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Mirrors > Home > ILE Home > Th. List > ibcval5 | 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 In this form, it is valid even for , although it is no longer valid for nonpositive .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 Jim Kingdon, 6-Nov-2021.) |
Ref | Expression |
---|---|
ibcval5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bcval2 9677 | . . . 4 | |
2 | 1 | adantl 271 | . . 3 |
3 | simprl 497 | . . . . . . . . 9 | |
4 | simprr 498 | . . . . . . . . 9 | |
5 | 3, 4 | mulcld 7139 | . . . . . . . 8 |
6 | simpr1 944 | . . . . . . . . 9 | |
7 | simpr2 945 | . . . . . . . . 9 | |
8 | simpr3 946 | . . . . . . . . 9 | |
9 | 6, 7, 8 | mulassd 7142 | . . . . . . . 8 |
10 | simpll 495 | . . . . . . . . . . . . 13 | |
11 | 10 | nn0zd 8467 | . . . . . . . . . . . 12 |
12 | simplr 496 | . . . . . . . . . . . . 13 | |
13 | 12 | nnzd 8468 | . . . . . . . . . . . 12 |
14 | 11, 13 | zsubcld 8474 | . . . . . . . . . . 11 |
15 | 14 | peano2zd 8472 | . . . . . . . . . 10 |
16 | 1red 7134 | . . . . . . . . . . . 12 | |
17 | 12 | nnred 8052 | . . . . . . . . . . . 12 |
18 | 10 | nn0red 8342 | . . . . . . . . . . . 12 |
19 | 12 | nnge1d 8081 | . . . . . . . . . . . 12 |
20 | 16, 17, 18, 19 | lesub2dd 7662 | . . . . . . . . . . 11 |
21 | 14 | zred 8469 | . . . . . . . . . . . 12 |
22 | leaddsub 7542 | . . . . . . . . . . . 12 | |
23 | 21, 16, 18, 22 | syl3anc 1169 | . . . . . . . . . . 11 |
24 | 20, 23 | mpbird 165 | . . . . . . . . . 10 |
25 | eluz2 8625 | . . . . . . . . . 10 | |
26 | 15, 11, 24, 25 | syl3anbrc 1122 | . . . . . . . . 9 |
27 | 26 | adantrr 462 | . . . . . . . 8 |
28 | cnex 7097 | . . . . . . . . 9 | |
29 | 28 | a1i 9 | . . . . . . . 8 |
30 | simprr 498 | . . . . . . . . 9 | |
31 | nnuz 8654 | . . . . . . . . 9 | |
32 | 30, 31 | syl6eleq 2171 | . . . . . . . 8 |
33 | vex 2604 | . . . . . . . . . 10 | |
34 | fvi 5251 | . . . . . . . . . 10 | |
35 | 33, 34 | ax-mp 7 | . . . . . . . . 9 |
36 | eluzelcn 8630 | . . . . . . . . . 10 | |
37 | 36 | adantl 271 | . . . . . . . . 9 |
38 | 35, 37 | syl5eqel 2165 | . . . . . . . 8 |
39 | 5, 9, 27, 29, 32, 38 | iseqsplit 9458 | . . . . . . 7 |
40 | elfzuz3 9042 | . . . . . . . . . . 11 | |
41 | 40 | adantl 271 | . . . . . . . . . 10 |
42 | eluznn 8687 | . . . . . . . . . 10 | |
43 | 12, 41, 42 | syl2anc 403 | . . . . . . . . 9 |
44 | 43 | adantrr 462 | . . . . . . . 8 |
45 | facnn 9654 | . . . . . . . 8 | |
46 | 44, 45 | syl 14 | . . . . . . 7 |
47 | facnn 9654 | . . . . . . . . 9 | |
48 | 30, 47 | syl 14 | . . . . . . . 8 |
