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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 9170 | . . . . . . 7 | |
2 | 1 | nn0zd 9332 | . . . . . 6 |
3 | 2nn 9039 | . . . . . 6 | |
4 | znq 9583 | . . . . . 6 | |
5 | 2, 3, 4 | sylancl 411 | . . . . 5 |
6 | flqle 10234 | . . . . 5 | |
7 | 5, 6 | syl 14 | . . . 4 |
8 | 5 | flqcld 10233 | . . . . . 6 |
9 | 8 | zred 9334 | . . . . 5 |
10 | nn0readdcl 9194 | . . . . . 6 | |
11 | 10 | rehalfcld 9124 | . . . . 5 |
12 | nn0re 9144 | . . . . . 6 | |
13 | 12 | adantr 274 | . . . . 5 |
14 | letr 8002 | . . . . 5 | |
15 | 9, 11, 13, 14 | syl3anc 1233 | . . . 4 |
16 | 7, 15 | mpand 427 | . . 3 |
17 | 1 | nn0ge0d 9191 | . . . . . 6 |
18 | halfnneg2 9110 | . . . . . . 7 | |
19 | 10, 18 | syl 14 | . . . . . 6 |
20 | 17, 19 | mpbid 146 | . . . . 5 |
21 | flqge0nn0 10249 | . . . . 5 | |
22 | 5, 20, 21 | syl2anc 409 | . . . 4 |
23 | simpl 108 | . . . 4 | |
24 | facwordi 10674 | . . . . 5 | |
25 | 24 | 3exp 1197 | . . . 4 |
26 | 22, 23, 25 | sylc 62 | . . 3 |
27 | faccl 10669 | . . . . . . . 8 | |
28 | 27 | nncnd 8892 | . . . . . . 7 |
29 | 28 | mulid1d 7937 | . . . . . 6 |
30 | 29 | adantr 274 | . . . . 5 |
31 | faccl 10669 | . . . . . . . 8 | |
32 | 31 | nnred 8891 | . . . . . . 7 |
33 | 32 | adantl 275 | . . . . . 6 |
34 | 27 | nnred 8891 | . . . . . . . 8 |
35 | 27 | nnnn0d 9188 | . . . . . . . . 9 |
36 | 35 | nn0ge0d 9191 | . . . . . . . 8 |
37 | 34, 36 | jca 304 | . . . . . . 7 |
38 | 37 | adantr 274 | . . . . . 6 |
39 | 31 | nnge1d 8921 | . . . . . . 7 |
40 | 39 | adantl 275 | . . . . . 6 |
41 | 1re 7919 | . . . . . . 7 | |
42 | lemul2a 8775 | . . . . . . 7 | |
43 | 41, 42 | mp3anl1 1326 | . . . . . 6 |
44 | 33, 38, 40, 43 | syl21anc 1232 | . . . . 5 |
45 | 30, 44 | eqbrtrrd 4013 | . . . 4 |
46 | faccl 10669 | . . . . . . 7 | |
47 | 22, 46 | syl 14 | . . . . . 6 |
48 | 47 | nnred 8891 | . . . . 5 |
49 | 34 | adantr 274 | . . . . 5 |
50 | remulcl 7902 | . . . . . 6 | |
51 | 34, 32, 50 | syl2an 287 | . . . . 5 |
52 | letr 8002 | . . . . 5 | |
53 | 48, 49, 51, 52 | syl3anc 1233 | . . . 4 |
54 | 45, 53 | mpan2d 426 | . . 3 |
55 | 16, 26, 54 | 3syld 57 | . 2 |
56 | nn0re 9144 | . . . . . 6 | |
57 | 56 | adantl 275 | . . . . 5 |
58 | letr 8002 | . . . . 5 | |
59 | 9, 11, 57, 58 | syl3anc 1233 | . . . 4 |
60 | 7, 59 | mpand 427 | . . 3 |
61 | simpr 109 | . . . 4 | |
62 | facwordi 10674 | . . . . 5 | |
63 | 62 | 3exp 1197 | . . . 4 |
64 | 22, 61, 63 | sylc 62 | . . 3 |
65 | 31 | nncnd 8892 | . . . . . . 7 |
66 | 65 | mulid2d 7938 | . . . . . 6 |
67 | 66 | adantl 275 | . . . . 5 |
68 | 31 | nnnn0d 9188 | . . . . . . . . 9 |
69 | 68 | nn0ge0d 9191 | . . . . . . . 8 |
70 | 32, 69 | jca 304 | . . . . . . 7 |
71 | 70 | adantl 275 | . . . . . 6 |
72 | 27 | nnge1d 8921 | . . . . . . 7 |
73 | 72 | adantr 274 | . . . . . 6 |
74 | lemul1a 8774 | . . . . . . 7 | |
75 | 41, 74 | mp3anl1 1326 | . . . . . 6 |
76 | 49, 71, 73, 75 | syl21anc 1232 | . . . . 5 |
77 | 67, 76 | eqbrtrrd 4013 | . . . 4 |
78 | letr 8002 | . . . . 5 | |
79 | 48, 33, 51, 78 | syl3anc 1233 | . . . 4 |
80 | 77, 79 | mpan2d 426 | . . 3 |
81 | 60, 64, 80 | 3syld 57 | . 2 |
82 | 23 | nn0zd 9332 | . . . 4 |
83 | zq 9585 | . . . 4 | |
84 | 82, 83 | syl 14 | . . 3 |
85 | 61 | nn0zd 9332 | . . . 4 |
86 | zq 9585 | . . . 4 | |
87 | 85, 86 | syl 14 | . . 3 |
88 | qavgle 10215 | . . 3 | |
89 | 84, 87, 88 | syl2anc 409 | . 2 |
90 | 55, 81, 89 | mpjaod 713 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 703 wceq 1348 wcel 2141 class class class wbr 3989 cfv 5198 (class class class)co 5853 cr 7773 cc0 7774 c1 7775 caddc 7777 cmul 7779 cle 7955 cdiv 8589 cn 8878 c2 8929 cn0 9135 cz 9212 cq 9578 cfl 10224 cfa 10659 |
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 |
This theorem depends on definitions: df-bi 116 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-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-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 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-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fl 10226 df-seqfrec 10402 df-fac 10660 |
This theorem is referenced by: (None) |
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