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Theorem pcprod 12702

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 12655	. . . . . . . . 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 2579	. . . . . . 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 5953	. . . . . 6 ⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝑁) = (𝑝 pCnt 𝑁))
9	1, 5, 6, 7, 8	pcmpt 12699	. . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑝 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑝 ≤ 𝑁, (𝑝 pCnt 𝑁), 0))
10		iftrue 3576	. . . . . . 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 3579	. . . . . . . 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 12466	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
15		dvdsle 12188	. . . . . . . . . 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 12678	. . . . . . . . 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 2241	. . . . . 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 9496	. . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
24		zdcle 9451	. . . . . . . 8 ⊢ ((𝑝 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑝 ≤ 𝑁)
25	22, 23, 24	syl2anc 411	. . . . . . 7 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → DECID 𝑝 ≤ 𝑁)
26		exmiddc 838	. . . . . . 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 800	. . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ) → if(𝑝 ≤ 𝑁, (𝑝 pCnt 𝑁), 0) = (𝑝 pCnt 𝑁))
29	9, 28	eqtrd 2238	. . . 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 2579	. 2 ⊢ (𝑁 ∈ ℕ → ∀𝑝 ∈ ℙ (𝑝 pCnt (seq1( · , 𝐹)‘𝑁)) = (𝑝 pCnt 𝑁))
32	1, 4	pcmptcl 12698	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝐹:ℕ⟶ℕ ∧ seq1( · , 𝐹):ℕ⟶ℕ))
33	32	simprd 114	. . . . 5 ⊢ (𝑁 ∈ ℕ → seq1( · , 𝐹):ℕ⟶ℕ)
34		ffvelcdm 5715	. . . . 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 9350	. . 3 ⊢ (𝑁 ∈ ℕ → (seq1( · , 𝐹)‘𝑁) ∈ ℕ₀)
37		nnnn0 9304	. . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
38		pc11 12687	. . 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 710 DECID wdc 836 = wceq 1373 ∈ wcel 2176 ∀wral 2484 ifcif 3571 class class class wbr 4045 ↦ cmpt 4106 ⟶wf 5268 ‘cfv 5272 (class class class)co 5946 0cc0 7927 1c1 7928 · cmul 7932 ≤ cle 8110 ℕcn 9038 ℕ₀cn0 9297 ℤcz 9374 seqcseq 10594 ↑cexp 10685 ∥ cdvds 12131 ℙcprime 12462 pCnt cpc 12640

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4160 ax-sep 4163 ax-nul 4171 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-iinf 4637 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-mulrcl 8026 ax-addcom 8027 ax-mulcom 8028 ax-addass 8029 ax-mulass 8030 ax-distr 8031 ax-i2m1 8032 ax-0lt1 8033 ax-1rid 8034 ax-0id 8035 ax-rnegex 8036 ax-precex 8037 ax-cnre 8038 ax-pre-ltirr 8039 ax-pre-ltwlin 8040 ax-pre-lttrn 8041 ax-pre-apti 8042 ax-pre-ltadd 8043 ax-pre-mulgt0 8044 ax-pre-mulext 8045 ax-arch 8046 ax-caucvg 8047

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4046 df-opab 4107 df-mpt 4108 df-tr 4144 df-id 4341 df-po 4344 df-iso 4345 df-iord 4414 df-on 4416 df-ilim 4417 df-suc 4419 df-iom 4640 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-rn 4687 df-res 4688 df-ima 4689 df-iota 5233 df-fun 5274 df-fn 5275 df-f 5276 df-f1 5277 df-fo 5278 df-f1o 5279 df-fv 5280 df-isom 5281 df-riota 5901 df-ov 5949 df-oprab 5950 df-mpo 5951 df-1st 6228 df-2nd 6229 df-recs 6393 df-frec 6479 df-1o 6504 df-2o 6505 df-er 6622 df-en 6830 df-fin 6832 df-sup 7088 df-inf 7089 df-pnf 8111 df-mnf 8112 df-xr 8113 df-ltxr 8114 df-le 8115 df-sub 8247 df-neg 8248 df-reap 8650 df-ap 8657 df-div 8748 df-inn 9039 df-2 9097 df-3 9098 df-4 9099 df-n0 9298 df-xnn0 9361 df-z 9375 df-uz 9651 df-q 9743 df-rp 9778 df-fz 10133 df-fzo 10267 df-fl 10415 df-mod 10470 df-seqfrec 10595 df-exp 10686 df-cj 11186 df-re 11187 df-im 11188 df-rsqrt 11342 df-abs 11343 df-dvds 12132 df-gcd 12308 df-prm 12463 df-pc 12641

This theorem is referenced by: (None)

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