Theorem fprodfac 11780

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 5558	. . 3 |- ( w = 0 -> ( ! ` w ) = ( ! ` 0 ) )
2		oveq2 5930	. . . 4 |- ( w = 0 -> ( 1 ... w ) = ( 1 ... 0 ) )
3	2	prodeq1d 11729	. . 3 |- ( w = 0 -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... 0 ) k )
4	1, 3	eqeq12d 2211	. 2 |- ( w = 0 -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` 0 ) = prod_ k e. ( 1 ... 0 ) k ) )
5		fveq2 5558	. . 3 |- ( w = x -> ( ! ` w ) = ( ! ` x ) )
6		oveq2 5930	. . . 4 |- ( w = x -> ( 1 ... w ) = ( 1 ... x ) )
7	6	prodeq1d 11729	. . 3 |- ( w = x -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... x ) k )
8	5, 7	eqeq12d 2211	. 2 |- ( w = x -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` x ) = prod_ k e. ( 1 ... x ) k ) )
9		fveq2 5558	. . 3 |- ( w = ( x + 1 ) -> ( ! ` w ) = ( ! ` ( x + 1 ) ) )
10		oveq2 5930	. . . 4 |- ( w = ( x + 1 ) -> ( 1 ... w ) = ( 1 ... ( x + 1 ) ) )
11	10	prodeq1d 11729	. . 3 |- ( w = ( x + 1 ) -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... ( x + 1 ) ) k )
12	9, 11	eqeq12d 2211	. 2 |- ( w = ( x + 1 ) -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` ( x + 1 ) ) = prod_ k e. ( 1 ... ( x + 1 ) ) k ) )
13		fveq2 5558	. . 3 |- ( w = A -> ( ! ` w ) = ( ! ` A ) )
14		oveq2 5930	. . . 4 |- ( w = A -> ( 1 ... w ) = ( 1 ... A ) )
15	14	prodeq1d 11729	. . 3 |- ( w = A -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... A ) k )
16	13, 15	eqeq12d 2211	. 2 |- ( w = A -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` A ) = prod_ k e. ( 1 ... A ) k ) )
17		prod0 11750	. . 3 |- prod_ k e. (/) k = 1
18		fz10 10121	. . . 4 |- ( 1 ... 0 ) = (/)
19	18	prodeq1i 11726	. . 3 |- prod_ k e. ( 1 ... 0 ) k = prod_ k e. (/) k
20		fac0 10820	. . 3 |- ( ! ` 0 ) = 1
21	17, 19, 20	3eqtr4ri 2228	. 2 |- ( ! ` 0 ) = prod_ k e. ( 1 ... 0 ) k
22		elnn0 9251	. . 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 5937	. . . . . 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 9256	. . . . . . . . 9 |- ( x e. NN -> x e. NN0 )
26		facp1 10822	. . . . . . . . 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 9638	. . . . . . . . . 10 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
29	28	biimpi 120	. . . . . . . . 9 |- ( x e. NN -> x e. ( ZZ>= ` 1 ) )
30		elfzelz 10100	. . . . . . . . . . 11 |- ( k e. ( 1 ... ( x + 1 ) ) -> k e. ZZ )
31	30	zcnd 9449	. . . . . . . . . 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 11765	. . . . . . . 8 |- ( x e. NN -> prod_ k e. ( 1 ... ( x + 1 ) ) k = ( prod_ k e. ( 1 ... x ) k x. ( x + 1 ) ) )
35	27, 34	eqeq12d 2211	. . . . . . 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 9353	. . . . . . 7 |- ( x = 0 -> 1 e. ZZ )
40		1cnd 8042	. . . . . . 7 |- ( x = 0 -> 1 e. CC )
41		id 19	. . . . . . . 8 |- ( k = 1 -> k = 1 )
42	41	fprod1 11759	. . . . . . 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 5929	. . . . . . . . 9 |- ( x = 0 -> ( x + 1 ) = ( 0 + 1 ) )
45		0p1e1 9104	. . . . . . . . 9 |- ( 0 + 1 ) = 1
46	44, 45	eqtrdi 2245	. . . . . . . 8 |- ( x = 0 -> ( x + 1 ) = 1 )
47	46	oveq2d 5938	. . . . . . 7 |- ( x = 0 -> ( 1 ... ( x + 1 ) ) = ( 1 ... 1 ) )
48	47	prodeq1d 11729	. . . . . 6 |- ( x = 0 -> prod_ k e. ( 1 ... ( x + 1 ) ) k = prod_ k e. ( 1 ... 1 ) k )
49		fv0p1e1 9105	. . . . . . 7 |- ( x = 0 -> ( ! ` ( x + 1 ) ) = ( ! ` 1 ) )
50		fac1 10821	. . . . . . 7 |- ( ! ` 1 ) = 1
51	49, 50	eqtrdi 2245	. . . . . 6 |- ( x = 0 -> ( ! ` ( x + 1 ) ) = 1 )
52	43, 48, 51	3eqtr4rd 2240	. . . . 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 717	. . 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 9440	1 |- ( A e. NN0 -> ( ! ` A ) = prod_ k e. ( 1 ... A ) k )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2167   (/)c0 3450   ` cfv 5258  (class class class)co 5922   CCcc 7877   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   NNcn 8990   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083   !cfa 10817   prod_cprod 11715

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-fac 10818  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-proddc 11716

This theorem is referenced by:  gausslemma2dlem1  15302  gausslemma2dlem6  15308