Theorem bcc0 38539

Description: The generalized binomial coefficient C choose K is zero iff C is an integer between zero and ( K - 1 ) inclusive. (Contributed by Steve Rodriguez, 22-Apr-2020.)

Hypotheses

Ref	Expression
bccval.c	|- ( ph -> C e. CC )
bccval.k	|- ( ph -> K e. NN0 )

Assertion

Ref	Expression
bcc0	|- ( ph -> ( ( CC𝑐 K ) = 0 <-> C e. ( 0 ... ( K - 1 ) ) ) )

Proof of Theorem bcc0

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bccval.c	. . . 4 |- ( ph -> C e. CC )
2		bccval.k	. . . 4 |- ( ph -> K e. NN0 )
3	1, 2	bccval 38537	. . 3 |- ( ph -> ( CC𝑐 K ) = ( ( C FallFac K ) / ( ! ` K ) ) )
4	3	eqeq1d 2624	. 2 |- ( ph -> ( ( CC𝑐 K ) = 0 <-> ( ( C FallFac K ) / ( ! ` K ) ) = 0 ) )
5		fallfaccl 14747	. . . 4 |- ( ( C e. CC /\ K e. NN0 ) -> ( C FallFac K ) e. CC )
6	1, 2, 5	syl2anc 693	. . 3 |- ( ph -> ( C FallFac K ) e. CC )
7		faccl 13070	. . . . 5 |- ( K e. NN0 -> ( ! ` K ) e. NN )
8	2, 7	syl 17	. . . 4 |- ( ph -> ( ! ` K ) e. NN )
9	8	nncnd 11036	. . 3 |- ( ph -> ( ! ` K ) e. CC )
10		facne0 13073	. . . 4 |- ( K e. NN0 -> ( ! ` K ) =/= 0 )
11	2, 10	syl 17	. . 3 |- ( ph -> ( ! ` K ) =/= 0 )
12	6, 9, 11	diveq0ad 10811	. 2 |- ( ph -> ( ( ( C FallFac K ) / ( ! ` K ) ) = 0 <-> ( C FallFac K ) = 0 ) )
13		fallfacval 14740	. . . . 5 |- ( ( C e. CC /\ K e. NN0 ) -> ( C FallFac K ) = prod_ k e. ( 0 ... ( K - 1 ) ) ( C - k ) )
14	1, 2, 13	syl2anc 693	. . . 4 |- ( ph -> ( C FallFac K ) = prod_ k e. ( 0 ... ( K - 1 ) ) ( C - k ) )
15	14	eqeq1d 2624	. . 3 |- ( ph -> ( ( C FallFac K ) = 0 <-> prod_ k e. ( 0 ... ( K - 1 ) ) ( C - k ) = 0 ) )
16		elfzuz3 12339	. . . . . . 7 |- ( C e. ( 0 ... ( K - 1 ) ) -> ( K - 1 ) e. ( ZZ>= ` C ) )
17	16	adantl 482	. . . . . 6 |- ( ( ph /\ C e. ( 0 ... ( K - 1 ) ) ) -> ( K - 1 ) e. ( ZZ>= ` C ) )
18		nn0uz 11722	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
19		elfznn0 12433	. . . . . . . 8 |- ( C e. ( 0 ... ( K - 1 ) ) -> C e. NN0 )
20	19	adantl 482	. . . . . . 7 |- ( ( ph /\ C e. ( 0 ... ( K - 1 ) ) ) -> C e. NN0 )
21	1	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ C e. ( 0 ... ( K - 1 ) ) ) /\ k e. NN0 ) -> C e. CC )
22		nn0cn 11302	. . . . . . . . 9 |- ( k e. NN0 -> k e. CC )
23	22	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ C e. ( 0 ... ( K - 1 ) ) ) /\ k e. NN0 ) -> k e. CC )
24	21, 23	subcld 10392	. . . . . . 7 |- ( ( ( ph /\ C e. ( 0 ... ( K - 1 ) ) ) /\ k e. NN0 ) -> ( C - k ) e. CC )
25	1	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ C e. ( 0 ... ( K - 1 ) ) ) /\ k = C ) -> C e. CC )
26		eqcom 2629	. . . . . . . . . 10 |- ( k = C <-> C = k )
27	26	biimpi 206	. . . . . . . . 9 |- ( k = C -> C = k )
28	27	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ C e. ( 0 ... ( K - 1 ) ) ) /\ k = C ) -> C = k )
