ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  1arith GIF version

Theorem 1arith 12348
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 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prmex 12096 . . . . . 6 ℙ ∈ V
21mptex 5738 . . . . 5 (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ V
3 1arith.1 . . . . 5 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
42, 3fnmpti 5340 . . . 4 𝑀 Fn ℕ
531arithlem3 12346 . . . . . . 7 (𝑥 ∈ ℕ → (𝑀𝑥):ℙ⟶ℕ0)
6 nn0ex 9171 . . . . . . . 8 0 ∈ V
76, 1elmap 6671 . . . . . . 7 ((𝑀𝑥) ∈ (ℕ0𝑚 ℙ) ↔ (𝑀𝑥):ℙ⟶ℕ0)
85, 7sylibr 134 . . . . . 6 (𝑥 ∈ ℕ → (𝑀𝑥) ∈ (ℕ0𝑚 ℙ))
9 1zzd 9269 . . . . . . . 8 (𝑥 ∈ ℕ → 1 ∈ ℤ)
10 nnz 9261 . . . . . . . 8 (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
119, 10fzfigd 10417 . . . . . . 7 (𝑥 ∈ ℕ → (1...𝑥) ∈ Fin)
12 ffn 5361 . . . . . . . . . 10 ((𝑀𝑥):ℙ⟶ℕ0 → (𝑀𝑥) Fn ℙ)
13 elpreima 5631 . . . . . . . . . 10 ((𝑀𝑥) Fn ℙ → (𝑞 ∈ ((𝑀𝑥) “ ℕ) ↔ (𝑞 ∈ ℙ ∧ ((𝑀𝑥)‘𝑞) ∈ ℕ)))
145, 12, 133syl 17 . . . . . . . . 9 (𝑥 ∈ ℕ → (𝑞 ∈ ((𝑀𝑥) “ ℕ) ↔ (𝑞 ∈ ℙ ∧ ((𝑀𝑥)‘𝑞) ∈ ℕ)))
1531arithlem2 12345 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
1615eleq1d 2246 . . . . . . . . . . 11 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) ∈ ℕ ↔ (𝑞 pCnt 𝑥) ∈ ℕ))
17 prmz 12094 . . . . . . . . . . . . 13 (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
18 id 19 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
19 dvdsle 11833 . . . . . . . . . . . . 13 ((𝑞 ∈ ℤ ∧ 𝑥 ∈ ℕ) → (𝑞𝑥𝑞𝑥))
2017, 18, 19syl2anr 290 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞𝑥𝑞𝑥))
21 pcelnn 12303 . . . . . . . . . . . . 13 ((𝑞 ∈ ℙ ∧ 𝑥 ∈ ℕ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞𝑥))
2221ancoms 268 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞𝑥))
23 prmnn 12093 . . . . . . . . . . . . . 14 (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
24 nnuz 9552 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
2523, 24eleqtrdi 2270 . . . . . . . . . . . . 13 (𝑞 ∈ ℙ → 𝑞 ∈ (ℤ‘1))
26 elfz5 10003 . . . . . . . . . . . . 13 ((𝑞 ∈ (ℤ‘1) ∧ 𝑥 ∈ ℤ) → (𝑞 ∈ (1...𝑥) ↔ 𝑞𝑥))
2725, 10, 26syl2anr 290 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∈ (1...𝑥) ↔ 𝑞𝑥))
2820, 22, 273imtr4d 203 . . . . . . . . . . 11 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
2916, 28sylbid 150 . . . . . . . . . 10 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) ∈ ℕ → 𝑞 ∈ (1...𝑥)))
3029expimpd 363 . . . . . . . . 9 (𝑥 ∈ ℕ → ((𝑞 ∈ ℙ ∧ ((𝑀𝑥)‘𝑞) ∈ ℕ) → 𝑞 ∈ (1...𝑥)))
3114, 30sylbid 150 . . . . . . . 8 (𝑥 ∈ ℕ → (𝑞 ∈ ((𝑀𝑥) “ ℕ) → 𝑞 ∈ (1...𝑥)))
3231ssrdv 3161 . . . . . . 7 (𝑥 ∈ ℕ → ((𝑀𝑥) “ ℕ) ⊆ (1...𝑥))
33 elfznn 10040 . . . . . . . . . . . . . 14 (𝑗 ∈ (1...𝑥) → 𝑗 ∈ ℕ)
3433adantl 277 . . . . . . . . . . . . 13 ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → 𝑗 ∈ ℕ)
