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Mirrors > Home > ILE Home > Th. List > pcidlem | GIF version |
Description: The prime count of a prime power. (Contributed by Mario Carneiro, 12-Mar-2014.) |
Ref | Expression |
---|---|
pcidlem | ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℙ) | |
2 | prmnn 12068 | . . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
3 | 1, 2 | syl 14 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℕ) |
4 | simpr 109 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝐴 ∈ ℕ0) | |
5 | 3, 4 | nnexpcld 10635 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃↑𝐴) ∈ ℕ) |
6 | 1, 5 | pccld 12258 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃↑𝐴)) ∈ ℕ0) |
7 | 6 | nn0red 9193 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃↑𝐴)) ∈ ℝ) |
8 | 7 | leidd 8437 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃↑𝐴)) ≤ (𝑃 pCnt (𝑃↑𝐴))) |
9 | 5 | nnzd 9337 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃↑𝐴) ∈ ℤ) |
10 | pcdvdsb 12277 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃↑𝐴) ∈ ℤ ∧ (𝑃 pCnt (𝑃↑𝐴)) ∈ ℕ0) → ((𝑃 pCnt (𝑃↑𝐴)) ≤ (𝑃 pCnt (𝑃↑𝐴)) ↔ (𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ∥ (𝑃↑𝐴))) | |
11 | 1, 9, 6, 10 | syl3anc 1234 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → ((𝑃 pCnt (𝑃↑𝐴)) ≤ (𝑃 pCnt (𝑃↑𝐴)) ↔ (𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ∥ (𝑃↑𝐴))) |
12 | 8, 11 | mpbid 146 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ∥ (𝑃↑𝐴)) |
13 | 3, 6 | nnexpcld 10635 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ∈ ℕ) |
14 | 13 | nnzd 9337 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ∈ ℤ) |
15 | dvdsle 11808 | . . . . 5 ⊢ (((𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ∈ ℤ ∧ (𝑃↑𝐴) ∈ ℕ) → ((𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ∥ (𝑃↑𝐴) → (𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ≤ (𝑃↑𝐴))) | |
16 | 14, 5, 15 | syl2anc 409 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → ((𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ∥ (𝑃↑𝐴) → (𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ≤ (𝑃↑𝐴))) |
17 | 12, 16 | mpd 13 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ≤ (𝑃↑𝐴)) |
18 | 3 | nnred 8895 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℝ) |
19 | prmuz2 12089 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2)) | |
20 | eluz2gt1 9565 | . . . . 5 ⊢ (𝑃 ∈ (ℤ≥‘2) → 1 < 𝑃) | |
21 | 1, 19, 20 | 3syl 17 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 1 < 𝑃) |
22 | nn0leexp2 10649 | . . . 4 ⊢ (((𝑃 ∈ ℝ ∧ (𝑃 pCnt (𝑃↑𝐴)) ∈ ℕ0 ∧ 𝐴 ∈ ℕ0) ∧ 1 < 𝑃) → ((𝑃 pCnt (𝑃↑𝐴)) ≤ 𝐴 ↔ (𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ≤ (𝑃↑𝐴))) | |
23 | 18, 6, 4, 21, 22 | syl31anc 1237 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → ((𝑃 pCnt (𝑃↑𝐴)) ≤ 𝐴 ↔ (𝑃↑(𝑃 pCnt (𝑃↑𝐴))) ≤ (𝑃↑𝐴))) |
24 | 17, 23 | mpbird 166 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃↑𝐴)) ≤ 𝐴) |
25 | iddvds 11770 | . . . 4 ⊢ ((𝑃↑𝐴) ∈ ℤ → (𝑃↑𝐴) ∥ (𝑃↑𝐴)) | |
26 | 9, 25 | syl 14 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃↑𝐴) ∥ (𝑃↑𝐴)) |
27 | pcdvdsb 12277 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃↑𝐴) ∈ ℤ ∧ 𝐴 ∈ ℕ0) → (𝐴 ≤ (𝑃 pCnt (𝑃↑𝐴)) ↔ (𝑃↑𝐴) ∥ (𝑃↑𝐴))) | |
