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Mirrors > Home > ILE Home > Th. List > bccmpl | Unicode version |
Description: "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.) |
Ref | Expression |
---|---|
bccmpl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bcval2 10496 | . . . 4 | |
2 | fznn0sub2 9905 | . . . . . 6 | |
3 | bcval2 10496 | . . . . . 6 | |
4 | 2, 3 | syl 14 | . . . . 5 |
5 | elfznn0 9894 | . . . . . . . . . . 11 | |
6 | 5 | faccld 10482 | . . . . . . . . . 10 |
7 | 6 | nncnd 8734 | . . . . . . . . 9 |
8 | 2, 7 | syl 14 | . . . . . . . 8 |
9 | elfznn0 9894 | . . . . . . . . . 10 | |
10 | 9 | faccld 10482 | . . . . . . . . 9 |
11 | 10 | nncnd 8734 | . . . . . . . 8 |
12 | 8, 11 | mulcomd 7787 | . . . . . . 7 |
13 | elfz3nn0 9895 | . . . . . . . . . 10 | |
14 | elfzelz 9806 | . . . . . . . . . 10 | |
15 | nn0cn 8987 | . . . . . . . . . . 11 | |
16 | zcn 9059 | . . . . . . . . . . 11 | |
17 | nncan 7991 | . . . . . . . . . . 11 | |
18 | 15, 16, 17 | syl2an 287 | . . . . . . . . . 10 |
19 | 13, 14, 18 | syl2anc 408 | . . . . . . . . 9 |
20 | 19 | fveq2d 5425 | . . . . . . . 8 |
21 | 20 | oveq1d 5789 | . . . . . . 7 |
22 | 12, 21 | eqtr4d 2175 | . . . . . 6 |
23 | 22 | oveq2d 5790 | . . . . 5 |
24 | 4, 23 | eqtr4d 2175 | . . . 4 |
25 | 1, 24 | eqtr4d 2175 | . . 3 |
26 | 25 | adantl 275 | . 2 |
27 | bcval3 10497 | . . . 4 | |
28 | simp1 981 | . . . . 5 | |
29 | nn0z 9074 | . . . . . . 7 | |
30 | zsubcl 9095 | . . . . . . 7 | |
31 | 29, 30 | sylan 281 | . . . . . 6 |
32 | 31 | 3adant3 1001 | . . . . 5 |
33 | fznn0sub2 9905 | . . . . . . . 8 | |
34 | 18 | eleq1d 2208 | . . . . . . . 8 |
35 | 33, 34 | syl5ib 153 | . . . . . . 7 |
36 | 35 | con3d 620 | . . . . . 6 |
37 | 36 | 3impia 1178 | . . . . 5 |
38 | bcval3 10497 | . . . . 5 | |
39 | 28, 32, 37, 38 | syl3anc 1216 | . . . 4 |
40 | 27, 39 | eqtr4d 2175 | . . 3 |
41 | 40 | 3expa 1181 | . 2 |
42 | simpr 109 | . . . 4 | |
43 | 0zd 9066 | . . . 4 | |
44 | 29 | adantr 274 | . . . 4 |
45 | fzdcel 9820 | . . . 4 DECID | |
46 | 42, 43, 44, 45 | syl3anc 1216 | . . 3 DECID |
47 | exmiddc 821 | . . 3 DECID | |
48 | 46, 47 | syl 14 | . 2 |
49 | 26, 41, 48 | mpjaodan 787 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 w3a 962 wceq 1331 wcel 1480 cfv 5123 (class class class)co 5774 cc 7618 cc0 7620 cmul 7625 cmin 7933 cdiv 8432 cn0 8977 cz 9054 cfz 9790 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: bcnn 10503 bcnp1n 10505 bcp1m1 10511 bcnm1 10518 |
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