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Theorem bcval 11550
 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 11551 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 6048 . . . 4
21eleq2d 2471 . . 3
3 fveq2 5687 . . . 4
4 oveq1 6047 . . . . . 6
54fveq2d 5691 . . . . 5
65oveq1d 6055 . . . 4
73, 6oveq12d 6058 . . 3
8 eqidd 2405 . . 3
92, 7, 8ifbieq12d 3721 . 2
10 eleq1 2464 . . 3
11 oveq2 6048 . . . . . 6
1211fveq2d 5691 . . . . 5
13 fveq2 5687 . . . . 5
1412, 13oveq12d 6058 . . . 4
1514oveq2d 6056 . . 3
16 eqidd 2405 . . 3
1710, 15, 16ifbieq12d 3721 . 2
18 df-bc 11549 . 2
19 ovex 6065 . . 3
20 c0ex 9041 . . 3
2119, 20ifex 3757 . 2
229, 17, 18, 21ovmpt2 6168 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1649   wcel 1721  cif 3699  cfv 5413  (class class class)co 6040  cc0 8946   cmul 8951   cmin 9247   cdiv 9633  cn0 10177  cz 10238  cfz 10999  cfa 11521   cbc 11548 This theorem is referenced by:  bcval2  11551  bcval3  11552 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-mulcl 9008  ax-i2m1 9014 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-bc 11549
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