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Theorem bcval 11311
 Description: Value of the binomial coefficient, choose . Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when does not hold. See bcval2 11312 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
Assertion
Ref Expression
bcval
Dummy variables are mutually distinct and distinct from all other variables.

Proof of Theorem bcval
StepHypRef Expression
1 oveq2 5827 . . . 4
21eleq2d 2351 . . 3
3 fveq2 5485 . . . 4
4 oveq1 5826 . . . . . 6
54fveq2d 5489 . . . . 5
65oveq1d 5834 . . . 4
73, 6oveq12d 5837 . . 3
8 eqidd 2285 . . 3
92, 7, 8ifbieq12d 3588 . 2
10 eleq1 2344 . . 3
11 oveq2 5827 . . . . . 6
1211fveq2d 5489 . . . . 5
13 fveq2 5485 . . . . 5
1412, 13oveq12d 5837 . . . 4
1514oveq2d 5835 . . 3
16 eqidd 2285 . . 3
1710, 15, 16ifbieq12d 3588 . 2
18 df-bc 11310 . 2
19 ovex 5844 . . 3
20 c0ex 8827 . . 3
2119, 20ifex 3624 . 2
229, 17, 18, 21ovmpt2 5944 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360   wceq 1624   wcel 1685  cif 3566  cfv 5221  (class class class)co 5819  cc0 8732   cmul 8737   cmin 9032   cdiv 9418  cn0 9960  cz 10019  cfz 10776  cfa 11282   cbc 11309 This theorem is referenced by:  bcval2  11312  bcval3  11313 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-mulcl 8794  ax-i2m1 8800 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-bc 11310
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