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Mirrors > Home > ILE Home > Th. List > pcid | Unicode version |
Description: The prime count of a prime power. (Contributed by Mario Carneiro, 9-Sep-2014.) |
Ref | Expression |
---|---|
pcid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elznn0nn 9187 | . 2 | |
2 | pcidlem 12212 | . . 3 | |
3 | prmnn 12003 | . . . . . . . 8 | |
4 | 3 | adantr 274 | . . . . . . 7 |
5 | 4 | nncnd 8853 | . . . . . 6 |
6 | 4 | nnap0d 8885 | . . . . . 6 # |
7 | simprl 521 | . . . . . . 7 | |
8 | 7 | recnd 7909 | . . . . . 6 |
9 | nnnn0 9103 | . . . . . . 7 | |
10 | 9 | ad2antll 483 | . . . . . 6 |
11 | expineg2 10438 | . . . . . 6 # | |
12 | 5, 6, 8, 10, 11 | syl22anc 1221 | . . . . 5 |
13 | 12 | oveq2d 5843 | . . . 4 |
14 | simpl 108 | . . . . . 6 | |
15 | 1zzd 9200 | . . . . . 6 | |
16 | 1ne0 8907 | . . . . . . 7 | |
17 | 16 | a1i 9 | . . . . . 6 |
18 | 4, 10 | nnexpcld 10583 | . . . . . 6 |
19 | pcdiv 12193 | . . . . . 6 | |
20 | 14, 15, 17, 18, 19 | syl121anc 1225 | . . . . 5 |
21 | pc1 12196 | . . . . . . . 8 | |
22 | 21 | adantr 274 | . . . . . . 7 |
23 | pcidlem 12212 | . . . . . . . 8 | |
24 | 10, 23 | syldan 280 | . . . . . . 7 |
25 | 22, 24 | oveq12d 5845 | . . . . . 6 |
26 | df-neg 8054 | . . . . . . 7 | |
27 | 8 | negnegd 8182 | . . . . . . 7 |
28 | 26, 27 | eqtr3id 2204 | . . . . . 6 |
29 | 25, 28 | eqtrd 2190 | . . . . 5 |
30 | 20, 29 | eqtrd 2190 | . . . 4 |
31 | 13, 30 | eqtrd 2190 | . . 3 |
32 | 2, 31 | jaodan 787 | . 2 |
33 | 1, 32 | sylan2b 285 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1335 wcel 2128 wne 2327 class class class wbr 3967 (class class class)co 5827 cc 7733 cr 7734 cc0 7735 c1 7736 cmin 8051 cneg 8052 # cap 8461 cdiv 8550 cn 8839 cn0 9096 cz 9173 cexp 10428 cprime 12000 cpc 12175 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4082 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-iinf 4550 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-mulrcl 7834 ax-addcom 7835 ax-mulcom 7836 ax-addass 7837 ax-mulass 7838 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-1rid 7842 ax-0id 7843 ax-rnegex 7844 ax-precex 7845 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-apti 7850 ax-pre-ltadd 7851 ax-pre-mulgt0 7852 ax-pre-mulext 7853 ax-arch 7854 ax-caucvg 7855 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-tr 4066 df-id 4256 df-po 4259 df-iso 4260 df-iord 4329 df-on 4331 df-ilim 4332 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-isom 5182 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-1st 6091 df-2nd 6092 df-recs 6255 df-frec 6341 df-1o 6366 df-2o 6367 df-er 6483 df-en 6689 df-sup 6931 df-inf 6932 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-reap 8455 df-ap 8462 df-div 8551 df-inn 8840 df-2 8898 df-3 8899 df-4 8900 df-n0 9097 df-z 9174 df-uz 9446 df-q 9536 df-rp 9568 df-fz 9920 df-fzo 10052 df-fl 10179 df-mod 10232 df-seqfrec 10355 df-exp 10429 df-cj 10754 df-re 10755 df-im 10756 df-rsqrt 10910 df-abs 10911 df-dvds 11696 df-gcd 11843 df-prm 12001 df-pc 12176 |
This theorem is referenced by: (None) |
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