Theorem fprodfac 12175

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 5639	. . 3 |- ( w = 0 -> ( ! ` w ) = ( ! ` 0 ) )
2		oveq2 6025	. . . 4 |- ( w = 0 -> ( 1 ... w ) = ( 1 ... 0 ) )
3	2	prodeq1d 12124	. . 3 |- ( w = 0 -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... 0 ) k )
4	1, 3	eqeq12d 2246	. 2 |- ( w = 0 -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` 0 ) = prod_ k e. ( 1 ... 0 ) k ) )
5		fveq2 5639	. . 3 |- ( w = x -> ( ! ` w ) = ( ! ` x ) )
6		oveq2 6025	. . . 4 |- ( w = x -> ( 1 ... w ) = ( 1 ... x ) )
7	6	prodeq1d 12124	. . 3 |- ( w = x -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... x ) k )
8	5, 7	eqeq12d 2246	. 2 |- ( w = x -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` x ) = prod_ k e. ( 1 ... x ) k ) )
9		fveq2 5639	. . 3 |- ( w = ( x + 1 ) -> ( ! ` w ) = ( ! ` ( x + 1 ) ) )
10		oveq2 6025	. . . 4 |- ( w = ( x + 1 ) -> ( 1 ... w ) = ( 1 ... ( x + 1 ) ) )
11	10	prodeq1d 12124	. . 3 |- ( w = ( x + 1 ) -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... ( x + 1 ) ) k )
12	9, 11	eqeq12d 2246	. 2 |- ( w = ( x + 1 ) -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` ( x + 1 ) ) = prod_ k e. ( 1 ... ( x + 1 ) ) k ) )
13		fveq2 5639	. . 3 |- ( w = A -> ( ! ` w ) = ( ! ` A ) )
14		oveq2 6025	. . . 4 |- ( w = A -> ( 1 ... w ) = ( 1 ... A ) )
15	14	prodeq1d 12124	. . 3 |- ( w = A -> prod_ k e. ( 1 ... w ) k = prod_ k e. ( 1 ... A ) k )
16	13, 15	eqeq12d 2246	. 2 |- ( w = A -> ( ( ! ` w ) = prod_ k e. ( 1 ... w ) k <-> ( ! ` A ) = prod_ k e. ( 1 ... A ) k ) )
17		prod0 12145	. . 3 |- prod_ k e. (/) k = 1
18		fz10 10280	. . . 4 |- ( 1 ... 0 ) = (/)
19	18	prodeq1i 12121	. . 3 |- prod_ k e. ( 1 ... 0 ) k = prod_ k e. (/) k
20		fac0 10989	. . 3 |- ( ! ` 0 ) = 1
21	17, 19, 20	3eqtr4ri 2263	. 2 |- ( ! ` 0 ) = prod_ k e. ( 1 ... 0 ) k
22		elnn0 9403	. . 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 6032	. . . . . 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 9408	. . . . . . . . 9 |- ( x e. NN -> x e. NN0 )
26		facp1 10991	. . . . . . . . 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 9792	. . . . . . . . . 10 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
29	28	biimpi 120	. . . . . . . . 9 |- ( x e. NN -> x e. ( ZZ>= ` 1 ) )
30		elfzelz 10259	. . . . . . . . . . 11 |- ( k e. ( 1 ... ( x + 1 ) ) -> k e. ZZ )
31	30	zcnd 9602	. . . . . . . . . 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 12160	. . . . . . . 8 |- ( x e. NN -> prod_ k e. ( 1 ... ( x + 1 ) ) k = ( prod_ k e. ( 1 ... x ) k x. ( x + 1 ) ) )
35	27, 34	eqeq12d 2246	. . . . . . 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 9505	. . . . . . 7 |- ( x = 0 -> 1 e. ZZ )
40		1cnd 8194	. . . . . . 7 |- ( x = 0 -> 1 e. CC )
41		id 19	. . . . . . . 8 |- ( k = 1 -> k = 1 )
42	41	fprod1 12154	. . . . . . 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 6024	. . . . . . . . 9 |- ( x = 0 -> ( x + 1 ) = ( 0 + 1 ) )
45		0p1e1 9256	. . . . . . . . 9 |- ( 0 + 1 ) = 1
46	44, 45	eqtrdi 2280	. . . . . . . 8 |- ( x = 0 -> ( x + 1 ) = 1 )
47	46	oveq2d 6033	. . . . . . 7 |- ( x = 0 -> ( 1 ... ( x + 1 ) ) = ( 1 ... 1 ) )
48	47	prodeq1d 12124	. . . . . 6 |- ( x = 0 -> prod_ k e. ( 1 ... ( x + 1 ) ) k = prod_ k e. ( 1 ... 1 ) k )
49		fv0p1e1 9257	. . . . . . 7 |- ( x = 0 -> ( ! ` ( x + 1 ) ) = ( ! ` 1 ) )
50		fac1 10990	. . . . . . 7 |- ( ! ` 1 ) = 1
51	49, 50	eqtrdi 2280	. . . . . 6 |- ( x = 0 -> ( ! ` ( x + 1 ) ) = 1 )
52	43, 48, 51	3eqtr4rd 2275	. . . . 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 723	. . 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 9593	1 |- ( A e. NN0 -> ( ! ` A ) = prod_ k e. ( 1 ... A ) k )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   = wceq 1397   e. wcel 2202   (/)c0 3494   ` cfv 5326  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   NNcn 9142   NN0cn0 9401   ZZcz 9478   ZZ>=cuz 9754   ...cfz 10242   !cfa 10986   prod_cprod 12110

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-exp 10800  df-fac 10987  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-proddc 12111

This theorem is referenced by:  gausslemma2dlem1  15789  gausslemma2dlem6  15795