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Mirrors > Home > ILE Home > Th. List > facavg | Unicode version |
Description: The product of two factorials is greater than or equal to the factorial of (the floor of) their average. (Contributed by NM, 9-Dec-2005.) |
Ref | Expression |
---|---|
facavg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0addcl 9125 | . . . . . . 7 | |
2 | 1 | nn0zd 9284 | . . . . . 6 |
3 | 2nn 8994 | . . . . . 6 | |
4 | znq 9533 | . . . . . 6 | |
5 | 2, 3, 4 | sylancl 410 | . . . . 5 |
6 | flqle 10177 | . . . . 5 | |
7 | 5, 6 | syl 14 | . . . 4 |
8 | 5 | flqcld 10176 | . . . . . 6 |
9 | 8 | zred 9286 | . . . . 5 |
10 | nn0readdcl 9149 | . . . . . 6 | |
11 | 10 | rehalfcld 9079 | . . . . 5 |
12 | nn0re 9099 | . . . . . 6 | |
13 | 12 | adantr 274 | . . . . 5 |
14 | letr 7960 | . . . . 5 | |
15 | 9, 11, 13, 14 | syl3anc 1220 | . . . 4 |
16 | 7, 15 | mpand 426 | . . 3 |
17 | 1 | nn0ge0d 9146 | . . . . . 6 |
18 | halfnneg2 9065 | . . . . . . 7 | |
19 | 10, 18 | syl 14 | . . . . . 6 |
20 | 17, 19 | mpbid 146 | . . . . 5 |
21 | flqge0nn0 10192 | . . . . 5 | |
22 | 5, 20, 21 | syl2anc 409 | . . . 4 |
23 | simpl 108 | . . . 4 | |
24 | facwordi 10614 | . . . . 5 | |
25 | 24 | 3exp 1184 | . . . 4 |
26 | 22, 23, 25 | sylc 62 | . . 3 |
27 | faccl 10609 | . . . . . . . 8 | |
28 | 27 | nncnd 8847 | . . . . . . 7 |
29 | 28 | mulid1d 7895 | . . . . . 6 |
30 | 29 | adantr 274 | . . . . 5 |
31 | faccl 10609 | . . . . . . . 8 | |
32 | 31 | nnred 8846 | . . . . . . 7 |
33 | 32 | adantl 275 | . . . . . 6 |
34 | 27 | nnred 8846 | . . . . . . . 8 |
35 | 27 | nnnn0d 9143 | . . . . . . . . 9 |
36 | 35 | nn0ge0d 9146 | . . . . . . . 8 |
37 | 34, 36 | jca 304 | . . . . . . 7 |
38 | 37 | adantr 274 | . . . . . 6 |
39 | 31 | nnge1d 8876 | . . . . . . 7 |
40 | 39 | adantl 275 | . . . . . 6 |
41 | 1re 7877 | . . . . . . 7 | |
42 | lemul2a 8730 | . . . . . . 7 | |
43 | 41, 42 | mp3anl1 1313 | . . . . . 6 |
44 | 33, 38, 40, 43 | syl21anc 1219 | . . . . 5 |
45 | 30, 44 | eqbrtrrd 3988 | . . . 4 |
46 | faccl 10609 | . . . . . . 7 | |
47 | 22, 46 | syl 14 | . . . . . 6 |
48 | 47 | nnred 8846 | . . . . 5 |
49 | 34 | adantr 274 | . . . . 5 |
50 | remulcl 7860 | . . . . . 6 | |
51 | 34, 32, 50 | syl2an 287 | . . . . 5 |
52 | letr 7960 | . . . . 5 | |
53 | 48, 49, 51, 52 | syl3anc 1220 | . . . 4 |
54 | 45, 53 | mpan2d 425 | . . 3 |
55 | 16, 26, 54 | 3syld 57 | . 2 |
56 | nn0re 9099 | . . . . . 6 | |
57 | 56 | adantl 275 | . . . . 5 |
58 | letr 7960 | . . . . 5 | |
59 | 9, 11, 57, 58 | syl3anc 1220 | . . . 4 |
60 | 7, 59 | mpand 426 | . . 3 |
61 | simpr 109 | . . . 4 | |
62 | facwordi 10614 | . . . . 5 | |
63 | 62 | 3exp 1184 | . . . 4 |
64 | 22, 61, 63 | sylc 62 | . . 3 |
65 | 31 | nncnd 8847 | . . . . . . 7 |
66 | 65 | mulid2d 7896 | . . . . . 6 |
67 | 66 | adantl 275 | . . . . 5 |
68 | 31 | nnnn0d 9143 | . . . . . . . . 9 |
69 | 68 | nn0ge0d 9146 | . . . . . . . 8 |
70 | 32, 69 | jca 304 | . . . . . . 7 |
71 | 70 | adantl 275 | . . . . . 6 |
72 | 27 | nnge1d 8876 | . . . . . . 7 |
73 | 72 | adantr 274 | . . . . . 6 |
74 | lemul1a 8729 | . . . . . . 7 | |
75 | 41, 74 | mp3anl1 1313 | . . . . . 6 |
76 | 49, 71, 73, 75 | syl21anc 1219 | . . . . 5 |
77 | 67, 76 | eqbrtrrd 3988 | . . . 4 |
78 | letr 7960 | . . . . 5 | |
79 | 48, 33, 51, 78 | syl3anc 1220 | . . . 4 |
80 | 77, 79 | mpan2d 425 | . . 3 |
81 | 60, 64, 80 | 3syld 57 | . 2 |
82 | 23 | nn0zd 9284 | . . . 4 |
83 | zq 9535 | . . . 4 | |
84 | 82, 83 | syl 14 | . . 3 |
85 | 61 | nn0zd 9284 | . . . 4 |
86 | zq 9535 | . . . 4 | |
87 | 85, 86 | syl 14 | . . 3 |
88 | qavgle 10158 | . . 3 | |
89 | 84, 87, 88 | syl2anc 409 | . 2 |
90 | 55, 81, 89 | mpjaod 708 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wceq 1335 wcel 2128 class class class wbr 3965 cfv 5170 (class class class)co 5824 cr 7731 cc0 7732 c1 7733 caddc 7735 cmul 7737 cle 7913 cdiv 8545 cn 8833 c2 8884 cn0 9090 cz 9167 cq 9528 cfl 10167 cfa 10599 |
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 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 |
This theorem depends on definitions: df-bi 116 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 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-frec 6338 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-n0 9091 df-z 9168 df-uz 9440 df-q 9529 df-rp 9561 df-fl 10169 df-seqfrec 10345 df-fac 10600 |
This theorem is referenced by: (None) |
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