49 | 48 | oveq1d 5547 | . . . . . . 7 |
50 | 39, 46, 49 | 3eqtr4d 2123 | . . . . . 6 |
51 | 50 | expr 367 | . . . . 5 |
52 | 10 | faccld 9663 | . . . . . . . . 9 |
53 | 52 | nncnd 8053 | . . . . . . . 8 |
54 | 53 | mulid2d 7137 | . . . . . . 7 |
55 | 43, 45 | syl 14 | . . . . . . . 8 |
56 | 55 | oveq2d 5548 | . . . . . . 7 |
57 | 54, 56 | eqtr3d 2115 | . . . . . 6 |
58 | fveq2 5198 | . . . . . . . . 9 | |
59 | fac0 9655 | . . . . . . . . 9 | |
60 | 58, 59 | syl6eq 2129 | . . . . . . . 8 |
61 | oveq1 5539 | . . . . . . . . . . 11 | |
62 | 0p1e1 8153 | . . . . . . . . . . 11 | |
63 | 61, 62 | syl6eq 2129 | . . . . . . . . . 10 |
64 | iseqeq1 9434 | . . . . . . . . . 10 | |
65 | 63, 64 | syl 14 | . . . . . . . . 9 |
66 | 65 | fveq1d 5200 | . . . . . . . 8 |
67 | 60, 66 | oveq12d 5550 | . . . . . . 7 |
68 | 67 | eqeq2d 2092 | . . . . . 6 |
69 | 57, 68 | syl5ibrcom 155 | . . . . 5 |
70 | fznn0sub 9075 | . . . . . . 7 | |
71 | 70 | adantl 271 | . . . . . 6 |
72 | elnn0 8290 | . . . . . 6 | |
73 | 71, 72 | sylib 120 | . . . . 5 |
74 | 51, 69, 73 | mpjaod 670 | . . . 4 |
75 | 74 | oveq1d 5547 | . . 3 |
76 | 28 | a1i 9 | . . . . 5 |
77 | vex 2604 | . . . . . . 7 | |
78 | fvi 5251 | . . . . . . 7 | |
79 | 77, 78 | ax-mp 7 | . . . . . 6 |
80 | eluzelcn 8630 | . . . . . . 7 | |
81 | 80 | adantl 271 | . . . . . 6 |
82 | 79, 81 | syl5eqel 2165 | . . . . 5 |
83 | mulcl 7100 | . . . . . 6 | |
84 | 83 | adantl 271 | . . . . 5 |
85 | 26, 76, 82, 84 | iseqcl 9443 | . . . 4 |
86 | 12 | nnnn0d 8341 | . . . . . 6 |
87 | 86 | faccld 9663 | . . . . 5 |
88 | 87 | nncnd 8053 | . . . 4 |
89 | 71 | faccld 9663 | . . . . 5 |
90 | 89 | nncnd 8053 | . . . 4 |
91 | 87 | nnap0d 8084 | . . . 4 # |
92 | 89 | nnap0d 8084 | . . . 4 # |
93 | 85, 88, 90, 91, 92 | divcanap5d 7903 | . . 3 |
94 | 2, 75, 93 | 3eqtrd 2117 | . 2 |
95 | simplr 496 | . . . . . . 7 | |
96 | 95 | nnnn0d 8341 | . . . . . 6 |
97 | 96 | faccld 9663 | . . . . 5 |
98 | 97 | nncnd 8053 | . . . 4 |
99 | 97 | nnap0d 8084 | . . . 4 # |
100 | 98, 99 | div0apd 7875 | . . 3 |
101 | mulcl 7100 | . . . . . 6 | |
102 | 101 | adantl 271 | . . . . 5 |
103 | eluzelcn 8630 | . . . . . . 7 | |
104 | 103 | adantl 271 | . . . . . 6 |
105 | 35, 104 | syl5eqel 2165 | . . . . 5 |
106 | 28 | a1i 9 | . . . . 5 |
107 | simpr 108 | . . . . . 6 | |
108 | 107 | mul02d 7496 | . . . . 5 |
109 | 107 | mul01d 7497 | . . . . 5 |
110 | simpr 108 | . . . . . . . . 9 | |
111 | nn0uz 8653 | . . . . . . . . . . . 12 | |
112 | 96, 111 | syl6eleq 2171 | . . . . . . . . . . 11 |
113 | simpll 495 | . . . . . . . . . . . 12 | |
114 | 113 | nn0zd 8467 | . . . . . . . . . . 11 |
