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Mirrors > Home > ILE Home > Th. List > bcval5 | Unicode version |
Description: Write out the top and bottom parts of the binomial coefficient explicitly. In this form, it is valid even for , although it is no longer valid for nonpositive . (Contributed by Mario Carneiro, 22-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.) |
Ref | Expression |
---|---|
bcval5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bcval2 10496 | . . . 4 | |
2 | 1 | adantl 275 | . . 3 |
3 | simprl 520 | . . . . . . . . 9 | |
4 | simprr 521 | . . . . . . . . 9 | |
5 | 3, 4 | mulcld 7786 | . . . . . . . 8 |
6 | simpr1 987 | . . . . . . . . 9 | |
7 | simpr2 988 | . . . . . . . . 9 | |
8 | simpr3 989 | . . . . . . . . 9 | |
9 | 6, 7, 8 | mulassd 7789 | . . . . . . . 8 |
10 | simpll 518 | . . . . . . . . . . . . 13 | |
11 | 10 | nn0zd 9171 | . . . . . . . . . . . 12 |
12 | simplr 519 | . . . . . . . . . . . . 13 | |
13 | 12 | nnzd 9172 | . . . . . . . . . . . 12 |
14 | 11, 13 | zsubcld 9178 | . . . . . . . . . . 11 |
15 | 14 | peano2zd 9176 | . . . . . . . . . 10 |
16 | 1red 7781 | . . . . . . . . . . . 12 | |
17 | 12 | nnred 8733 | . . . . . . . . . . . 12 |
18 | 10 | nn0red 9031 | . . . . . . . . . . . 12 |
19 | 12 | nnge1d 8763 | . . . . . . . . . . . 12 |
20 | 16, 17, 18, 19 | lesub2dd 8324 | . . . . . . . . . . 11 |
21 | 14 | zred 9173 | . . . . . . . . . . . 12 |
22 | leaddsub 8200 | . . . . . . . . . . . 12 | |
23 | 21, 16, 18, 22 | syl3anc 1216 | . . . . . . . . . . 11 |
24 | 20, 23 | mpbird 166 | . . . . . . . . . 10 |
25 | eluz2 9332 | . . . . . . . . . 10 | |
26 | 15, 11, 24, 25 | syl3anbrc 1165 | . . . . . . . . 9 |
27 | 26 | adantrr 470 | . . . . . . . 8 |
28 | simprr 521 | . . . . . . . . 9 | |
29 | nnuz 9361 | . . . . . . . . 9 | |
30 | 28, 29 | eleqtrdi 2232 | . . . . . . . 8 |
31 | fvi 5478 | . . . . . . . . . 10 | |
32 | 31 | elv 2690 | . . . . . . . . 9 |
33 | eluzelcn 9337 | . . . . . . . . . 10 | |
34 | 33 | adantl 275 | . . . . . . . . 9 |
35 | 32, 34 | eqeltrid 2226 | . . . . . . . 8 |
36 | 5, 9, 27, 30, 35 | seq3split 10252 | . . . . . . 7 |
37 | elfzuz3 9803 | . . . . . . . . . . 11 | |
38 | 37 | adantl 275 | . . . . . . . . . 10 |
39 | eluznn 9394 | . . . . . . . . . 10 | |
40 | 12, 38, 39 | syl2anc 408 | . . . . . . . . 9 |
41 | 40 | adantrr 470 | . . . . . . . 8 |
42 | facnn 10473 | . . . . . . . 8 | |
43 | 41, 42 | syl 14 | . . . . . . 7 |
44 | facnn 10473 | . . . . . . . . 9 | |
45 | 28, 44 | syl 14 | . . . . . . . 8 |
46 | 45 | oveq1d 5789 | . . . . . . 7 |