35 prmdc 12113 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → DECID 𝑗 ∈ ℙ)
3634, 35syl 14 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → DECID 𝑗 ∈ ℙ)
3736adantr 276 . . . . . . . . . . 11 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → DECID 𝑗 ∈ ℙ)
385ad2antrr 488 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → (𝑀𝑥):ℙ⟶ℕ0)
39 simpr 110 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → 𝑗 ∈ ℙ)
4038, 39ffvelcdmd 5648 . . . . . . . . . . . . 13 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → ((𝑀𝑥)‘𝑗) ∈ ℕ0)
4140nn0zd 9362 . . . . . . . . . . . 12 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → ((𝑀𝑥)‘𝑗) ∈ ℤ)
42 elnndc 9601 . . . . . . . . . . . 12 (((𝑀𝑥)‘𝑗) ∈ ℤ → DECID ((𝑀𝑥)‘𝑗) ∈ ℕ)
4341, 42syl 14 . . . . . . . . . . 11 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → DECID ((𝑀𝑥)‘𝑗) ∈ ℕ)
44 dcan2 934 . . . . . . . . . . 11 (DECID 𝑗 ∈ ℙ → (DECID ((𝑀𝑥)‘𝑗) ∈ ℕ → DECID (𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ)))
4537, 43, 44sylc 62 . . . . . . . . . 10 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → DECID (𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ))
46 simpr 110 . . . . . . . . . . . . 13 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ¬ 𝑗 ∈ ℙ)
4746intnanrd 932 . . . . . . . . . . . 12 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ¬ (𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ))
4847olcd 734 . . . . . . . . . . 11 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → ((𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ) ∨ ¬ (𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ)))
49 df-dc 835 . . . . . . . . . . 11 (DECID (𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ) ↔ ((𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ) ∨ ¬ (𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ)))
5048, 49sylibr 134 . . . . . . . . . 10 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ ¬ 𝑗 ∈ ℙ) → DECID (𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ))
51 exmiddc 836 . . . . . . . . . . 11 (DECID 𝑗 ∈ ℙ → (𝑗 ∈ ℙ ∨ ¬ 𝑗 ∈ ℙ))
5236, 51syl 14 . . . . . . . . . 10 ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → (𝑗 ∈ ℙ ∨ ¬ 𝑗 ∈ ℙ))
5345, 50, 52mpjaodan 798 . . . . . . . . 9 ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → DECID (𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ))
54 elpreima 5631 . . . . . . . . . . . 12 ((𝑀𝑥) Fn ℙ → (𝑗 ∈ ((𝑀𝑥) “ ℕ) ↔ (𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ)))
555, 12, 543syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℕ → (𝑗 ∈ ((𝑀𝑥) “ ℕ) ↔ (𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ)))
5655dcbid 838 . . . . . . . . . 10 (𝑥 ∈ ℕ → (DECID 𝑗 ∈ ((𝑀𝑥) “ ℕ) ↔ DECID (𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ)))