28 | 1, 9, 4, 27 | syl3anc 1234 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝐴 ≤ (𝑃 pCnt (𝑃↑𝐴)) ↔ (𝑃↑𝐴) ∥ (𝑃↑𝐴))) |
29 | 26, 28 | mpbird 166 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝐴 ≤ (𝑃 pCnt (𝑃↑𝐴))) |
30 | nn0re 9148 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
31 | 30 | adantl 275 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝐴 ∈ ℝ) |
32 | 7, 31 | letri3d 8039 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → ((𝑃 pCnt (𝑃↑𝐴)) = 𝐴 ↔ ((𝑃 pCnt (𝑃↑𝐴)) ≤ 𝐴 ∧ 𝐴 ≤ (𝑃 pCnt (𝑃↑𝐴))))) |
33 | 24, 29, 32 | mpbir2and 940 | 1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1349 ∈ wcel 2142 class class class wbr 3990 ‘cfv 5200 (class class class)co 5857 ℝcr 7777 1c1 7779 < clt 7958 ≤ cle 7959 ℕcn 8882 2c2 8933 ℕ0cn0 9139 ℤcz 9216 ℤ≥cuz 9491 ↑cexp 10479 ∥ cdvds 11753 ℙcprime 12065 pCnt cpc 12242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 610 ax-in2 611 ax-io 705 ax-5 1441 ax-7 1442 ax-gen 1443 ax-ie1 1487 ax-ie2 1488 ax-8 1498 ax-10 1499 ax-11 1500 ax-i12 1501 ax-bndl 1503 ax-4 1504 ax-17 1520 ax-i9 1524 ax-ial 1528 ax-i5r 1529 ax-13 2144 ax-14 2145 ax-ext 2153 ax-coll 4105 ax-sep 4108 ax-nul 4116 ax-pow 4161 ax-pr 4195 ax-un 4419 ax-setind 4522 ax-iinf 4573 ax-cnex 7869 ax-resscn 7870 ax-1cn 7871 ax-1re 7872 ax-icn 7873 ax-addcl 7874 ax-addrcl 7875 ax-mulcl 7876 ax-mulrcl 7877 ax-addcom 7878 ax-mulcom 7879 ax-addass 7880 ax-mulass 7881 ax-distr 7882 ax-i2m1 7883 ax-0lt1 7884 ax-1rid 7885 ax-0id 7886 ax-rnegex 7887 ax-precex 7888 ax-cnre 7889 ax-pre-ltirr 7890 ax-pre-ltwlin 7891 ax-pre-lttrn 7892 ax-pre-apti 7893 ax-pre-ltadd 7894 ax-pre-mulgt0 7895 ax-pre-mulext 7896 ax-arch 7897 ax-caucvg 7898 |
This theorem depends on definitions: df-bi 116 df-stab 827 df-dc 831 df-3or 975 df-3an 976 df-tru 1352 df-fal 1355 df-nf 1455 df-sb 1757 df-eu 2023 df-mo 2024 df-clab 2158 df-cleq 2164 df-clel 2167 df-nfc 2302 df-ne 2342 df-nel 2437 df-ral 2454 df-rex 2455 df-reu 2456 df-rmo 2457 df-rab 2458 df-v 2733 df-sbc 2957 df-csb 3051 df-dif 3124 df-un 3126 df-in 3128 df-ss 3135 df-nul 3416 df-if 3528 df-pw 3569 df-sn 3590 df-pr 3591 df-op 3593 df-uni 3798 df-int 3833 df-iun 3876 df-br 3991 df-opab 4052 df-mpt 4053 df-tr 4089 df-id 4279 df-po 4282 df-iso 4283 df-iord 4352 df-on 4354 df-ilim 4355 df-suc 4357 df-iom 4576 df-xp 4618 df-rel 4619 df-cnv 4620 df-co 4621 df-dm 4622 df-rn 4623 df-res 4624 df-ima 4625 df-iota 5162 df-fun 5202 df-fn 5203 df-f 5204 df-f1 5205 df-fo 5206 df-f1o 5207 df-fv 5208 df-isom 5209 df-riota 5813 df-ov 5860 df-oprab 5861 df-mpo 5862 df-1st 6123 df-2nd 6124 df-recs 6288 df-frec 6374 df-1o 6399 df-2o 6400 df-er 6517 df-en 6723 df-sup 6965 df-inf 6966 df-pnf 7960 df-mnf 7961 df-xr 7962 df-ltxr 7963 df-le 7964 df-sub 8096 df-neg 8097 df-reap 8498 df-ap 8505 df-div 8594 df-inn 8883 df-2 8941 df-3 8942 df-4 8943 df-n0 9140 df-z 9217 df-uz 9492 df-q 9583 df-rp 9615 df-fz 9970 df-fzo 10103 df-fl 10230 df-mod 10283 df-seqfrec 10406 df-exp 10480 df-cj 10810 df-re 10811 df-im 10812 df-rsqrt 10966 df-abs 10967 df-dvds 11754 df-gcd 11902 df-prm 12066 df-pc 12243 |
This theorem is referenced by: pcid 12281 pcmpt 12299 |
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