47 | 36, 43, 46 | 3eqtr4d 2182 | . . . . . 6 |
48 | 47 | expr 372 | . . . . 5 |
49 | 10 | faccld 10482 | . . . . . . . . 9 |
50 | 49 | nncnd 8734 | . . . . . . . 8 |
51 | 50 | mulid2d 7784 | . . . . . . 7 |
52 | 40, 42 | syl 14 | . . . . . . . 8 |
53 | 52 | oveq2d 5790 | . . . . . . 7 |
54 | 51, 53 | eqtr3d 2174 | . . . . . 6 |
55 | fveq2 5421 | . . . . . . . . 9 | |
56 | fac0 10474 | . . . . . . . . 9 | |
57 | 55, 56 | syl6eq 2188 | . . . . . . . 8 |
58 | oveq1 5781 | . . . . . . . . . . 11 | |
59 | 0p1e1 8834 | . . . . . . . . . . 11 | |
60 | 58, 59 | syl6eq 2188 | . . . . . . . . . 10 |
61 | 60 | seqeq1d 10224 | . . . . . . . . 9 |
62 | 61 | fveq1d 5423 | . . . . . . . 8 |
63 | 57, 62 | oveq12d 5792 | . . . . . . 7 |
64 | 63 | eqeq2d 2151 | . . . . . 6 |
65 | 54, 64 | syl5ibrcom 156 | . . . . 5 |
66 | fznn0sub 9837 | . . . . . . 7 | |
67 | 66 | adantl 275 | . . . . . 6 |
68 | elnn0 8979 | . . . . . 6 | |
69 | 67, 68 | sylib 121 | . . . . 5 |
70 | 48, 65, 69 | mpjaod 707 | . . . 4 |
71 | 70 | oveq1d 5789 | . . 3 |
72 | eqid 2139 | . . . . . 6 | |
73 | fvi 5478 | . . . . . . . 8 | |
74 | 73 | elv 2690 | . . . . . . 7 |
75 | eluzelcn 9337 | . . . . . . . 8 | |
76 | 75 | adantl 275 | . . . . . . 7 |
77 | 74, 76 | eqeltrid 2226 | . . . . . 6 |
78 | mulcl 7747 | . . . . . . 7 | |
79 | 78 | adantl 275 | . . . . . 6 |
80 | 72, 15, 77, 79 | seqf 10234 | . . . . 5 |
81 | 80, 26 | ffvelrnd 5556 | . . . 4 |
82 | 12 | nnnn0d 9030 | . . . . . 6 |
83 | 82 | faccld 10482 | . . . . 5 |
84 | 83 | nncnd 8734 | . . . 4 |
85 | 67 | faccld 10482 | . . . . 5 |
86 | 85 | nncnd 8734 | . . . 4 |
87 | 83 | nnap0d 8766 | . . . 4 # |
88 | 85 | nnap0d 8766 | . . . 4 # |
89 | 81, 84, 86, 87, 88 | divcanap5d 8577 | . . 3 |
90 | 2, 71, 89 | 3eqtrd 2176 | . 2 |
91 | simplr 519 | . . . . . . 7 | |
92 | 91 | nnnn0d 9030 | . . . . . 6 |
93 | 92 | faccld 10482 | . . . . 5 |
94 | 93 | nncnd 8734 | . . . 4 |
95 | 93 | nnap0d 8766 | . . . 4 # |
96 | 94, 95 | div0apd 8547 | . . 3 |
97 | mulcl 7747 | . . . . . 6 | |
98 | 97 | adantl 275 | . . . . 5 |
99 | eluzelcn 9337 | . . . . . . 7 | |
100 | 99 | adantl 275 | . . . . . 6 |
101 | 32, 100 | eqeltrid 2226 | . . . . 5 |
102 | simpr 109 | . . . . . 6 | |
103 | 102 | mul02d 8154 | . . . . 5 |
104 | 102 | mul01d 8155 | . . . . 5 |
105 | simpr 109 | . . . . . . . . 9 | |
106 | nn0uz 9360 | . . . . . . . . . . . 12 | |
107 | 92, 106 | eleqtrdi 2232 | . . . . . . . . . . 11 |
108 | simpll 518 | . . . . . . . . . . . 12 | |
109 | 108 | nn0zd 9171 | . . . . . . . . . . 11 |
110 | elfz5 9798 | . . . . . . . . . . 11 | |
111 | 107, 109, 110 | syl2anc 408 | . . . . . . . . . 10 |
