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| Mirrors > Home > ILE Home > Th. List > bcm1k | Unicode version | ||
| Description: The proportion of one
binomial coefficient to another with |
| Ref | Expression |
|---|---|
| bcm1k |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfzuz2 10133 |
. . . . . . . . 9
| |
| 2 | nnuz 9666 |
. . . . . . . . 9
| |
| 3 | 1, 2 | eleqtrrdi 2298 |
. . . . . . . 8
|
| 4 | 3 | nnnn0d 9330 |
. . . . . . 7
|
| 5 | 4 | faccld 10862 |
. . . . . 6
|
| 6 | 5 | nncnd 9032 |
. . . . 5
|
| 7 | fznn0sub 10161 |
. . . . . . . . . 10
| |
| 8 | nn0p1nn 9316 |
. . . . . . . . . 10
| |
| 9 | 7, 8 | syl 14 |
. . . . . . . . 9
|
| 10 | 9 | nnnn0d 9330 |
. . . . . . . 8
|
| 11 | 10 | faccld 10862 |
. . . . . . 7
|
| 12 | elfznn 10158 |
. . . . . . . 8
| |
| 13 | nnm1nn0 9318 |
. . . . . . . 8
| |
| 14 | faccl 10861 |
. . . . . . . 8
| |
| 15 | 12, 13, 14 | 3syl 17 |
. . . . . . 7
|
| 16 | 11, 15 | nnmulcld 9067 |
. . . . . 6
|
| 17 | 16 | nncnd 9032 |
. . . . 5
|
| 18 | 9 | nncnd 9032 |
. . . . 5
|
| 19 | 12 | nncnd 9032 |
. . . . 5
|
| 20 | 16 | nnap0d 9064 |
. . . . 5
|
| 21 | 12 | nnap0d 9064 |
. . . . 5
|
| 22 | 6, 17, 18, 19, 20, 21 | divmuldivapd 8887 |
. . . 4
|
| 23 | elfzel2 10127 |
. . . . . . . . . 10
| |
| 24 | 23 | zcnd 9478 |
. . . . . . . . 9
|
| 25 | 1cnd 8070 |
. . . . . . . . 9
| |
| 26 | 24, 19, 25 | subsubd 8393 |
. . . . . . . 8
|
| 27 | 26 | fveq2d 5574 |
. . . . . . 7
|
| 28 | 27 | oveq1d 5949 |
. . . . . 6
|
| 29 | 28 | oveq2d 5950 |
. . . . 5
|
| 30 | 26 | oveq1d 5949 |
. . . . 5
|
| 31 | 29, 30 | oveq12d 5952 |
. . . 4
|
| 32 | facp1 10856 |
. . . . . . . . 9
| |
| 33 | 7, 32 | syl 14 |
. . . . . . . 8
|
| 34 | 33 | eqcomd 2210 |
. . . . . . 7
|
| 35 | facnn2 10860 |
. . . . . . . 8
| |
| 36 | 12, 35 | syl 14 |
. . . . . . 7
|
| 37 | 34, 36 | oveq12d 5952 |
. . . . . 6
|
| 38 | 7 | faccld 10862 |
. . . . . . . 8
|
| 39 | 38 | nncnd 9032 |
. . . . . . 7
|
| 40 | 12 | nnnn0d 9330 |
. . . . . . . . 9
|
| 41 | 40 | faccld 10862 |
. . . . . . . 8
|
| 42 | 41 | nncnd 9032 |
. . . . . . 7
|
| 43 | 39, 42, 18 | mul32d 8207 |
. . . . . 6
|
| 44 | 11 | nncnd 9032 |
. . . . . . 7
|
| 45 | 15 | nncnd 9032 |
. . . . . . 7
|
| 46 | 44, 45, 19 | mulassd 8078 |
. . . . . 6
|
| 47 | 37, 43, 46 | 3eqtr4d 2247 |
. . . . 5
|
| 48 | 47 | oveq2d 5950 |
. . . 4
|
| 49 | 22, 31, 48 | 3eqtr4d 2247 |
. . 3
|
| 50 | 6, 18 | mulcomd 8076 |
. . . 4
|
| 51 | 38, 41 | nnmulcld 9067 |
. . . . . 6
|
| 52 | 51 | nncnd 9032 |
. . . . 5
|
| 53 | 52, 18 | mulcomd 8076 |
. . . 4
|
| 54 | 50, 53 | oveq12d 5952 |
. . 3
|
| 55 | 51 | nnap0d 9064 |
. . . 4
|
| 56 | 9 | nnap0d 9064 |
. . . 4
|
| 57 | 6, 52, 18, 55, 56 | divcanap5d 8872 |
. . 3
|
| 58 | 49, 54, 57 | 3eqtrrd 2242 |
. 2
|
| 59 | 0p1e1 9132 |
. . . . . 6
| |
| 60 | 59 | oveq1i 5944 |
. . . . 5
|
| 61 | 0z 9365 |
. . . . . 6
| |
| 62 | fzp1ss 10177 |
. . . . . 6
| |
| 63 | 61, 62 | ax-mp 5 |
. . . . 5
|
| 64 | 60, 63 | eqsstrri 3225 |
. . . 4
|
| 65 | 64 | sseli 3188 |
. . 3
|
| 66 | bcval2 10876 |
. . 3
| |
| 67 | 65, 66 | syl 14 |
. 2
|
| 68 | ax-1cn 8000 |
. . . . . . . 8
| |
| 69 | npcan 8263 |
. . . . . . . 8
| |
| 70 | 24, 68, 69 | sylancl 413 |
. . . . . . 7
|
| 71 | peano2zm 9392 |
. . . . . . . 8
| |
| 72 | uzid 9644 |
. . . . . . . 8
| |
| 73 | peano2uz 9686 |
. . . . . . . 8
| |
| 74 | 23, 71, 72, 73 | 4syl 18 |
. . . . . . 7
|
| 75 | 70, 74 | eqeltrrd 2282 |
. . . . . 6
|
| 76 | fzss2 10168 |
. . . . . 6
| |
| 77 | 75, 76 | syl 14 |
. . . . 5
|
| 78 | elfzmlbm 10235 |
. . . . 5
| |
| 79 | 77, 78 | sseldd 3193 |
. . . 4
|
| 80 | bcval2 10876 |
. . . 4
| |
| 81 | 79, 80 | syl 14 |
. . 3
|
| 82 | 81 | oveq1d 5949 |
. 2
|
| 83 | 58, 67, 82 | 3eqtr4d 2247 |
1
|
| 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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-iinf 4634 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-mulrcl 8006 ax-addcom 8007 ax-mulcom 8008 ax-addass 8009 ax-mulass 8010 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-1rid 8014 ax-0id 8015 ax-rnegex 8016 ax-precex 8017 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-apti 8022 ax-pre-ltadd 8023 ax-pre-mulgt0 8024 ax-pre-mulext 8025 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4338 df-po 4341 df-iso 4342 df-iord 4411 df-on 4413 df-ilim 4414 df-suc 4416 df-iom 4637 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-res 4685 df-ima 4686 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-f1 5273 df-fo 5274 df-f1o 5275 df-fv 5276 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-1st 6216 df-2nd 6217 df-recs 6381 df-frec 6467 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-reap 8630 df-ap 8637 df-div 8728 df-inn 9019 df-n0 9278 df-z 9355 df-uz 9631 df-q 9723 df-fz 10113 df-seqfrec 10574 df-fac 10852 df-bc 10874 |
| This theorem is referenced by: bcp1nk 10888 bcpasc 10892 |
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