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Mirrors > Home > ILE Home > Th. List > bcm1k | Unicode version |
Description: The proportion of one binomial coefficient to another with decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.) |
Ref | Expression |
---|---|
bcm1k |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz2 9816 | . . . . . . . . 9 | |
2 | nnuz 9368 | . . . . . . . . 9 | |
3 | 1, 2 | eleqtrrdi 2233 | . . . . . . . 8 |
4 | 3 | nnnn0d 9037 | . . . . . . 7 |
5 | 4 | faccld 10489 | . . . . . 6 |
6 | 5 | nncnd 8741 | . . . . 5 |
7 | fznn0sub 9844 | . . . . . . . . . 10 | |
8 | nn0p1nn 9023 | . . . . . . . . . 10 | |
9 | 7, 8 | syl 14 | . . . . . . . . 9 |
10 | 9 | nnnn0d 9037 | . . . . . . . 8 |
11 | 10 | faccld 10489 | . . . . . . 7 |
12 | elfznn 9841 | . . . . . . . 8 | |
13 | nnm1nn0 9025 | . . . . . . . 8 | |
14 | faccl 10488 | . . . . . . . 8 | |
15 | 12, 13, 14 | 3syl 17 | . . . . . . 7 |
16 | 11, 15 | nnmulcld 8776 | . . . . . 6 |
17 | 16 | nncnd 8741 | . . . . 5 |
18 | 9 | nncnd 8741 | . . . . 5 |
19 | 12 | nncnd 8741 | . . . . 5 |
20 | 16 | nnap0d 8773 | . . . . 5 # |
21 | 12 | nnap0d 8773 | . . . . 5 # |
22 | 6, 17, 18, 19, 20, 21 | divmuldivapd 8599 | . . . 4 |
23 | elfzel2 9811 | . . . . . . . . . 10 | |
24 | 23 | zcnd 9181 | . . . . . . . . 9 |
25 | 1cnd 7789 | . . . . . . . . 9 | |
26 | 24, 19, 25 | subsubd 8108 | . . . . . . . 8 |
27 | 26 | fveq2d 5425 | . . . . . . 7 |
28 | 27 | oveq1d 5789 | . . . . . 6 |
29 | 28 | oveq2d 5790 | . . . . 5 |
30 | 26 | oveq1d 5789 | . . . . 5 |
31 | 29, 30 | oveq12d 5792 | . . . 4 |
32 | facp1 10483 | . . . . . . . . 9 | |
33 | 7, 32 | syl 14 | . . . . . . . 8 |
34 | 33 | eqcomd 2145 | . . . . . . 7 |
35 | facnn2 10487 | . . . . . . . 8 | |
36 | 12, 35 | syl 14 | . . . . . . 7 |
37 | 34, 36 | oveq12d 5792 | . . . . . 6 |
38 | 7 | faccld 10489 | . . . . . . . 8 |
39 | 38 | nncnd 8741 | . . . . . . 7 |
40 | 12 | nnnn0d 9037 | . . . . . . . . 9 |
41 | 40 | faccld 10489 | . . . . . . . 8 |
42 | 41 | nncnd 8741 | . . . . . . 7 |
43 | 39, 42, 18 | mul32d 7922 | . . . . . 6 |
44 | 11 | nncnd 8741 | . . . . . . 7 |
45 | 15 | nncnd 8741 | . . . . . . 7 |
46 | 44, 45, 19 | mulassd 7796 | . . . . . 6 |
47 | 37, 43, 46 | 3eqtr4d 2182 | . . . . 5 |
48 | 47 | oveq2d 5790 | . . . 4 |
49 | 22, 31, 48 | 3eqtr4d 2182 | . . 3 |
50 | 6, 18 | mulcomd 7794 | . . . 4 |
51 | 38, 41 | nnmulcld 8776 | . . . . . 6 |
52 | 51 | nncnd 8741 | . . . . 5 |
53 | 52, 18 | mulcomd 7794 | . . . 4 |
54 | 50, 53 | oveq12d 5792 | . . 3 |
55 | 51 | nnap0d 8773 | . . . 4 # |
56 | 9 | nnap0d 8773 | . . . 4 # |
57 | 6, 52, 18, 55, 56 | divcanap5d 8584 | . . 3 |
58 | 49, 54, 57 | 3eqtrrd 2177 | . 2 |
59 | 0p1e1 8841 | . . . . . 6 | |
60 | 59 | oveq1i 5784 | . . . . 5 |
61 | 0z 9072 | . . . . . 6 | |
62 | fzp1ss 9860 | . . . . . 6 | |
63 | 61, 62 | ax-mp 5 | . . . . 5 |
64 | 60, 63 | eqsstrri 3130 | . . . 4 |
65 | 64 | sseli 3093 | . . 3 |
66 | bcval2 10503 | . . 3 | |
67 | 65, 66 | syl 14 | . 2 |
68 | ax-1cn 7720 | . . . . . . . 8 | |
69 | npcan 7978 | . . . . . . . 8 | |
70 | 24, 68, 69 | sylancl 409 | . . . . . . 7 |
71 | peano2zm 9099 | . . . . . . . 8 | |
72 | uzid 9347 | . . . . . . . 8 | |
73 | peano2uz 9385 | . . . . . . . 8 | |
74 | 23, 71, 72, 73 | 4syl 18 | . . . . . . 7 |
75 | 70, 74 | eqeltrrd 2217 | . . . . . 6 |
76 | fzss2 9851 | . . . . . 6 | |
77 | 75, 76 | syl 14 | . . . . 5 |
78 | elfzmlbm 9915 | . . . . 5 | |
79 | 77, 78 | sseldd 3098 | . . . 4 |
80 | bcval2 10503 | . . . 4 | |
81 | 79, 80 | syl 14 | . . 3 |
82 | 81 | oveq1d 5789 | . 2 |
83 | 58, 67, 82 | 3eqtr4d 2182 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 wss 3071 cfv 5123 (class class class)co 5774 cc 7625 cc0 7627 c1 7628 caddc 7630 cmul 7632 cmin 7940 cdiv 8439 cn 8727 cn0 8984 cz 9061 cuz 9333 cfz 9797 cfa 10478 cbc 10500 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-mulrcl 7726 ax-addcom 7727 ax-mulcom 7728 ax-addass 7729 ax-mulass 7730 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-1rid 7734 ax-0id 7735 ax-rnegex 7736 ax-precex 7737 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-apti 7742 ax-pre-ltadd 7743 ax-pre-mulgt0 7744 ax-pre-mulext 7745 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-reap 8344 df-ap 8351 df-div 8440 df-inn 8728 df-n0 8985 df-z 9062 df-uz 9334 df-q 9419 df-fz 9798 df-seqfrec 10226 df-fac 10479 df-bc 10501 |
This theorem is referenced by: bcp1nk 10515 bcpasc 10519 |
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