112 | nn0re 8986 | . . . . . . . . . . . 12 | |
113 | 112 | ad2antrr 479 | . . . . . . . . . . 11 |
114 | nnre 8727 | . . . . . . . . . . . 12 | |
115 | 114 | ad2antlr 480 | . . . . . . . . . . 11 |
116 | 113, 115 | subge0d 8297 | . . . . . . . . . 10 |
117 | 111, 116 | bitr4d 190 | . . . . . . . . 9 |
118 | 105, 117 | mtbid 661 | . . . . . . . 8 |
119 | simpl 108 | . . . . . . . . . . . 12 | |
120 | 119 | nn0zd 9171 | . . . . . . . . . . 11 |
121 | simpr 109 | . . . . . . . . . . . 12 | |
122 | 121 | nnzd 9172 | . . . . . . . . . . 11 |
123 | 120, 122 | zsubcld 9178 | . . . . . . . . . 10 |
124 | 123 | adantr 274 | . . . . . . . . 9 |
125 | 0z 9065 | . . . . . . . . 9 | |
126 | zltnle 9100 | . . . . . . . . 9 | |
127 | 124, 125, 126 | sylancl 409 | . . . . . . . 8 |
128 | 118, 127 | mpbird 166 | . . . . . . 7 |
129 | zltp1le 9108 | . . . . . . . 8 | |
130 | 124, 125, 129 | sylancl 409 | . . . . . . 7 |
131 | 128, 130 | mpbid 146 | . . . . . 6 |
132 | nn0ge0 9002 | . . . . . . 7 | |
133 | 132 | ad2antrr 479 | . . . . . 6 |
134 | 0zd 9066 | . . . . . . 7 | |
135 | 124 | peano2zd 9176 | . . . . . . 7 |
136 | elfz 9796 | . . . . . . 7 | |
137 | 134, 135, 109, 136 | syl3anc 1216 | . . . . . 6 |
138 | 131, 133, 137 | mpbir2and 928 | . . . . 5 |
139 | 0cn 7758 | . . . . . 6 | |
140 | fvi 5478 | . . . . . 6 | |
141 | 139, 140 | mp1i 10 | . . . . 5 |
142 | 98, 101, 103, 104, 138, 141 | seq3z 10284 | . . . 4 |
143 | 142 | oveq1d 5789 | . . 3 |
144 | nnz 9073 | . . . . 5 | |
145 | bcval3 10497 | . . . . 5 | |
146 | 144, 145 | syl3an2 1250 | . . . 4 |
147 | 146 | 3expa 1181 | . . 3 |
148 | 96, 143, 147 | 3eqtr4rd 2183 | . 2 |
149 | 0zd 9066 | . . . 4 | |
150 | fzdcel 9820 | . . . 4 DECID | |
151 | 122, 149, 120, 150 | syl3anc 1216 | . . 3 DECID |
152 | exmiddc 821 | . . 3 DECID | |
153 | 151, 152 | syl 14 | . 2 |
154 | 90, 148, 153 | mpjaodan 787 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 w3a 962 wceq 1331 wcel 1480 cvv 2686 class class class wbr 3929 cid 4210 cfv 5123 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 caddc 7623 cmul 7625 clt 7800 cle 7801 cmin 7933 cdiv 8432 cn 8720 cn0 8977 cz 9054 cuz 9326 cfz 9790 cseq 10218 cfa 10471 cbc 10493 |
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 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
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 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-fz 9791 df-seqfrec 10219 df-fac 10472 df-bc 10494 |
This theorem is referenced by: bcn2 10510 |
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