Theorem fprodfac 12126

Description: Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)

Assertion

Ref	Expression
fprodfac	|- ( A e. NN0 -> ( ! ` A ) = prod_ k e. ( 1 ... A ) k )

Distinct variable group:   A, k

Proof of Theorem fprodfac

Dummy variables w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5627	. . 3 |- ( w = 0 -> ( ! ` w ) = ( ! ` 0 ) )
2		oveq2 6009	. . . 4 |- ( w = 0 -> ( 1 ... w ) = ( 1 ... 0 ) )
3	2	prodeq1d 12075	. . 3 |- ( w = 0 -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... 0 ) k )
4	1, 3	eqeq12d 2244	. 2 |- ( w = 0 -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` 0 ) = prod_ k e. ( 1 ... 0 ) k ) )
5		fveq2 5627	. . 3 |- ( w = x -> ( ! ` w ) = ( ! ` x ) )
6		oveq2 6009	. . . 4 |- ( w = x -> ( 1 ... w ) = ( 1 ... x ) )
7	6	prodeq1d 12075	. . 3 |- ( w = x -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... x ) k )
8	5, 7	eqeq12d 2244	. 2 |- ( w = x -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` x ) = prod_ k e. ( 1 ... x ) k ) )
9		fveq2 5627	. . 3 |- ( w = ( x + 1 ) -> ( ! ` w ) = ( ! ` ( x + 1 ) ) )
10		oveq2 6009	. . . 4 |- ( w = ( x + 1 ) -> ( 1 ... w ) = ( 1 ... ( x + 1 ) ) )
11	10	prodeq1d 12075	. . 3 |- ( w = ( x + 1 ) -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... ( x + 1 ) ) k )
12	9, 11	eqeq12d 2244	. 2 |- ( w = ( x + 1 ) -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` ( x + 1 ) ) = prod_ k e. ( 1 ... ( x + 1 ) ) k ) )
13		fveq2 5627	. . 3 |- ( w = A -> ( ! ` w ) = ( ! ` A ) )
14		oveq2 6009	. . . 4 |- ( w = A -> ( 1 ... w ) = ( 1 ... A ) )
15	14	prodeq1d 12075	. . 3 |- ( w = A -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... A ) k )
16	13, 15	eqeq12d 2244	. 2 |- ( w = A -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` A ) = prod_ k e. ( 1 ... A ) k ) )
17		prod0 12096	. . 3 |- prod_ k e. (/) k = 1
18		fz10 10242	. . . 4 |- ( 1 ... 0 ) = (/)
19	18	prodeq1i 12072	. . 3 |- prod_ k e. ( 1 ... 0 ) k = prod_ k e. (/) k
20		fac0 10950	. . 3 |- ( ! ` 0 ) = 1
21	17, 19, 20	3eqtr4ri 2261	. 2 |- ( ! ` 0 ) = prod_ k e. ( 1 ... 0 ) k
22		elnn0 9371	. . 3 |- ( x e. NN0 <-> ( x e. NN \/ x = 0 ) )
23		simpr 110	. . . . . . 7 |- ( ( x e. NN /\ ( ! ` x ) = prod_ k e. ( 1 ... x ) k ) -> ( ! ` x ) = prod_ k e. ( 1 ... x ) k )
24	23	oveq1d 6016	. . . . . 6 |- ( ( x e. NN /\ ( ! ` x ) = prod_ k e. ( 1 ... x ) k ) -> ( ( ! ` x ) x. ( x + 1 ) ) = ( prod_ k e. ( 1 ... x ) k x. ( x + 1 ) ) )
25		nnnn0 9376	. . . . . . . . 9 |- ( x e. NN -> x e. NN0 )
26		facp1 10952	. . . . . . . . 9 |- ( x e. NN0 -> ( ! ` ( x + 1 ) ) = ( ( ! ` x ) x. ( x + 1 ) ) )
27	25, 26	syl 14	. . . . . . . 8 |- ( x e. NN -> ( ! ` ( x + 1 ) ) = ( ( ! ` x ) x. ( x + 1 ) ) )
28		elnnuz 9759	. . . . . . . . . 10 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
29	28	biimpi 120	. . . . . . . . 9 |- ( x e. NN -> x e. ( ZZ>= ` 1 ) )
30		elfzelz 10221	. . . . . . . . . . 11 |- ( k e. ( 1 ... ( x + 1 ) ) -> k e. ZZ )
31	30	zcnd 9570	. . . . . . . . . 10 |- ( k e. ( 1 ... ( x + 1 ) ) -> k e. CC )
