑅𝑀:ℕ–onto𝑅))
12961, 127, 128mpbir2an 709 1 𝑀:ℕ–1-1-onto𝑅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  wral 3050  wrex 3059  {crab 3418  wss 3944  ifcif 4530   class class class wbr 5149  cmpt 5232  ccnv 5677  dom cdm 5678  cima 5681   Fn wfn 6544  wf 6545  1-1wf1 6546  ontowfo 6547  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  m cmap 8845  Fincfn 8964  cr 11139  0cc0 11140  1c1 11141   + caddc 11143   < clt 11280  cle 11281  cn 12245  0cn0 12505  cz 12591  cuz 12855  ...cfz 13519  cfl 13791  cexp 14062  cdvds 16234  cprime 16645   pCnt cpc 16808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-q 12966  df-rp 13010  df-fz 13520  df-fl 13793  df-mod 13871  df-seq 14003  df-exp 14063  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-dvds 16235  df-gcd 16473  df-prm 16646  df-pc 16809
This theorem is referenced by:  1arith2  16900  sqff1o  27159
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