Theorem fprodfac 12360

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 5690	. . 3 |- ( w = 0 -> ( ! ` w ) = ( ! ` 0 ) )
2		oveq2 6083	. . . 4 |- ( w = 0 -> ( 1 ... w ) = ( 1 ... 0 ) )
3	2	prodeq1d 12309	. . 3 |- ( w = 0 -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... 0 ) k )
4	1, 3	eqeq12d 2253	. 2 |- ( w = 0 -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` 0 ) = prod_ k e. ( 1 ... 0 ) k ) )
5		fveq2 5690	. . 3 |- ( w = x -> ( ! ` w ) = ( ! ` x ) )
6		oveq2 6083	. . . 4 |- ( w = x -> ( 1 ... w ) = ( 1 ... x ) )
7	6	prodeq1d 12309	. . 3 |- ( w = x -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... x ) k )
8	5, 7	eqeq12d 2253	. 2 |- ( w = x -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` x ) = prod_ k e. ( 1 ... x ) k ) )
9		fveq2 5690	. . 3 |- ( w = ( x + 1 ) -> ( ! ` w ) = ( ! ` ( x + 1 ) ) )
10		oveq2 6083	. . . 4 |- ( w = ( x + 1 ) -> ( 1 ... w ) = ( 1 ... ( x + 1 ) ) )
11	10	prodeq1d 12309	. . 3 |- ( w = ( x + 1 ) -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... ( x + 1 ) ) k )
12	9, 11	eqeq12d 2253	. 2 |- ( w = ( x + 1 ) -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` ( x + 1 ) ) = prod_ k e. ( 1 ... ( x + 1 ) ) k ) )
13		fveq2 5690	. . 3 |- ( w = A -> ( ! ` w ) = ( ! ` A ) )
14		oveq2 6083	. . . 4 |- ( w = A -> ( 1 ... w ) = ( 1 ... A ) )
15	14	prodeq1d 12309	. . 3 |- ( w = A -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... A ) k )
16	13, 15	eqeq12d 2253	. 2 |- ( w = A -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` A ) = prod_ k e. ( 1 ... A ) k ) )
17		prod0 12330	. . 3 |- prod_ k e. (/) k = 1
18		fz10 10429	. . . 4 |- ( 1 ... 0 ) = (/)
19	18	prodeq1i 12306	. . 3 |- prod_ k e. ( 1 ... 0 ) k = prod_ k e. (/) k
20		fac0 11144	. . 3 |- ( ! ` 0 ) = 1
21	17, 19, 20	3eqtr4ri 2270	. 2 |- ( ! ` 0 ) = prod_ k e. ( 1 ... 0 ) k
22		elnn0 9544	. . 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 6090	. . . . . 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 9549	. . . . . . . . 9 |- ( x e. NN -> x e. NN0 )
26		facp1 11146	. . . . . . . . 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 9938	. . . . . . . . . 10 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
29	28	biimpi 120	. . . . . . . . 9 |- ( x e. NN -> x e. ( ZZ>= ` 1 ) )
30		elfzelz 10407	. . . . . . . . . . 11 |- ( k e. ( 1 ... ( x + 1 ) ) -> k e. ZZ )
31	30	zcnd 9748	. . . . . . . . . 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 12345	. . . . . . . 8 |- ( x e. NN -> prod_ k e. ( 1 ... ( x + 1 ) ) k = ( prod_ k e. ( 1 ... x ) k x. ( x + 1 ) ) )
35	27, 34	eqeq12d 2253	. . . . . . 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 9650	. . . . . . 7 |- ( x = 0 -> 1 e. ZZ )
40		1cnd 8332	. . . . . . 7 |- ( x = 0 -> 1 e. CC )
41		id 19	. . . . . . . 8 |- ( k = 1 -> k = 1 )
42	41	fprod1 12339	. . . . . . 7 |- ( ( 1 e. ZZ /\ 1 e. CC ) -> prod_ k e. ( 1 ... 1 ) k = 1 )
43	39, 40, 42	syl2anc 415	. . . . . 6 |- ( x = 0 -> prod_ k e. ( 1 ... 1 ) k = 1 )
44		oveq1 6082	. . . . . . . . 9 |- ( x = 0 -> ( x + 1 ) = ( 0 + 1 ) )
45		0p1e1 9397	. . . . . . . . 9 |- ( 0 + 1 ) = 1
46	44, 45	eqtrdi 2287	. . . . . . . 8 |- ( x = 0 -> ( x + 1 ) = 1 )
47	46	oveq2d 6091	. . . . . . 7 |- ( x = 0 -> ( 1 ... ( x + 1 ) ) = ( 1 ... 1 ) )
48	47	prodeq1d 12309	. . . . . 6 |- ( x = 0 -> prod_ k e. ( 1 ... ( x + 1 ) ) k = prod_ k e. ( 1 ... 1 ) k )
49		fv0p1e1 9398	. . . . . . 7 |- ( x = 0 -> ( ! ` ( x + 1 ) ) = ( ! ` 1 ) )
50		fac1 11145	. . . . . . 7 |- ( ! ` 1 ) = 1
51	49, 50	eqtrdi 2287	. . . . . 6 |- ( x = 0 -> ( ! ` ( x + 1 ) ) = 1 )
52	43, 48, 51	3eqtr4rd 2282	. . . . 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 728	. . 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 9739	1 |- ( A e. NN0 -> ( ! ` A ) = prod_ k e. ( 1 ... A ) k )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   (/)c0 3520   ` cfv 5372  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   NNcn 9283   NN0cn0 9542   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   !cfa 11141   prod_cprod 12295

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-fac 11142  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-proddc 12296

This theorem is referenced by:  gausslemma2dlem1  16094  gausslemma2dlem6  16100