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Mirrors > Home > ILE Home > Th. List > pcgcd1 | Unicode version |
Description: The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.) |
Ref | Expression |
---|---|
pcgcd1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5861 | . . . 4 | |
2 | 1 | oveq2d 5869 | . . 3 |
3 | simp2 993 | . . . . . . 7 | |
4 | gcdid0 11935 | . . . . . . 7 | |
5 | 3, 4 | syl 14 | . . . . . 6 |
6 | 5 | oveq2d 5869 | . . . . 5 |
7 | zq 9585 | . . . . . . 7 | |
8 | pcabs 12279 | . . . . . . 7 | |
9 | 7, 8 | sylan2 284 | . . . . . 6 |
10 | 9 | 3adant3 1012 | . . . . 5 |
11 | 6, 10 | eqtrd 2203 | . . . 4 |
12 | 11 | adantr 274 | . . 3 |
13 | 2, 12 | sylan9eqr 2225 | . 2 |
14 | simpl1 995 | . . . . 5 | |
15 | 3 | adantr 274 | . . . . . . 7 |
16 | simpl3 997 | . . . . . . 7 | |
17 | simprr 527 | . . . . . . . 8 | |
18 | simpr 109 | . . . . . . . . 9 | |
19 | 18 | necon3ai 2389 | . . . . . . . 8 |
20 | 17, 19 | syl 14 | . . . . . . 7 |
21 | gcdn0cl 11917 | . . . . . . 7 | |
22 | 15, 16, 20, 21 | syl21anc 1232 | . . . . . 6 |
23 | 22 | nnzd 9333 | . . . . 5 |
24 | gcddvds 11918 | . . . . . . 7 | |
25 | 15, 16, 24 | syl2anc 409 | . . . . . 6 |
26 | 25 | simpld 111 | . . . . 5 |
27 | pcdvdstr 12280 | . . . . 5 | |
28 | 14, 23, 15, 26, 27 | syl13anc 1235 | . . . 4 |
29 | 15, 7 | syl 14 | . . . . . . . . . 10 |
30 | pcxcl 12265 | . . . . . . . . . 10 | |
31 | 14, 29, 30 | syl2anc 409 | . . . . . . . . 9 |
32 | pczcl 12252 | . . . . . . . . . . 11 | |
33 | 14, 16, 17, 32 | syl12anc 1231 | . . . . . . . . . 10 |
34 | 33 | nn0red 9189 | . . . . . . . . 9 |
35 | pcge0 12266 | . . . . . . . . . . 11 | |
36 | 14, 15, 35 | syl2anc 409 | . . . . . . . . . 10 |
37 | ge0gtmnf 9780 | . . . . . . . . . 10 | |
38 | 31, 36, 37 | syl2anc 409 | . . . . . . . . 9 |
39 | simprl 526 | . . . . . . . . 9 | |
40 | xrre 9777 | . . . . . . . . 9 | |
41 | 31, 34, 38, 39, 40 | syl22anc 1234 | . . . . . . . 8 |
42 | pnfnre 7961 | . . . . . . . . . . . 12 | |
43 | 42 | neli 2437 | . . . . . . . . . . 11 |
44 | pc0 12258 | . . . . . . . . . . . . 13 | |
45 | 14, 44 | syl 14 | . . . . . . . . . . . 12 |
46 | 45 | eleq1d 2239 | . . . . . . . . . . 11 |
47 | 43, 46 | mtbiri 670 | . . . . . . . . . 10 |
48 | oveq2 5861 | . . . . . . . . . . . 12 | |
49 | 48 | eleq1d 2239 | . . . . . . . . . . 11 |
50 | 49 | notbid 662 | . . . . . . . . . 10 |
51 | 47, 50 | syl5ibrcom 156 | . . . . . . . . 9 |
52 | 51 | necon2ad 2397 | . . . . . . . 8 |
53 | 41, 52 | mpd 13 | . . . . . . 7 |
54 | pczdvds 12267 | . . . . . . 7 | |
55 | 14, 15, 53, 54 | syl12anc 1231 | . . . . . 6 |
56 | pczcl 12252 | . . . . . . . . 9 | |
57 | 14, 15, 53, 56 | syl12anc 1231 | . . . . . . . 8 |
58 | pcdvdsb 12273 | . . . . . . . 8 | |
59 | 14, 16, 57, 58 | syl3anc 1233 | . . . . . . 7 |
60 | 39, 59 | mpbid 146 | . . . . . 6 |
61 | prmnn 12064 | . . . . . . . . . 10 | |
62 | 14, 61 | syl 14 | . . . . . . . . 9 |
63 | 62, 57 | nnexpcld 10631 | . . . . . . . 8 |
64 | 63 | nnzd 9333 | . . . . . . 7 |
65 | dvdsgcd 11967 | . . . . . . 7 | |
66 | 64, 15, 16, 65 | syl3anc 1233 | . . . . . 6 |
67 | 55, 60, 66 | mp2and 431 | . . . . 5 |
68 | pcdvdsb 12273 | . . . . . 6 | |
69 | 14, 23, 57, 68 | syl3anc 1233 | . . . . 5 |
70 | 67, 69 | mpbird 166 | . . . 4 |
71 | 14, 22 | pccld 12254 | . . . . . 6 |
72 | 71 | nn0red 9189 | . . . . 5 |
73 | 72, 41 | letri3d 8035 | . . . 4 |
74 | 28, 70, 73 | mpbir2and 939 | . . 3 |
75 | 74 | anassrs 398 | . 2 |
76 | simpl3 997 | . . . 4 | |
77 | 0zd 9224 | . . . 4 | |
78 | zdceq 9287 | . . . 4 DECID | |
79 | 76, 77, 78 | syl2anc 409 | . . 3 DECID |
80 | dcne 2351 | . . 3 DECID | |
81 | 79, 80 | sylib 121 | . 2 |
82 | 13, 75, 81 | mpjaodan 793 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 DECID wdc 829 w3a 973 wceq 1348 wcel 2141 wne 2340 class class class wbr 3989 cfv 5198 (class class class)co 5853 cr 7773 cc0 7774 cpnf 7951 cmnf 7952 cxr 7953 clt 7954 cle 7955 cn 8878 cn0 9135 cz 9212 cq 9578 cexp 10475 cabs 10961 cdvds 11749 cgcd 11897 cprime 12061 cpc 12238 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-1o 6395 df-2o 6396 df-er 6513 df-en 6719 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-dvds 11750 df-gcd 11898 df-prm 12062 df-pc 12239 |
This theorem is referenced by: pcgcd 12282 |
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