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Mirrors > Home > ILE Home > Th. List > pcpremul | Unicode version |
Description: Multiplicative property of the prime count pre-function. Note that the primality of is essential for this property; but . Since this is needed to show uniqueness for the real prime count function (over ), we don't bother to define it off the primes. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Ref | Expression |
---|---|
pcpremul.1 | |
pcpremul.2 | |
pcpremul.3 |
Ref | Expression |
---|---|
pcpremul |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3213 | . . . . . 6 | |
2 | nn0ssz 9191 | . . . . . 6 | |
3 | 1, 2 | sstri 3137 | . . . . 5 |
4 | 3 | a1i 9 | . . . 4 |
5 | prmuz2 12024 | . . . . . 6 | |
6 | 5 | 3ad2ant1 1003 | . . . . 5 |
7 | zmulcl 9226 | . . . . . . 7 | |
8 | 7 | ad2ant2r 501 | . . . . . 6 |
9 | 8 | 3adant1 1000 | . . . . 5 |
10 | simp2l 1008 | . . . . . . . 8 | |
11 | 10 | zcnd 9293 | . . . . . . 7 |
12 | simp3l 1010 | . . . . . . . 8 | |
13 | 12 | zcnd 9293 | . . . . . . 7 |
14 | simp2r 1009 | . . . . . . . 8 | |
15 | 0zd 9185 | . . . . . . . . 9 | |
16 | zapne 9244 | . . . . . . . . 9 # | |
17 | 10, 15, 16 | syl2anc 409 | . . . . . . . 8 # |
18 | 14, 17 | mpbird 166 | . . . . . . 7 # |
19 | simp3r 1011 | . . . . . . . 8 | |
20 | zapne 9244 | . . . . . . . . 9 # | |
21 | 12, 15, 20 | syl2anc 409 | . . . . . . . 8 # |
22 | 19, 21 | mpbird 166 | . . . . . . 7 # |
23 | 11, 13, 18, 22 | mulap0d 8537 | . . . . . 6 # |
24 | zapne 9244 | . . . . . . 7 # | |
25 | 9, 15, 24 | syl2anc 409 | . . . . . 6 # |
26 | 23, 25 | mpbid 146 | . . . . 5 |
27 | eqid 2157 | . . . . . 6 | |
28 | 27 | pclemdc 12179 | . . . . 5 DECID |
29 | 6, 9, 26, 28 | syl12anc 1218 | . . . 4 DECID |
30 | 27 | pclemub 12178 | . . . . 5 |
31 | 6, 9, 26, 30 | syl12anc 1218 | . . . 4 |
32 | oveq2 5835 | . . . . . . 7 | |
33 | 32 | breq1d 3977 | . . . . . 6 |
34 | eqid 2157 | . . . . . . . . . 10 | |
35 | pcpremul.1 | . . . . . . . . . 10 | |
36 | 34, 35 | pcprecl 12180 | . . . . . . . . 9 |
37 | 6, 10, 14, 36 | syl12anc 1218 | . . . . . . . 8 |
38 | 37 | simpld 111 | . . . . . . 7 |
39 | eqid 2157 | . . . . . . . . . 10 | |
40 | pcpremul.2 | . . . . . . . . . 10 | |
41 | 39, 40 | pcprecl 12180 | . . . . . . . . 9 |
42 | 6, 12, 19, 41 | syl12anc 1218 | . . . . . . . 8 |
43 | 42 | simpld 111 | . . . . . . 7 |
44 | 38, 43 | nn0addcld 9153 | . . . . . 6 |
45 | prmnn 12003 | . . . . . . . . . 10 | |
46 | 45 | 3ad2ant1 1003 | . . . . . . . . 9 |
47 | 46, 44 | nnexpcld 10583 | . . . . . . . 8 |
48 | 47 | nnzd 9291 | . . . . . . 7 |
49 | 46, 43 | nnexpcld 10583 | . . . . . . . . 9 |
50 | 49 | nnzd 9291 | . . . . . . . 8 |
51 | 10, 50 | zmulcld 9298 | . . . . . . 7 |
52 | 46 | nncnd 8853 | . . . . . . . . 9 |
53 | 52, 43, 38 | expaddd 10563 | . . . . . . . 8 |
54 | 37 | simprd 113 | . . . . . . . . 9 |
55 | 46, 38 | nnexpcld 10583 | . . . . . . . . . . 11 |
56 | 55 | nnzd 9291 | . . . . . . . . . 10 |
57 | dvdsmulc 11727 | . . . . . . . . . 10 | |
58 | 56, 10, 50, 57 | syl3anc 1220 | . . . . . . . . 9 |
59 | 54, 58 | mpd 13 | . . . . . . . 8 |
60 | 53, 59 | eqbrtrd 3989 | . . . . . . 7 |
61 | 42 | simprd 113 | . . . . . . . 8 |
62 | dvdscmul 11726 | . . . . . . . . 9 | |
63 | 50, 12, 10, 62 | syl3anc 1220 | . . . . . . . 8 |
64 | 61, 63 | mpd 13 | . . . . . . 7 |
65 | 48, 51, 9, 60, 64 | dvdstrd 11737 | . . . . . 6 |
66 | 33, 44, 65 | elrabd 2870 | . . . . 5 |
67 | oveq2 5835 | . . . . . . 7 | |
68 | 67 | breq1d 3977 | . . . . . 6 |
69 | 68 | cbvrabv 2711 | . . . . 5 |
70 | 66, 69 | eleqtrdi 2250 | . . . 4 |
71 | 4, 29, 31, 70 | suprzubdc 11852 | . . 3 |
72 | pcpremul.3 | . . 3 | |