115 | elfz5 9037 | . . . . . . . . . . 11 | |
116 | 112, 114, 115 | syl2anc 403 | . . . . . . . . . 10 |
117 | nn0re 8297 | . . . . . . . . . . . 12 | |
118 | 117 | ad2antrr 471 | . . . . . . . . . . 11 |
119 | nnre 8046 | . . . . . . . . . . . 12 | |
120 | 119 | ad2antlr 472 | . . . . . . . . . . 11 |
121 | 118, 120 | subge0d 7635 | . . . . . . . . . 10 |
122 | 116, 121 | bitr4d 189 | . . . . . . . . 9 |
123 | 110, 122 | mtbid 629 | . . . . . . . 8 |
124 | simpl 107 | . . . . . . . . . . . 12 | |
125 | 124 | nn0zd 8467 | . . . . . . . . . . 11 |
126 | simpr 108 | . . . . . . . . . . . 12 | |
127 | 126 | nnzd 8468 | . . . . . . . . . . 11 |
128 | 125, 127 | zsubcld 8474 | . . . . . . . . . 10 |
129 | 128 | adantr 270 | . . . . . . . . 9 |
130 | 0z 8362 | . . . . . . . . 9 | |
131 | zltnle 8397 | . . . . . . . . 9 | |
132 | 129, 130, 131 | sylancl 404 | . . . . . . . 8 |
133 | 123, 132 | mpbird 165 | . . . . . . 7 |
134 | zltp1le 8405 | . . . . . . . 8 | |
135 | 129, 130, 134 | sylancl 404 | . . . . . . 7 |
136 | 133, 135 | mpbid 145 | . . . . . 6 |
137 | nn0ge0 8313 | . . . . . . 7 | |
138 | 137 | ad2antrr 471 | . . . . . 6 |
139 | 0zd 8363 | . . . . . . 7 | |
140 | 129 | peano2zd 8472 | . . . . . . 7 |
141 | elfz 9035 | . . . . . . 7 | |
142 | 139, 140, 114, 141 | syl3anc 1169 | . . . . . 6 |
143 | 136, 138, 142 | mpbir2and 885 | . . . . 5 |
144 | elex 2610 | . . . . . 6 | |
145 | 144 | ad2antrr 471 | . . . . 5 |
146 | 0cn 7111 | . . . . . 6 | |
147 | fvi 5251 | . . . . . 6 | |
148 | 146, 147 | mp1i 10 | . . . . 5 |
149 | 102, 105, 106, 108, 109, 143, 145, 148 | iseqz 9469 | . . . 4 |
150 | 149 | oveq1d 5547 | . . 3 |
151 | nnz 8370 | . . . . 5 | |
152 | bcval3 9678 | . . . . 5 | |
153 | 151, 152 | syl3an2 1203 | . . . 4 |
154 | 153 | 3expa 1138 | . . 3 |
155 | 100, 150, 154 | 3eqtr4rd 2124 | . 2 |
156 | 0zd 8363 | . . . 4 | |
157 | fzdcel 9059 | . . . 4 DECID | |
158 | 127, 156, 125, 157 | syl3anc 1169 | . . 3 DECID |
159 | exmiddc 777 | . . 3 DECID | |
160 | 158, 159 | syl 14 | . 2 |
161 | 94, 155, 160 | mpjaodan 744 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 DECID wdc 775 w3a 919 wceq 1284 wcel 1433 cvv 2601 class class class wbr 3785 cid 4043 cfv 4922 (class class class)co 5532 cc 6979 cr 6980 cc0 6981 c1 6982 caddc 6984 cmul 6986 clt 7153 cle 7154 cmin 7279 cdiv 7760 cn 8039 cn0 8288 cz 8351 cuz 8619 cfz 9029 cseq 9431 cfa 9652 cbc 9674 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-q 8705 df-fz 9030 df-iseq 9432 df-fac 9653 df-bc 9675 |
This theorem is referenced by: bcn2 9691 |
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