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Mirrors > Home > ILE Home > Th. List > bcn1 | Unicode version |
Description: Binomial coefficient:
![]() ![]() |
Ref | Expression |
---|---|
bcn1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 9209 |
. 2
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2 | 1eluzge0 9606 |
. . . . . . 7
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3 | 2 | a1i 9 |
. . . . . 6
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4 | elnnuz 9596 |
. . . . . . 7
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5 | 4 | biimpi 120 |
. . . . . 6
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6 | elfzuzb 10051 |
. . . . . 6
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7 | 3, 5, 6 | sylanbrc 417 |
. . . . 5
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8 | bcval2 10765 |
. . . . 5
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9 | 7, 8 | syl 14 |
. . . 4
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10 | facnn2 10749 |
. . . . 5
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11 | fac1 10744 |
. . . . . . 7
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12 | 11 | oveq2i 5908 |
. . . . . 6
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13 | nnm1nn0 9248 |
. . . . . . . . 9
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14 | 13 | faccld 10751 |
. . . . . . . 8
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15 | 14 | nncnd 8964 |
. . . . . . 7
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16 | 15 | mulridd 8005 |
. . . . . 6
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17 | 12, 16 | eqtrid 2234 |
. . . . 5
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18 | 10, 17 | oveq12d 5915 |
. . . 4
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19 | nncn 8958 |
. . . . 5
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20 | 14 | nnap0d 8996 |
. . . . 5
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21 | 19, 15, 20 | divcanap3d 8783 |
. . . 4
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22 | 9, 18, 21 | 3eqtrd 2226 |
. . 3
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23 | 0nn0 9222 |
. . . . 5
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24 | 1z 9310 |
. . . . 5
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25 | 0lt1 8115 |
. . . . . 6
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26 | 25 | olci 733 |
. . . . 5
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27 | bcval4 10767 |
. . . . 5
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28 | 23, 24, 26, 27 | mp3an 1348 |
. . . 4
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29 | oveq1 5904 |
. . . . 5
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30 | eqeq12 2202 |
. . . . 5
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31 | 29, 30 | mpancom 422 |
. . . 4
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32 | 28, 31 | mpbiri 168 |
. . 3
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33 | 22, 32 | jaoi 717 |
. 2
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34 | 1, 33 | sylbi 121 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-frec 6417 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-n0 9208 df-z 9285 df-uz 9560 df-q 9652 df-fz 10041 df-seqfrec 10479 df-fac 10741 df-bc 10763 |
This theorem is referenced by: bcnp1n 10774 bcn2m1 10784 bcn2p1 10785 bcnm1 10787 |
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