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Theorem pcprod 12542

Description: The product of the primes taken to their respective powers reconstructs the original number. (Contributed by Mario Carneiro, 12-Mar-2014.)

**Hypothesis**
Ref	Expression
pcprod.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝑁)), 1))

**Assertion**
Ref	Expression
pcprod	⊢ (𝑁 ∈ ℕ → (seq1( · , 𝐹)‘𝑁) = 𝑁)

Distinct variable group: 𝑛,𝑁 Allowed substitution hint: 𝐹(𝑛)

Proof of Theorem pcprod Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pcprod.1	. . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝑁)), 1))
2		pccl 12495	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑛 pCnt 𝑁) ∈ ℕ₀)
3	2	ancoms 268	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑁) ∈ ℕ₀)
4	3	ralrimiva 2570	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ∀𝑛 ∈ ℙ (𝑛 pCnt 𝑁) ∈ ℕ₀)
5	4	adantl 277	. . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ∀𝑛 ∈ ℙ (𝑛 pCnt 𝑁) ∈ ℕ₀)
6		simpr 110	. . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
7		simpl 109	. . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → 𝑝 ∈ ℙ)
8		oveq1 5932	. . . . . 6 ⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝑁) = (𝑝 pCnt 𝑁))
9	1, 5, 6, 7, 8	pcmpt 12539	. . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑝 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑝 ≤ 𝑁, (𝑝 pCnt 𝑁), 0))
10		iftrue 3567	. . . . . . 7 ⊢ (𝑝 ≤ 𝑁 → if(𝑝 ≤ 𝑁, (𝑝 pCnt 𝑁), 0) = (𝑝 pCnt 𝑁))
11	10	adantl 277	. . . . . 6 ⊢ (((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) ∧ 𝑝 ≤ 𝑁) → if(𝑝 ≤ 𝑁, (𝑝 pCnt 𝑁), 0) = (𝑝 pCnt 𝑁))
12		iffalse 3570	. . . . . . . 8 ⊢ (¬ 𝑝 ≤ 𝑁 → if(𝑝 ≤ 𝑁, (𝑝 pCnt 𝑁), 0) = 0)
13	12	adantl 277	. . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑝 ≤ 𝑁) → if(𝑝 ≤ 𝑁, (𝑝 pCnt 𝑁), 0) = 0)
14		prmz 12306	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
15		dvdsle 12028	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑝 ∥ 𝑁 → 𝑝 ≤ 𝑁))
16	14, 15	sylan 283	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑝 ∥ 𝑁 → 𝑝 ≤ 𝑁))
17	16	con3dimp 636	. . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑝 ≤ 𝑁) → ¬ 𝑝 ∥ 𝑁)
18		pceq0 12518	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ((𝑝 pCnt 𝑁) = 0 ↔ ¬ 𝑝 ∥ 𝑁))
19	18	adantr 276	. . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑝 ≤ 𝑁) → ((𝑝 pCnt 𝑁) = 0 ↔ ¬ 𝑝 ∥ 𝑁))
20	17, 19	mpbird 167	. . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑝 ≤ 𝑁) → (𝑝 pCnt 𝑁) = 0)
21	13, 20	eqtr4d 2232	. . . . . 6 ⊢ (((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑝 ≤ 𝑁) → if(𝑝 ≤ 𝑁, (𝑝 pCnt 𝑁), 0) = (𝑝 pCnt 𝑁))
22	14	adantr 276	. . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → 𝑝 ∈ ℤ)
23	6	nnzd 9466	. . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
24		zdcle 9421	. . . . . . . 8 ⊢ ((𝑝 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑝 ≤ 𝑁)
25	22, 23, 24	syl2anc 411	. . . . . . 7 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → DECID 𝑝 ≤ 𝑁)
26		exmiddc 837	. . . . . . 7 ⊢ (DECID 𝑝 ≤ 𝑁 → (𝑝 ≤ 𝑁 ∨ ¬ 𝑝 ≤ 𝑁))
27	25, 26	syl 14	. . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑝 ≤ 𝑁 ∨ ¬ 𝑝 ≤ 𝑁))
28	11, 21, 27	mpjaodan 799	. . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → if(𝑝 ≤ 𝑁, (𝑝 pCnt 𝑁), 0) = (𝑝 pCnt 𝑁))
29	9, 28	eqtrd 2229	. . . 4 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑝 pCnt (seq1( · , 𝐹)‘𝑁)) = (𝑝 pCnt 𝑁))
30	29	ancoms 268	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (seq1( · , 𝐹)‘𝑁)) = (𝑝 pCnt 𝑁))
31	30	ralrimiva 2570	. 2 ⊢ (𝑁 ∈ ℕ → ∀𝑝 ∈ ℙ (𝑝 pCnt (seq1( · , 𝐹)‘𝑁)) = (𝑝 pCnt 𝑁))
32	1, 4	pcmptcl 12538	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝐹:ℕ⟶ℕ ∧ seq1( · , 𝐹):ℕ⟶ℕ))
33	32	simprd 114	. . . . 5 ⊢ (𝑁 ∈ ℕ → seq1( · , 𝐹):ℕ⟶ℕ)
34		ffvelcdm 5698	. . . . 5 ⊢ ((seq1( · , 𝐹):ℕ⟶ℕ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘𝑁) ∈ ℕ)
35	33, 34	mpancom 422	. . . 4 ⊢ (𝑁 ∈ ℕ → (seq1( · , 𝐹)‘𝑁) ∈ ℕ)
36	35	nnnn0d 9321	. . 3 ⊢ (𝑁 ∈ ℕ → (seq1( · , 𝐹)‘𝑁) ∈ ℕ₀)
37		nnnn0 9275	. . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
38		pc11 12527	. . 3 ⊢ (((seq1( · , 𝐹)‘𝑁) ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((seq1( · , 𝐹)‘𝑁) = 𝑁 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (seq1( · , 𝐹)‘𝑁)) = (𝑝 pCnt 𝑁)))
39	36, 37, 38	syl2anc 411	. 2 ⊢ (𝑁 ∈ ℕ → ((seq1( · , 𝐹)‘𝑁) = 𝑁 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (seq1( · , 𝐹)‘𝑁)) = (𝑝 pCnt 𝑁)))
40	31, 39	mpbird 167	1 ⊢ (𝑁 ∈ ℕ → (seq1( · , 𝐹)‘𝑁) = 𝑁)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 DECID wdc 835 = wceq 1364 ∈ wcel 2167 ∀wral 2475 ifcif 3562 class class class wbr 4034 ↦ cmpt 4095 ⟶wf 5255 ‘cfv 5259 (class class class)co 5925 0cc0 7898 1c1 7899 · cmul 7903 ≤ cle 8081 ℕcn 9009 ℕ₀cn0 9268 ℤcz 9345 seqcseq 10558 ↑cexp 10649 ∥ cdvds 11971 ℙcprime 12302 pCnt cpc 12480

