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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 8980 | . . . . . . 7 | |
2 | 1 | nn0zd 9139 | . . . . . 6 |
3 | 2nn 8849 | . . . . . 6 | |
4 | znq 9384 | . . . . . 6 | |
5 | 2, 3, 4 | sylancl 409 | . . . . 5 |
6 | flqle 10019 | . . . . 5 | |
7 | 5, 6 | syl 14 | . . . 4 |
8 | 5 | flqcld 10018 | . . . . . 6 |
9 | 8 | zred 9141 | . . . . 5 |
10 | nn0readdcl 9004 | . . . . . 6 | |
11 | 10 | rehalfcld 8934 | . . . . 5 |
12 | nn0re 8954 | . . . . . 6 | |
13 | 12 | adantr 274 | . . . . 5 |
14 | letr 7815 | . . . . 5 | |
15 | 9, 11, 13, 14 | syl3anc 1201 | . . . 4 |
16 | 7, 15 | mpand 425 | . . 3 |
17 | 1 | nn0ge0d 9001 | . . . . . 6 |
18 | halfnneg2 8920 | . . . . . . 7 | |
19 | 10, 18 | syl 14 | . . . . . 6 |
20 | 17, 19 | mpbid 146 | . . . . 5 |
21 | flqge0nn0 10034 | . . . . 5 | |
22 | 5, 20, 21 | syl2anc 408 | . . . 4 |
23 | simpl 108 | . . . 4 | |
24 | facwordi 10454 | . . . . 5 | |
25 | 24 | 3exp 1165 | . . . 4 |
26 | 22, 23, 25 | sylc 62 | . . 3 |
27 | faccl 10449 | . . . . . . . 8 | |
28 | 27 | nncnd 8702 | . . . . . . 7 |
29 | 28 | mulid1d 7751 | . . . . . 6 |
30 | 29 | adantr 274 | . . . . 5 |
31 | faccl 10449 | . . . . . . . 8 | |
32 | 31 | nnred 8701 | . . . . . . 7 |
33 | 32 | adantl 275 | . . . . . 6 |
34 | 27 | nnred 8701 | . . . . . . . 8 |
35 | 27 | nnnn0d 8998 | . . . . . . . . 9 |
36 | 35 | nn0ge0d 9001 | . . . . . . . 8 |
37 | 34, 36 | jca 304 | . . . . . . 7 |
38 | 37 | adantr 274 | . . . . . 6 |
39 | 31 | nnge1d 8731 | . . . . . . 7 |
40 | 39 | adantl 275 | . . . . . 6 |
41 | 1re 7733 | . . . . . . 7 | |
42 | lemul2a 8585 | . . . . . . 7 | |
43 | 41, 42 | mp3anl1 1294 | . . . . . 6 |
44 | 33, 38, 40, 43 | syl21anc 1200 | . . . . 5 |
45 | 30, 44 | eqbrtrrd 3922 | . . . 4 |
46 | faccl 10449 | . . . . . . 7 | |
47 | 22, 46 | syl 14 | . . . . . 6 |
48 | 47 | nnred 8701 | . . . . 5 |
49 | 34 | adantr 274 | . . . . 5 |
50 | remulcl 7716 | . . . . . 6 | |
51 | 34, 32, 50 | syl2an 287 | . . . . 5 |
52 | letr 7815 | . . . . 5 | |
53 | 48, 49, 51, 52 | syl3anc 1201 | . . . 4 |
54 | 45, 53 | mpan2d 424 | . . 3 |
55 | 16, 26, 54 | 3syld 57 | . 2 |
56 | nn0re 8954 | . . . . . 6 | |
57 | 56 | adantl 275 | . . . . 5 |
58 | letr 7815 | . . . . 5 | |
59 | 9, 11, 57, 58 | syl3anc 1201 | . . . 4 |
60 | 7, 59 | mpand 425 | . . 3 |
61 | simpr 109 | . . . 4 | |
62 | facwordi 10454 | . . . . 5 | |
63 | 62 | 3exp 1165 | . . . 4 |
64 | 22, 61, 63 | sylc 62 | . . 3 |
65 | 31 | nncnd 8702 | . . . . . . 7 |
66 | 65 | mulid2d 7752 | . . . . . 6 |
67 | 66 | adantl 275 | . . . . 5 |
68 | 31 | nnnn0d 8998 | . . . . . . . . 9 |
69 | 68 | nn0ge0d 9001 | . . . . . . . 8 |
70 | 32, 69 | jca 304 | . . . . . . 7 |
71 | 70 | adantl 275 | . . . . . 6 |
72 | 27 | nnge1d 8731 | . . . . . . 7 |
73 | 72 | adantr 274 | . . . . . 6 |
74 | lemul1a 8584 | . . . . . . 7 | |
75 | 41, 74 | mp3anl1 1294 | . . . . . 6 |
76 | 49, 71, 73, 75 | syl21anc 1200 | . . . . 5 |
77 | 67, 76 | eqbrtrrd 3922 | . . . 4 |
78 | letr 7815 | . . . . 5 | |
79 | 48, 33, 51, 78 | syl3anc 1201 | . . . 4 |
80 | 77, 79 | mpan2d 424 | . . 3 |
81 | 60, 64, 80 | 3syld 57 | . 2 |
82 | 23 | nn0zd 9139 | . . . 4 |
83 | zq 9386 | . . . 4 | |
84 | 82, 83 | syl 14 | . . 3 |
85 | 61 | nn0zd 9139 | . . . 4 |
86 | zq 9386 | . . . 4 | |
87 | 85, 86 | syl 14 | . . 3 |
88 | qavgle 10004 | . . 3 | |
89 | 84, 87, 88 | syl2anc 408 | . 2 |
90 | 55, 81, 89 | mpjaod 692 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 682 wceq 1316 wcel 1465 class class class wbr 3899 cfv 5093 (class class class)co 5742 cr 7587 cc0 7588 c1 7589 caddc 7591 cmul 7593 cle 7769 cdiv 8400 cn 8688 c2 8739 cn0 8945 cz 9022 cq 9379 cfl 10009 cfa 10439 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-n0 8946 df-z 9023 df-uz 9295 df-q 9380 df-rp 9410 df-fl 10011 df-seqfrec 10187 df-fac 10440 |
This theorem is referenced by: (None) |
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