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Theorem 1arith 12330
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 12078 . . . . . 6 ℙ ∈ V
21mptex 5734 . . . . 5 (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) ∈ V
3 1arith.1 . . . . 5 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))
42, 3fnmpti 5336 . . . 4 𝑀 Fn ℕ
531arithlem3 12328 . . . . . . 7 (𝑥 ∈ ℕ → (𝑀𝑥):ℙ⟶ℕ0)
6 nn0ex 9153 . . . . . . . 8 0 ∈ V
76, 1elmap 6667 . . . . . . 7 ((𝑀𝑥) ∈ (ℕ0𝑚 ℙ) ↔ (𝑀𝑥):ℙ⟶ℕ0)
85, 7sylibr 134 . . . . . 6 (𝑥 ∈ ℕ → (𝑀𝑥) ∈ (ℕ0𝑚 ℙ))
9 1zzd 9251 . . . . . . . 8 (𝑥 ∈ ℕ → 1 ∈ ℤ)
10 nnz 9243 . . . . . . . 8 (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
119, 10fzfigd 10399 . . . . . . 7 (𝑥 ∈ ℕ → (1...𝑥) ∈ Fin)
12 ffn 5357 . . . . . . . . . 10 ((𝑀𝑥):ℙ⟶ℕ0 → (𝑀𝑥) Fn ℙ)
13 elpreima 5627 . . . . . . . . . 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 12327 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑀𝑥)‘𝑞) = (𝑞 pCnt 𝑥))
1615eleq1d 2244 . . . . . . . . . . 11 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) ∈ ℕ ↔ (𝑞 pCnt 𝑥) ∈ ℕ))
17 prmz 12076 . . . . . . . . . . . . 13 (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
18 id 19 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ)
19 dvdsle 11815 . . . . . . . . . . . . 13 ((𝑞 ∈ ℤ ∧ 𝑥 ∈ ℕ) → (𝑞𝑥𝑞𝑥))
2017, 18, 19syl2anr 290 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞𝑥𝑞𝑥))
21 pcelnn 12285 . . . . . . . . . . . . 13 ((𝑞 ∈ ℙ ∧ 𝑥 ∈ ℕ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞𝑥))
2221ancoms 268 . . . . . . . . . . . 12 ((𝑥 ∈ ℕ ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt 𝑥) ∈ ℕ ↔ 𝑞𝑥))
23 prmnn 12075 . . . . . . . . . . . . . 14 (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
24 nnuz 9534 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
2523, 24eleqtrdi 2268 . . . . . . . . . . . . 13 (𝑞 ∈ ℙ → 𝑞 ∈ (ℤ‘1))
26 elfz5 9985 . . . . . . . . . . . . 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 3159 . . . . . . 7 (𝑥 ∈ ℕ → ((𝑀𝑥) “ ℕ) ⊆ (1...𝑥))
33 elfznn 10022 . . . . . . . . . . . . . 14 (𝑗 ∈ (1...𝑥) → 𝑗 ∈ ℕ)
3433adantl 277 . . . . . . . . . . . . 13 ((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) → 𝑗 ∈ ℕ)
35 prmdc 12095 . . . . . . . . . . . . 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 5644 . . . . . . . . . . . . 13 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → ((𝑀𝑥)‘𝑗) ∈ ℕ0)
4140nn0zd 9344 . . . . . . . . . . . 12 (((𝑥 ∈ ℕ ∧ 𝑗 ∈ (1...𝑥)) ∧ 𝑗 ∈ ℙ) → ((𝑀𝑥)‘𝑗) ∈ ℤ)
42 elnndc 9583 . . . . . . . . . . . 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 5627 . . . . . . . . . . . 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 2548 . . . . . . 7 (𝑥 ∈ ℕ → ∀𝑗 ∈ (1...𝑥)DECID 𝑗 ∈ ((𝑀𝑥) “ ℕ))
60 ssfidc 6924 . . . . . . 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 4794 . . . . . . . . 9 (𝑒 = (𝑀𝑥) → 𝑒 = (𝑀𝑥))
6362imaeq1d 4962 . . . . . . . 8 (𝑒 = (𝑀𝑥) → (𝑒 “ ℕ) = ((𝑀𝑥) “ ℕ))
6463eleq1d 2244 . . . . . . 7 (𝑒 = (𝑀𝑥) → ((𝑒 “ ℕ) ∈ Fin ↔ ((𝑀𝑥) “ ℕ) ∈ Fin))
65 1arith.2 . . . . . . 7 𝑅 = {𝑒 ∈ (ℕ0𝑚 ℙ) ∣ (𝑒 “ ℕ) ∈ Fin}
6664, 65elrab2 2894 . . . . . 6 ((𝑀𝑥) ∈ 𝑅 ↔ ((𝑀𝑥) ∈ (ℕ0𝑚 ℙ) ∧ ((𝑀𝑥) “ ℕ) ∈ Fin))
678, 61, 66sylanbrc 417 . . . . 5 (𝑥 ∈ ℕ → (𝑀𝑥) ∈ 𝑅)
6867rgen 2528 . . . 4 𝑥 ∈ ℕ (𝑀𝑥) ∈ 𝑅
69 ffnfv 5666 . . . 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 12327 . . . . . . . . 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 2190 . . . . . . 7 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑞 ∈ ℙ) → (((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞) ↔ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
7574ralbidva 2471 . . . . . 6 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑞 ∈ ℙ ((𝑀𝑥)‘𝑞) = ((𝑀𝑦)‘𝑞) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt 𝑥) = (𝑞 pCnt 𝑦)))
7631arithlem3 12328 . . . . . . 7 (𝑦 ∈ ℕ → (𝑀𝑦):ℙ⟶ℕ0)