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7989 ax-resscn 7990 ax-1cn 7991 ax-1re 7992 ax-icn 7993 ax-addcl 7994 ax-addrcl 7995 ax-mulcl 7996 ax-mulrcl 7997 ax-addcom 7998 ax-mulcom 7999 ax-addass 8000 ax-mulass 8001 ax-distr 8002 ax-i2m1 8003 ax-0lt1 8004 ax-1rid 8005 ax-0id 8006 ax-rnegex 8007 ax-precex 8008 ax-cnre 8009 ax-pre-ltirr 8010 ax-pre-ltwlin 8011 ax-pre-lttrn 8012 ax-pre-apti 8013 ax-pre-ltadd 8014 ax-pre-mulgt0 8015 ax-pre-mulext 8016 ax-arch 8017 ax-caucvg 8018

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-1o 6483 df-2o 6484 df-er 6601 df-en 6809 df-fin 6811 df-sup 7059 df-inf 7060 df-pnf 8082 df-mnf 8083 df-xr 8084 df-ltxr 8085 df-le 8086 df-sub 8218 df-neg 8219 df-reap 8621 df-ap 8628 df-div 8719 df-inn 9010 df-2 9068 df-3 9069 df-4 9070 df-n0 9269 df-xnn0 9332 df-z 9346 df-uz 9621 df-q 9713 df-rp 9748 df-fz 10103 df-fzo 10237 df-fl 10379 df-mod 10434 df-seqfrec 10559 df-exp 10650 df-cj 11026 df-re 11027 df-im 11028 df-rsqrt 11182 df-abs 11183 df-dvds 11972 df-gcd 12148 df-prm 12303 df-pc 12481

This theorem is referenced by: (None)

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