32	31	adantl 277	. . . . . . . . 9 |- ( ( x e. NN /\ k e. ( 1 ... ( x + 1 ) ) ) -> k e. CC )
33		id 19	. . . . . . . . 9 |- ( k = ( x + 1 ) -> k = ( x + 1 ) )
34	29, 32, 33	fprodp1 12111	. . . . . . . 8 |- ( x e. NN -> prod_ k e. ( 1 ... ( x + 1 ) ) k = ( prod_ k e. ( 1 ... x ) k x. ( x + 1 ) ) )
35	27, 34	eqeq12d 2244	. . . . . . 7 |- ( x e. NN -> ( ( ! ` ( x + 1 ) ) = prod_ k e. ( 1 ... ( x + 1 ) ) k <-> ( ( ! ` x ) x. ( x + 1 ) ) = ( prod_ k e. ( 1 ... x ) k x. ( x + 1 ) ) ) )
36	35	adantr 276	. . . . . 6 |- ( ( x e. NN /\ ( ! ` x ) = prod_ k e. ( 1 ... x ) k ) -> ( ( ! ` ( x + 1 ) ) = prod_ k e. ( 1 ... ( x + 1 ) ) k <-> ( ( ! ` x ) x. ( x + 1 ) ) = ( prod_ k e. ( 1 ... x ) k x. ( x + 1 ) ) ) )
37	24, 36	mpbird 167	. . . . 5 |- ( ( x e. NN /\ ( ! ` x ) = prod_ k e. ( 1 ... x ) k ) -> ( ! ` ( x + 1 ) ) = prod_ k e. ( 1 ... ( x + 1 ) ) k )
38	37	ex 115	. . . 4 |- ( x e. NN -> ( ( ! ` x ) = prod_ k e. ( 1 ... x ) k -> ( ! ` ( x + 1 ) ) = prod_ k e. ( 1 ... ( x + 1 ) ) k ) )
39		1zzd 9473	. . . . . . 7 |- ( x = 0 -> 1 e. ZZ )
40		1cnd 8162	. . . . . . 7 |- ( x = 0 -> 1 e. CC )
41		id 19	. . . . . . . 8 |- ( k = 1 -> k = 1 )
42	41	fprod1 12105	. . . . . . 7 |- ( ( 1 e. ZZ /\ 1 e. CC ) -> prod_ k e. ( 1 ... 1 ) k = 1 )
43	39, 40, 42	syl2anc 411	. . . . . 6 |- ( x = 0 -> prod_ k e. ( 1 ... 1 ) k = 1 )
44		oveq1 6008	. . . . . . . . 9 |- ( x = 0 -> ( x + 1 ) = ( 0 + 1 ) )
45		0p1e1 9224	. . . . . . . . 9 |- ( 0 + 1 ) = 1
46	44, 45	eqtrdi 2278	. . . . . . . 8 |- ( x = 0 -> ( x + 1 ) = 1 )
47	46	oveq2d 6017	. . . . . . 7 |- ( x = 0 -> ( 1 ... ( x + 1 ) ) = ( 1 ... 1 ) )
48	47	prodeq1d 12075	. . . . . 6 |- ( x = 0 -> prod_ k e. ( 1 ... ( x + 1 ) ) k = prod_ k e. ( 1 ... 1 ) k )
49		fv0p1e1 9225	. . . . . . 7 |- ( x = 0 -> ( ! ` ( x + 1 ) ) = ( ! ` 1 ) )
50		fac1 10951	. . . . . . 7 |- ( ! ` 1 ) = 1
51	49, 50	eqtrdi 2278	. . . . . 6 |- ( x = 0 -> ( ! ` ( x + 1 ) ) = 1 )
52	43, 48, 51	3eqtr4rd 2273	. . . . 5 |- ( x = 0 -> ( ! ` ( x + 1 ) ) = prod_ k e. ( 1 ... ( x + 1 ) ) k )
53	52	a1d 22	. . . 4 |- ( x = 0 -> ( ( ! ` x ) = prod_ k e. ( 1 ... x ) k -> ( ! ` ( x + 1 ) ) = prod_ k e. ( 1 ... ( x + 1 ) ) k ) )
54	38, 53	jaoi 721	. . 3 |- ( ( x e. NN \/ x = 0 ) -> ( ( ! ` x ) = prod_ k e. ( 1 ... x ) k -> ( ! ` ( x + 1 ) ) = prod_ k e. ( 1 ... ( x + 1 ) ) k ) )
55	22, 54	sylbi 121	. 2 |- ( x e. NN0 -> ( ( ! ` x ) = prod_ k e. ( 1 ... x ) k -> ( ! ` ( x + 1 ) ) = prod_ k e. ( 1 ... ( x + 1 ) ) k ) )
56	4, 8, 12, 16, 21, 55	nn0ind 9561	1 |- ( A e. NN0 -> ( ! ` A ) = prod_ k e. ( 1 ... A ) k )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   (/)c0 3491   ` cfv 5318  (class class class)co 6001   CCcc 7997   0cc0 7999   1c1 8000   + caddc 8002   x. cmul 8004   NNcn 9110   NN0cn0 9369   ZZcz 9446   ZZ>=cuz 9722   ...cfz 10204   !cfa 10947   prod_cprod 12061

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-exp 10761  df-fac 10948  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-proddc 12062

This theorem is referenced by:  gausslemma2dlem1  15740  gausslemma2dlem6  15746