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Theorem bcval 12428
 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 12429 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
Assertion
Ref Expression
bcval

Proof of Theorem bcval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6288 . . . 4
21eleq2d 2474 . . 3
3 fveq2 5851 . . . 4
4 oveq1 6287 . . . . . 6
54fveq2d 5855 . . . . 5
65oveq1d 6295 . . . 4
73, 6oveq12d 6298 . . 3
82, 7ifbieq1d 3910 . 2
9 eleq1 2476 . . 3
10 oveq2 6288 . . . . . 6
1110fveq2d 5855 . . . . 5
12 fveq2 5851 . . . . 5
1311, 12oveq12d 6298 . . . 4
1413oveq2d 6296 . . 3
159, 14ifbieq1d 3910 . 2
16 df-bc 12427 . 2
17 ovex 6308 . . 3
18 c0ex 9622 . . 3
1917, 18ifex 3955 . 2
208, 15, 16, 19ovmpt2 6421 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1407   wcel 1844  cif 3887  cfv 5571  (class class class)co 6280  cc0 9524   cmul 9529   cmin 9843   cdiv 10249  cn0 10838  cz 10907  cfz 11728  cfa 12399   cbc 12426 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-mulcl 9586  ax-i2m1 9592 This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-iota 5535  df-fun 5573  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-bc 12427 This theorem is referenced by:  bcval2  12429  bcval3  12430  bcneg1  29958  bccolsum  29961  fwddifnp1  30516
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