77 ffn 5357 . . . . . . . 8 ((𝑀𝑦):ℙ⟶ℕ0 → (𝑀𝑦) Fn ℙ)
78 eqfnfv 5605 . . . . . . . 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 9154 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
82 nnnn0 9154 . . . . . . 7 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
83 pc11 12295 . . . . . . 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 2561 . . 3 𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ((𝑀𝑥) = (𝑀𝑦) → 𝑥 = 𝑦)
88 dff13 5759 . . 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 2175 . . . . . 6 (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓𝑔)), 1)) = (𝑔 ∈ ℕ ↦ if(𝑔 ∈ ℙ, (𝑔↑(𝑓𝑔)), 1))
91 cnveq 4794 . . . . . . . . . . . 12 (𝑒 = 𝑓𝑒 = 𝑓)
9291imaeq1d 4962 . . . . . . . . . . 11 (𝑒 = 𝑓 → (𝑒 “ ℕ) = (𝑓 “ ℕ))
9392eleq1d 2244 . . . . . . . . . 10 (𝑒 = 𝑓 → ((𝑒 “ ℕ) ∈ Fin ↔ (𝑓 “ ℕ) ∈ Fin))
9493, 65elrab2 2894 . . . . . . . . 9 (𝑓𝑅 ↔ (𝑓 ∈ (ℕ0𝑚 ℙ) ∧ (𝑓 “ ℕ) ∈ Fin))
9594simplbi 274 . . . . . . . 8 (𝑓𝑅𝑓 ∈ (ℕ0𝑚 ℙ))
966, 1elmap 6667 . . . . . . . 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 8905 . . . . . 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 8903 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 ∈ ℝ)
103 peano2re 8067 . . . . . . . . . . 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 8903 . . . . . . . . . 10 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑞 ∈ ℝ)
107102ltp1d 8858 . . . . . . . . . 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 8354 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → 𝑦 < 𝑞)
110101nnzd 9345 . . . . . . . . . 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 9270 . . . . . . . . . 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 5357 . . . . . . . . . . 11 (𝑓:ℙ⟶ℕ0𝑓 Fn ℙ)
119 elpreima 5627 . . . . . . . . . . 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 4001 . . . . . . . . . . 11 (𝑘 = 𝑞 → (𝑘𝑦𝑞𝑦))
123122rspccv 2836 . . . . . . . . . 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 5644 . . . . . . . . 9 ((((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) ∧ (𝑞 ∈ ℙ ∧ (𝑦 + 1) ≤ 𝑞)) → (𝑓𝑞) ∈ ℕ0)
128 elnn0 9149 . . . . . . . . 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 12329 . . . . 5 (((𝑓𝑅𝑦 ∈ ℕ) ∧ ∀𝑘 ∈ (𝑓 “ ℕ)𝑘𝑦) → ∃𝑥 ∈ ℕ 𝑓 = (𝑀𝑥))
133 cnvimass 4984 . . . . . . 7 (𝑓 “ ℕ) ⊆ dom 𝑓
13497fdmd 5364 . . . . . . . 8 (𝑓𝑅 → dom 𝑓 = ℙ)
135 prmssnn 12077 . . . . . . . 8 ℙ ⊆ ℕ
136134, 135eqsstrdi 3205 . . . . . . 7 (𝑓𝑅 → dom 𝑓 ⊆ ℕ)
137133, 136sstrid 3164 . . . . . 6 (𝑓𝑅 → (𝑓 “ ℕ) ⊆ ℕ)
13894simprbi 275 . . . . . 6 (𝑓𝑅 → (𝑓 “ ℕ) ∈ Fin)
139 fiubnn 10776 . . . . . 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 2618 . . . 4 (𝑓𝑅 → ∃𝑥 ∈ ℕ 𝑓 = (𝑀𝑥))
142141rgen 2528 . . 3 𝑓𝑅𝑥 ∈ ℕ 𝑓 = (𝑀𝑥)
143 dffo3 5655 . . 3 (𝑀:ℕ–onto𝑅 ↔ (𝑀:ℕ⟶𝑅 ∧ ∀𝑓𝑅𝑥 ∈ ℕ 𝑓 = (𝑀𝑥)))
14470, 142, 143mpbir2an 942 . 2 𝑀:ℕ–onto𝑅
145 df-f1o 5215 . 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 2146  wral 2453  wrex 2454  {crab 2457  wss 3127  ifcif 3532   class class class wbr 3998  cmpt 4059  ccnv 4619  dom cdm 4620  cima 4623   Fn wfn 5203  wf 5204  1-1wf1 5205  ontowfo 5206  1-1-ontowf1o 5207  cfv 5208  (class class class)co 5865  𝑚 cmap 6638  Fincfn 6730  cr 7785  0cc0 7786  1c1 7787   + caddc 7789   < clt 7966  cle 7967  cn 8890  0cn0 9147  cz 9224  cuz 9499  ...cfz 9977  cexp 10487  cdvds 11760  cprime 12072   pCnt cpc 12249
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-1o 6407  df-2o 6408  df-er 6525  df-map 6640  df-en 6731  df-fin 6733  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-xnn0 9211  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-fl 10238  df-mod 10291  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-dvds 11761  df-gcd 11909  df-prm 12073  df-pc 12250
This theorem is referenced by:  1arith2  12331
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