5756adantr 276 . . . . . . . . 9 ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → (DECID 𝑗 ∈ ((𝑀𝑥) “ ℕ) ↔ DECID (𝑗 ∈ ℙ ∧ ((𝑀𝑥)‘𝑗) ∈ ℕ)))
5853, 57mpbird 167 . . . . . . . 8 ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → DECID 𝑗 ∈ ((𝑀𝑥) “ ℕ))
5958ralrimiva 2550 . . . . . . 7 (𝑥 ∈ ℕ → ∀𝑗 ∈ (1...𝑥)DECID 𝑗 ∈ ((𝑀𝑥) “ ℕ))
60 ssfidc 6928 . . . . . . 7 (((1...𝑥) ∈ Fin ∧ ((𝑀𝑥) “ ℕ) ⊆ (1...𝑥) ∧ ∀𝑗 ∈ (1...𝑥)DECID 𝑗 ∈ ((𝑀𝑥) “ ℕ)) → ((𝑀𝑥) “ ℕ) ∈ Fin)
6111, 32, 59, 60syl3anc 1238 . . . . . 6 (𝑥 ∈ ℕ → ((𝑀𝑥) “ ℕ) ∈ Fin)
62 cnveq 4797 . . . . . . . . 9 (𝑒 = (𝑀𝑥) → 𝑒 = (𝑀𝑥))
6362imaeq1d 4965 . . . . . . . 8 (𝑒 = (𝑀𝑥) → (𝑒 “ ℕ) = ((𝑀𝑥) “ ℕ))
6463eleq1d 2246 . . . . . . 7 (𝑒 = (𝑀𝑥) → ((𝑒 “ ℕ) ∈ Fin ↔ ((𝑀𝑥) “ ℕ) ∈ Fin))
65 1arith.2 . . . . . . 7 𝑅 = {𝑒 ∈ (ℕ0𝑚 ℙ) ∣ (𝑒 “ ℕ) ∈ Fin}
6664, 65elrab2 2896 . . . . . 6 ((𝑀𝑥) ∈ 𝑅 ↔ ((𝑀𝑥) ∈ (ℕ0𝑚 ℙ) ∧ ((𝑀𝑥) “ ℕ) ∈ Fin))
678, 61, 66sylanbrc 417 . . . . 5 (𝑥 ∈ ℕ → (𝑀𝑥) ∈ 𝑅)
6867rgen 2530 . . . 4 𝑥 ∈ ℕ (𝑀𝑥) ∈ 𝑅
69 ffnfv 5670 . . . 4 (𝑀:ℕ⟶𝑅 ↔ (𝑀 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝑀𝑥) ∈ 𝑅))
704, 68, 69mpbir2an 942 . . 3 𝑀:ℕ⟶𝑅
7115adantlr 477 . . . . . . . 8 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
7231arithlem2 12345 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
7372adantll 476 . . . . . . . 8 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → ((𝑀𝑦)‘𝑞) = (𝑞 pCnt 𝑦))
7471, 73eqeq12d 2192 . . . . . . 7 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞) ↔ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
7574ralbidva 2473 . . . . . 6 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
7631arithlem3 12346 . . . . . . 7 (𝑦 ∈ ℕ → (𝑀𝑦):ℙ⟶ℕ0)
77 ffn 5361 . . . . . . . 8 ((𝑀𝑦):ℙ⟶ℕ0 → (𝑀𝑦) Fn ℙ)
78 eqfnfv 5609 . . . . . . . 8 (((𝑀𝑥) Fn ℙ ∧ (𝑀𝑦) Fn ℙ) → ((𝑀𝑥) = (𝑀𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞)))
7912, 77, 78syl2an 289 . . . . . . 7 (((𝑀𝑥):ℙ⟶ℕ0 ∧ (𝑀𝑦):ℙ⟶ℕ0) → ((𝑀𝑥) = (𝑀𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞)))
805, 76, 79syl2an 289 . . . . . 6 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀𝑥) = (𝑀𝑦) ↔ ∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞)))
81 nnnn0 9172 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
82 nnnn0 9172 . . . . . . 7 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
83 pc11 12313 . . . . . . 7 ((𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑥 = 𝑦 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
8481, 82, 83syl2an 289 . . . . . 6 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 = 𝑦 ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
8575, 80, 843bitr4d 220 . . . . 5 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀𝑥) = (𝑀𝑦) ↔ 𝑥 = 𝑦))
8685biimpd 144 . . . 4 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦))
8786rgen2 2563 . . 3 𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦)