73 | 71, 72 | breqtrrdi 4009 | . 2 |
74 | 34, 35 | pcprendvds2 12182 | . . . . . 6 |
75 | 6, 10, 14, 74 | syl12anc 1218 | . . . . 5 |
76 | 39, 40 | pcprendvds2 12182 | . . . . . 6 |
77 | 6, 12, 19, 76 | syl12anc 1218 | . . . . 5 |
78 | ioran 742 | . . . . 5 | |
79 | 75, 77, 78 | sylanbrc 414 | . . . 4 |
80 | simp1 982 | . . . . 5 | |
81 | 55 | nnne0d 8884 | . . . . . . 7 |
82 | dvdsval2 11698 | . . . . . . 7 | |
83 | 56, 81, 10, 82 | syl3anc 1220 | . . . . . 6 |
84 | 54, 83 | mpbid 146 | . . . . 5 |
85 | 49 | nnne0d 8884 | . . . . . . 7 |
86 | dvdsval2 11698 | . . . . . . 7 | |
87 | 50, 85, 12, 86 | syl3anc 1220 | . . . . . 6 |
88 | 61, 87 | mpbid 146 | . . . . 5 |
89 | euclemma 12037 | . . . . 5 | |
90 | 80, 84, 88, 89 | syl3anc 1220 | . . . 4 |
91 | 79, 90 | mtbird 663 | . . 3 |
92 | 27, 72 | pcprecl 12180 | . . . . . . 7 |
93 | 6, 9, 26, 92 | syl12anc 1218 | . . . . . 6 |
94 | 93 | simpld 111 | . . . . 5 |
95 | nn0ltp1le 9235 | . . . . 5 | |
96 | 44, 94, 95 | syl2anc 409 | . . . 4 |
97 | 46 | nnzd 9291 | . . . . . . 7 |
98 | peano2nn0 9136 | . . . . . . . 8 | |
99 | 44, 98 | syl 14 | . . . . . . 7 |
100 | dvdsexp 11766 | . . . . . . . 8 | |
101 | 100 | 3expia 1187 | . . . . . . 7 |
102 | 97, 99, 101 | syl2anc 409 | . . . . . 6 |
103 | 93 | simprd 113 | . . . . . . 7 |
104 | 46, 99 | nnexpcld 10583 | . . . . . . . . 9 |
105 | 104 | nnzd 9291 | . . . . . . . 8 |
106 | 46, 94 | nnexpcld 10583 | . . . . . . . . 9 |
107 | 106 | nnzd 9291 | . . . . . . . 8 |
108 | dvdstr 11736 | . . . . . . . 8 | |
109 | 105, 107, 9, 108 | syl3anc 1220 | . . . . . . 7 |
110 | 103, 109 | mpan2d 425 | . . . . . 6 |
111 | 102, 110 | syld 45 | . . . . 5 |
112 | 99 | nn0zd 9290 | . . . . . 6 |
113 | 94 | nn0zd 9290 | . . . . . 6 |
114 | eluz 9458 | . . . . . 6 | |
115 | 112, 113, 114 | syl2anc 409 | . . . . 5 |
116 | 52, 44 | expp1d 10562 | . . . . . . 7 |
117 | 11, 13 | mulcld 7901 | . . . . . . . . 9 |
118 | 47 | nncnd 8853 | . . . . . . . . 9 |
119 | 47 | nnap0d 8885 | . . . . . . . . 9 # |
120 | 117, 118, 119 | divcanap2d 8670 | . . . . . . . 8 |
121 | 53 | oveq2d 5843 | . . . . . . . . . 10 |
122 | 55 | nncnd 8853 | . . . . . . . . . . 11 |
123 | 49 | nncnd 8853 | . . . . . . . . . . 11 |
124 | 55 | nnap0d 8885 | . . . . . . . . . . 11 # |
125 | 49 | nnap0d 8885 | . . . . . . . . . . 11 # |
126 | 11, 122, 13, 123, 124, 125 | divmuldivapd 8710 | . . . . . . . . . 10 |
127 | 121, 126 | eqtr4d 2193 | . . . . . . . . 9 |
128 | 127 | oveq2d 5843 | . . . . . . . 8 |
129 | 120, 128 | eqtr3d 2192 | . . . . . . 7 |
130 | 116, 129 | breq12d 3980 | . . . . . 6 |
131 | 84, 88 | zmulcld 9298 | . . . . . . 7 |
132 | 47 | nnne0d 8884 | . . . . . . 7 |
133 | dvdscmulr 11728 | . . . . . . 7 | |
134 | 97, 131, 48, 132, 133 | syl112anc 1224 | . . . . . 6 |
135 | 130, 134 | bitrd 187 | . . . . 5 |
136 | 111, 115, 135 | 3imtr3d 201 | . . . 4 |
137 | 96, 136 | sylbid 149 | . . 3 |
138 | 91, 137 | mtod 653 | . 2 |
139 | 44 | nn0red 9150 | . . 3 |
140 | 94 | nn0red 9150 | . . 3 |
141 | 139, 140 | eqleltd 7997 | . 2 |
142 | 73, 138, 141 | mpbir2and 929 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 820 w3a 963 wceq 1335 wcel 2128 wne 2327 wral 2435 wrex 2436 crab 2439 wss 3102 class class class wbr 3967 cfv 5173 (class class class)co 5827 csup 6929 cr 7734 cc0 7735 c1 7736 caddc 7738 cmul 7740 clt 7915 cle 7916 # cap 8461 cdiv 8550 cn 8839 c2 8890 cn0 9096 cz 9173 cuz 9445 cexp 10428 cdvds 11695 cprime 12000 |
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-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 |
This theorem is referenced by: pceulem 12185 pcmul 12192 |
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