29	25, 28	subeq0bd 10456	. . . . . . 7 |- ( ( ( ph /\ C e. ( 0 ... ( K - 1 ) ) ) /\ k = C ) -> ( C - k ) = 0 )
30	18, 20, 24, 29	fprodeq0 14705	. . . . . 6 |- ( ( ( ph /\ C e. ( 0 ... ( K - 1 ) ) ) /\ ( K - 1 ) e. ( ZZ>= ` C ) ) -> prod_ k e. ( 0 ... ( K - 1 ) ) ( C - k ) = 0 )
31	17, 30	mpdan 702	. . . . 5 |- ( ( ph /\ C e. ( 0 ... ( K - 1 ) ) ) -> prod_ k e. ( 0 ... ( K - 1 ) ) ( C - k ) = 0 )
32	31	ex 450	. . . 4 |- ( ph -> ( C e. ( 0 ... ( K - 1 ) ) -> prod_ k e. ( 0 ... ( K - 1 ) ) ( C - k ) = 0 ) )
33		fzfid 12772	. . . . . . 7 |- ( ( ph /\ -. C e. ( 0 ... ( K - 1 ) ) ) -> ( 0 ... ( K - 1 ) ) e. Fin )
34	1	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ -. C e. ( 0 ... ( K - 1 ) ) ) /\ k e. ( 0 ... ( K - 1 ) ) ) -> C e. CC )
35		elfznn0 12433	. . . . . . . . . 10 |- ( k e. ( 0 ... ( K - 1 ) ) -> k e. NN0 )
36	35	nn0cnd 11353	. . . . . . . . 9 |- ( k e. ( 0 ... ( K - 1 ) ) -> k e. CC )
37	36	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ -. C e. ( 0 ... ( K - 1 ) ) ) /\ k e. ( 0 ... ( K - 1 ) ) ) -> k e. CC )
38	34, 37	subcld 10392	. . . . . . 7 |- ( ( ( ph /\ -. C e. ( 0 ... ( K - 1 ) ) ) /\ k e. ( 0 ... ( K - 1 ) ) ) -> ( C - k ) e. CC )
39		nelne2 2891	. . . . . . . . . . 11 |- ( ( k e. ( 0 ... ( K - 1 ) ) /\ -. C e. ( 0 ... ( K - 1 ) ) ) -> k =/= C )
40	39	necomd 2849	. . . . . . . . . 10 |- ( ( k e. ( 0 ... ( K - 1 ) ) /\ -. C e. ( 0 ... ( K - 1 ) ) ) -> C =/= k )
41	40	ancoms 469	. . . . . . . . 9 |- ( ( -. C e. ( 0 ... ( K - 1 ) ) /\ k e. ( 0 ... ( K - 1 ) ) ) -> C =/= k )
42	41	adantll 750	. . . . . . . 8 |- ( ( ( ph /\ -. C e. ( 0 ... ( K - 1 ) ) ) /\ k e. ( 0 ... ( K - 1 ) ) ) -> C =/= k )
43	34, 37, 42	subne0d 10401	. . . . . . 7 |- ( ( ( ph /\ -. C e. ( 0 ... ( K - 1 ) ) ) /\ k e. ( 0 ... ( K - 1 ) ) ) -> ( C - k ) =/= 0 )
44	33, 38, 43	fprodn0 14709	. . . . . 6 |- ( ( ph /\ -. C e. ( 0 ... ( K - 1 ) ) ) -> prod_ k e. ( 0 ... ( K - 1 ) ) ( C - k ) =/= 0 )
45	44	ex 450	. . . . 5 |- ( ph -> ( -. C e. ( 0 ... ( K - 1 ) ) -> prod_ k e. ( 0 ... ( K - 1 ) ) ( C - k ) =/= 0 ) )
46	45	necon4bd 2814	. . . 4 |- ( ph -> ( prod_ k e. ( 0 ... ( K - 1 ) ) ( C - k ) = 0 -> C e. ( 0 ... ( K - 1 ) ) ) )
47	32, 46	impbid 202	. . 3 |- ( ph -> ( C e. ( 0 ... ( K - 1 ) ) <-> prod_ k e. ( 0 ... ( K - 1 ) ) ( C - k ) = 0 ) )
48	15, 47	bitr4d 271	. 2 |- ( ph -> ( ( C FallFac K ) = 0 <-> C e. ( 0 ... ( K - 1 ) ) ) )
49	4, 12, 48	3bitrd 294	1 |- ( ph -> ( ( CC𝑐 K ) = 0 <-> C e. ( 0 ... ( K - 1 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   !cfa 13060   prod_cprod 14635   FallFac cfallfac 14735  C_𝑐cbcc 38535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-fac 13061  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636  df-fallfac 14738  df-bcc 38536

This theorem is referenced by:  bccbc  38544  binomcxplemnn0  38548  binomcxplemfrat  38550  binomcxplemradcnv  38551