88 dff13 5763 . . 3 (𝑀:ℕ–1-1𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦)))
8970, 87, 88mpbir2an 942 . 2 𝑀:ℕ–1-1𝑅
90 eqid 2177 . . . . . 6 (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓𝑔)), 1)) = (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓𝑔)), 1))
91 cnveq 4797 . . . . . . . . . . . 12 (𝑒 = 𝑓𝑒 = 𝑓)
9291imaeq1d 4965 . . . . . . . . . . 11 (𝑒 = 𝑓 → (𝑒 “ ℕ) = (𝑓 “ ℕ))
9392eleq1d 2246 . . . . . . . . . 10 (𝑒 = 𝑓 → ((𝑒 “ ℕ) ∈ Fin ↔ (𝑓 “ ℕ) ∈ Fin))
9493, 65elrab2 2896 . . . . . . . . 9 (𝑓𝑅 ↔ (𝑓 ∈ (ℕ0𝑚 ℙ) ∧ (𝑓 “ ℕ) ∈ Fin))
9594simplbi 274 . . . . . . . 8 (𝑓𝑅𝑓 ∈ (ℕ0𝑚 ℙ))
966, 1elmap 6671 . . . . . . . 8 (𝑓 ∈ (ℕ0𝑚 ℙ) ↔ 𝑓:ℙ⟶ℕ0)
9795, 96sylib 122 . . . . . . 7 (𝑓𝑅𝑓:ℙ⟶ℕ0)
9897ad2antrr 488 . . . . . 6 (((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → 𝑓:ℙ⟶ℕ0)
99 simplr 528 . . . . . . 7 (((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → 𝑦 ∈ ℕ)
10099peano2nnd 8923 . . . . . 6 (((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → (𝑦 + 1) ∈ ℕ)
10199adantr 276 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 ∈ ℕ)
102101nnred 8921 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 ∈ ℝ)
103 peano2re 8083 . . . . . . . . . . 11 (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ)
104102, 103syl 14 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑦 + 1) ∈ ℝ)
10523ad2antrl 490 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℕ)
106105nnred 8921 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℝ)
107102ltp1d 8876 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 < (𝑦 + 1))
108 simprr 531 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑦 + 1) ≤ 𝑞)
109102, 104, 106, 107, 108ltletrd 8370 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 < 𝑞)
110101nnzd 9363 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 ∈ ℤ)
11117ad2antrl 490 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℤ)
112 zltnle 9288 . . . . . . . . . 10 ((𝑦 ∈ ℤ ∧ 𝑞 ∈ ℤ) → (𝑦 < 𝑞 ↔ ¬ 𝑞𝑦))
113110, 111, 112syl2anc 411 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑦 < 𝑞 ↔ ¬ 𝑞𝑦))
114109, 113mpbid 147 . . . . . . . 8 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ¬ 𝑞𝑦)
115 simprl 529 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℙ)
116115biantrurd 305 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ ↔ (𝑞 ∈ ℙ ∧ (𝑓𝑞) ∈ ℕ)))
11797ad3antrrr 492 . . . . . . . . . . 11 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑓:ℙ⟶ℕ0)
118 ffn 5361 . . . . . . . . . . 11 (𝑓:ℙ⟶ℕ0𝑓 Fn ℙ)
119 elpreima 5631 . . . . . . . . . . 11 (𝑓 Fn ℙ → (𝑞 ∈ (𝑓 “ ℕ) ↔ (𝑞 ∈ ℙ ∧ (𝑓𝑞) ∈ ℕ)))
120117, 118, 1193syl 17 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑞 ∈ (𝑓 “ ℕ) ↔ (𝑞 ∈ ℙ ∧ (𝑓𝑞) ∈ ℕ)))
121116, 120bitr4d 191 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ ↔ 𝑞 ∈ (𝑓 “ ℕ)))
122 breq1 4003 . . . . . . . . . . 11 (𝑘 = 𝑞 → (𝑘𝑦𝑞𝑦))
