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Theorem pcprod 12918

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 12871	. . . . . . . . 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 2605	. . . . . . 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 6024	. . . . . 6 ⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝑁) = (𝑝 pCnt 𝑁))
9	1, 5, 6, 7, 8	pcmpt 12915	. . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑝 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑝 ≤ 𝑁, (𝑝 pCnt 𝑁), 0))
10		iftrue 3610	. . . . . . 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 3613	. . . . . . . 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 12682	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
15		dvdsle 12404	. . . . . . . . . 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 640	. . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑝 ≤ 𝑁) → ¬ 𝑝 ∥ 𝑁)
18		pceq0 12894	. . . . . . . . 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 2267	. . . . . 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 9600	. . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
24		zdcle 9555	. . . . . . . 8 ⊢ ((𝑝 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑝 ≤ 𝑁)
25	22, 23, 24	syl2anc 411	. . . . . . 7 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → DECID 𝑝 ≤ 𝑁)
26		exmiddc 843	. . . . . . 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 805	. . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → if(𝑝 ≤ 𝑁, (𝑝 pCnt 𝑁), 0) = (𝑝 pCnt 𝑁))
29	9, 28	eqtrd 2264	. . . 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 2605	. 2 ⊢ (𝑁 ∈ ℕ → ∀𝑝 ∈ ℙ (𝑝 pCnt (seq1( · , 𝐹)‘𝑁)) = (𝑝 pCnt 𝑁))
32	1, 4	pcmptcl 12914	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝐹:ℕ⟶ℕ ∧ seq1( · , 𝐹):ℕ⟶ℕ))
33	32	simprd 114	. . . . 5 ⊢ (𝑁 ∈ ℕ → seq1( · , 𝐹):ℕ⟶ℕ)
34		ffvelcdm 5780	. . . . 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 9454	. . 3 ⊢ (𝑁 ∈ ℕ → (seq1( · , 𝐹)‘𝑁) ∈ ℕ₀)
37		nnnn0 9408	. . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
38		pc11 12903	. . 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 715 DECID wdc 841 = wceq 1397 ∈ wcel 2202 ∀wral 2510 ifcif 3605 class class class wbr 4088 ↦ cmpt 4150 ⟶wf 5322 ‘cfv 5326 (class class class)co 6017 0cc0 8031 1c1 8032 · cmul 8036 ≤ cle 8214 ℕcn 9142 ℕ₀cn0 9401 ℤcz 9478 seqcseq 10708 ↑cexp 10799 ∥ cdvds 12347 ℙcprime 12678 pCnt cpc 12856

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-frec 6556 df-1o 6581 df-2o 6582 df-er 6701 df-en 6909 df-fin 6911 df-sup 7182 df-inf 7183 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-n0 9402 df-xnn0 9465 df-z 9479 df-uz 9755 df-q 9853 df-rp 9888 df-fz 10243 df-fzo 10377 df-fl 10529 df-mod 10584 df-seqfrec 10709 df-exp 10800 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559 df-dvds 12348 df-gcd 12524 df-prm 12679 df-pc 12857

This theorem is referenced by: (None)

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