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Mirrors > Home > ILE Home > Th. List > bccl | Unicode version |
Description: A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.) |
Ref | Expression |
---|---|
bccl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5844 | . . . . 5 | |
2 | 1 | eleq1d 2233 | . . . 4 |
3 | 2 | ralbidv 2464 | . . 3 |
4 | oveq1 5844 | . . . . 5 | |
5 | 4 | eleq1d 2233 | . . . 4 |
6 | 5 | ralbidv 2464 | . . 3 |
7 | oveq1 5844 | . . . . 5 | |
8 | 7 | eleq1d 2233 | . . . 4 |
9 | 8 | ralbidv 2464 | . . 3 |
10 | oveq1 5844 | . . . . 5 | |
11 | 10 | eleq1d 2233 | . . . 4 |
12 | 11 | ralbidv 2464 | . . 3 |
13 | elfz1eq 9961 | . . . . . . 7 | |
14 | 13 | adantl 275 | . . . . . 6 |
15 | oveq2 5845 | . . . . . . 7 | |
16 | 0nn0 9121 | . . . . . . . . 9 | |
17 | bcn0 10658 | . . . . . . . . 9 | |
18 | 16, 17 | ax-mp 5 | . . . . . . . 8 |
19 | 1nn0 9122 | . . . . . . . 8 | |
20 | 18, 19 | eqeltri 2237 | . . . . . . 7 |
21 | 15, 20 | eqeltrdi 2255 | . . . . . 6 |
22 | 14, 21 | syl 14 | . . . . 5 |
23 | bcval3 10654 | . . . . . . 7 | |
24 | 16, 23 | mp3an1 1313 | . . . . . 6 |
25 | 24, 16 | eqeltrdi 2255 | . . . . 5 |
26 | 0zd 9195 | . . . . . 6 | |
27 | fzdcel 9966 | . . . . . . 7 DECID | |
28 | exmiddc 826 | . . . . . . 7 DECID | |
29 | 27, 28 | syl 14 | . . . . . 6 |
30 | 26, 26, 29 | mpd3an23 1328 | . . . . 5 |
31 | 22, 25, 30 | mpjaodan 788 | . . . 4 |
32 | 31 | rgen 2517 | . . 3 |
33 | oveq2 5845 | . . . . . 6 | |
34 | 33 | eleq1d 2233 | . . . . 5 |
35 | 34 | cbvralv 2690 | . . . 4 |
36 | bcpasc 10669 | . . . . . . . 8 | |
37 | 36 | adantlr 469 | . . . . . . 7 |
38 | oveq2 5845 | . . . . . . . . . . 11 | |
39 | 38 | eleq1d 2233 | . . . . . . . . . 10 |
40 | 39 | rspccva 2825 | . . . . . . . . 9 |
41 | peano2zm 9221 | . . . . . . . . . 10 | |
42 | oveq2 5845 | . . . . . . . . . . . 12 | |
43 | 42 | eleq1d 2233 | . . . . . . . . . . 11 |
44 | 43 | rspccva 2825 | . . . . . . . . . 10 |
45 | 41, 44 | sylan2 284 | . . . . . . . . 9 |
46 | 40, 45 | nn0addcld 9163 | . . . . . . . 8 |
47 | 46 | adantll 468 | . . . . . . 7 |
48 | 37, 47 | eqeltrrd 2242 | . . . . . 6 |
49 | 48 | ralrimiva 2537 | . . . . 5 |
50 | 49 | ex 114 | . . . 4 |
51 | 35, 50 | syl5bi 151 | . . 3 |
52 | 3, 6, 9, 12, 32, 51 | nn0ind 9297 | . 2 |
53 | oveq2 5845 | . . . 4 | |
54 | 53 | eleq1d 2233 | . . 3 |
55 | 54 | rspccva 2825 | . 2 |
56 | 52, 55 | sylan 281 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 DECID wdc 824 w3a 967 wceq 1342 wcel 2135 wral 2442 (class class class)co 5837 cc0 7745 c1 7746 caddc 7748 cmin 8061 cn0 9106 cz 9183 cfz 9936 cbc 10650 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-mulrcl 7844 ax-addcom 7845 ax-mulcom 7846 ax-addass 7847 ax-mulass 7848 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-1rid 7852 ax-0id 7853 ax-rnegex 7854 ax-precex 7855 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-apti 7860 ax-pre-ltadd 7861 ax-pre-mulgt0 7862 ax-pre-mulext 7863 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-if 3517 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-po 4269 df-iso 4270 df-iord 4339 df-on 4341 df-ilim 4342 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-frec 6351 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-reap 8465 df-ap 8472 df-div 8561 df-inn 8850 df-n0 9107 df-z 9184 df-uz 9459 df-q 9550 df-rp 9582 df-fz 9937 df-seqfrec 10372 df-fac 10629 df-bc 10651 |
This theorem is referenced by: bccl2 10671 bcn2m1 10672 bcn2p1 10673 binomlem 11414 bcxmas 11420 |
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