123122rspccv 2838 . . . . . . . . . 10 (∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦 → (𝑞 ∈ (𝑓 “ ℕ) → 𝑞𝑦))
124123ad2antlr 489 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑞 ∈ (𝑓 “ ℕ) → 𝑞𝑦))
125121, 124sylbid 150 . . . . . . . 8 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ → 𝑞𝑦))
126114, 125mtod 663 . . . . . . 7 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ¬ (𝑓𝑞) ∈ ℕ)
127117, 115ffvelcdmd 5648 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑓𝑞) ∈ ℕ0)
128 elnn0 9167 . . . . . . . . 9 ((𝑓𝑞) ∈ ℕ0 ↔ ((𝑓𝑞) ∈ ℕ ∨ (𝑓𝑞) = 0))
129127, 128sylib 122 . . . . . . . 8 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → ((𝑓𝑞) ∈ ℕ ∨ (𝑓𝑞) = 0))
130129ord 724 . . . . . . 7 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (¬ (𝑓𝑞) ∈ ℕ → (𝑓𝑞) = 0))
131126, 130mpd 13 . . . . . 6 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑓𝑞) = 0)
1323, 90, 98, 100, 1311arithlem4 12347 . . . . 5 (((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → ∃𝑥 ∈ ℕ 𝑓 = (𝑀𝑥))
133 cnvimass 4987 . . . . . . 7 (𝑓 “ ℕ) ⊆ dom 𝑓
13497fdmd 5368 . . . . . . . 8 (𝑓𝑅 → dom 𝑓 = ℙ)
135 prmssnn 12095 . . . . . . . 8 ℙ ⊆ ℕ
136134, 135eqsstrdi 3207 . . . . . . 7 (𝑓𝑅 → dom 𝑓 ⊆ ℕ)
137133, 136sstrid 3166 . . . . . 6 (𝑓𝑅 → (𝑓 “ ℕ) ⊆ ℕ)
13894simprbi 275 . . . . . 6 (𝑓𝑅 → (𝑓 “ ℕ) ∈ Fin)
139 fiubnn 10794 . . . . . 6 (((𝑓 “ ℕ) ⊆ ℕ ∧ (𝑓 “ ℕ) ∈ Fin) → ∃𝑦 ∈ ℕ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦)
140137, 138, 139syl2anc 411 . . . . 5 (𝑓𝑅 → ∃𝑦 ∈ ℕ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦)
141132, 140r19.29a 2620 . . . 4 (𝑓𝑅 → ∃𝑥 ∈ ℕ 𝑓 = (𝑀𝑥))
142141rgen 2530 . . 3 𝑓𝑅𝑥 ∈ ℕ 𝑓 = (𝑀𝑥)
143 dffo3 5659 . . 3 (𝑀:ℕ–onto𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑓𝑅𝑥 ∈ ℕ 𝑓 = (𝑀𝑥)))
14470, 142, 143mpbir2an 942 . 2 𝑀:ℕ–onto𝑅
145 df-f1o 5219 . 2 (𝑀:ℕ–1-1-onto𝑅 ↔ (𝑀:ℕ–1-1𝑅𝑀:ℕ–onto𝑅))
14689, 144, 145mpbir2an 942 1 𝑀:ℕ–1-1-onto𝑅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834   = wceq 1353  wcel 2148  wral 2455  wrex 2456  {crab 2459  wss 3129  ifcif 3534   class class class wbr 4000  cmpt 4061  ccnv 4622  dom cdm 4623  cima 4626   Fn wfn 5207  wf 5208  1-1wf1 5209  ontowfo 5210  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  𝑚 cmap 6642  Fincfn 6734  cr 7801  0cc0 7802  1c1 7803   + caddc 7805   < clt 7982  cle 7983  cn 8908  0cn0 9165  cz 9242  cuz 9517  ...cfz 9995  cexp 10505  cdvds 11778  cprime 12090   pCnt cpc 12267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-1o 6411  df-2o 6412  df-er 6529  df-map 6644  df-en 6735  df-fin 6737  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-xnn0 9229  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927  df-prm 12091  df-pc 12268
This theorem is referenced by:  1arith2  12349
  Copyright terms: Public domain W3C validator