Theorem facp1 9657

Description: The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Assertion

Ref	Expression
facp1	|- ( N e. NN0 -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )

Proof of Theorem facp1

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn0 8290	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		elnnuz 8655	. . . . . . 7 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
3	2	biimpi 118	. . . . . 6 |- ( N e. NN -> N e. ( ZZ>= ` 1 ) )
4		cnex 7097	. . . . . . 7 |- CC e. _V
5	4	a1i 9	. . . . . 6 |- ( N e. NN -> CC e. _V )
6		fvi 5251	. . . . . . . 8 |- ( f e. ( ZZ>= ` 1 ) -> ( _I ` f ) = f )
7		eluzelcn 8630	. . . . . . . 8 |- ( f e. ( ZZ>= ` 1 ) -> f e. CC )
8	6, 7	eqeltrd 2155	. . . . . . 7 |- ( f e. ( ZZ>= ` 1 ) -> ( _I ` f ) e. CC )
9	8	adantl 271	. . . . . 6 |- ( ( N e. NN /\ f e. ( ZZ>= ` 1 ) ) -> ( _I ` f ) e. CC )
10		mulcl 7100	. . . . . . 7 |- ( ( f e. CC /\ g e. CC ) -> ( f x. g ) e. CC )
11	10	adantl 271	. . . . . 6 |- ( ( N e. NN /\ ( f e. CC /\ g e. CC ) ) -> ( f x. g ) e. CC )
12	3, 5, 9, 11	iseqp1 9445	. . . . 5 |- ( N e. NN -> ( seq 1 ( x. , _I , CC ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , _I , CC ) ` N ) x. ( _I ` ( N + 1 ) ) ) )
13		peano2nn 8051	. . . . . . 7 |- ( N e. NN -> ( N + 1 ) e. NN )
14		fvi 5251	. . . . . . 7 |- ( ( N + 1 ) e. NN -> ( _I ` ( N + 1 ) ) = ( N + 1 ) )
15	13, 14	syl 14	. . . . . 6 |- ( N e. NN -> ( _I ` ( N + 1 ) ) = ( N + 1 ) )
16	15	oveq2d 5548	. . . . 5 |- ( N e. NN -> ( ( seq 1 ( x. , _I , CC ) ` N ) x. ( _I ` ( N + 1 ) ) ) = ( ( seq 1 ( x. , _I , CC ) ` N ) x. ( N + 1 ) ) )
17	12, 16	eqtrd 2113	. . . 4 |- ( N e. NN -> ( seq 1 ( x. , _I , CC ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , _I , CC ) ` N ) x. ( N + 1 ) ) )
18		facnn 9654	. . . . 5 |- ( ( N + 1 ) e. NN -> ( ! ` ( N + 1 ) ) = ( seq 1 ( x. , _I , CC ) ` ( N + 1 ) ) )
19	13, 18	syl 14	. . . 4 |- ( N e. NN -> ( ! ` ( N + 1 ) ) = ( seq 1 ( x. , _I , CC ) ` ( N + 1 ) ) )
20		facnn 9654	. . . . 5 |- ( N e. NN -> ( ! ` N ) = ( seq 1 ( x. , _I , CC ) ` N ) )
21	20	oveq1d 5547	. . . 4 |- ( N e. NN -> ( ( ! ` N ) x. ( N + 1 ) ) = ( ( seq 1 ( x. , _I , CC ) ` N ) x. ( N + 1 ) ) )
22	17, 19, 21	3eqtr4d 2123	. . 3 |- ( N e. NN -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
23		0p1e1 8153	. . . . . 6 |- ( 0 + 1 ) = 1
24	23	fveq2i 5201	. . . . 5 |- ( ! ` ( 0 + 1 ) ) = ( ! ` 1 )
25		fac1 9656	. . . . 5 |- ( ! ` 1 ) = 1
26	24, 25	eqtri 2101	. . . 4 |- ( ! ` ( 0 + 1 ) ) = 1
27		oveq1 5539	. . . . 5 |- ( N = 0 -> ( N + 1 ) = ( 0 + 1 ) )
28	27	fveq2d 5202	. . . 4 |- ( N = 0 -> ( ! ` ( N + 1 ) ) = ( ! ` ( 0 + 1 ) ) )
29		fveq2 5198	. . . . . 6 |- ( N = 0 -> ( ! ` N ) = ( ! ` 0 ) )
30	29, 27	oveq12d 5550	. . . . 5 |- ( N = 0 -> ( ( ! ` N ) x. ( N + 1 ) ) = ( ( ! ` 0 ) x. ( 0 + 1 ) ) )
31		fac0 9655	. . . . . . 7 |- ( ! ` 0 ) = 1
32	31, 23	oveq12i 5544	. . . . . 6 |- ( ( ! ` 0 ) x. ( 0 + 1 ) ) = ( 1 x. 1 )
33		1t1e1 8184	. . . . . 6 |- ( 1 x. 1 ) = 1
34	32, 33	eqtri 2101	. . . . 5 |- ( ( ! ` 0 ) x. ( 0 + 1 ) ) = 1
35	30, 34	syl6eq 2129	. . . 4 |- ( N = 0 -> ( ( ! ` N ) x. ( N + 1 ) ) = 1 )
36	26, 28, 35	3eqtr4a 2139	. . 3 |- ( N = 0 -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
37	22, 36	jaoi 668	. 2 |- ( ( N e. NN \/ N = 0 ) -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
38	1, 37	sylbi 119	1 |- ( N e. NN0 -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   = wceq 1284   e. wcel 1433   _Vcvv 2601   _I cid 4043   ` cfv 4922  (class class class)co 5532   CCcc 6979   0cc0 6981   1c1 6982   + caddc 6984   x. cmul 6986   NNcn 8039   NN0cn0 8288   ZZ>=cuz 8619   seqcseq 9431   !cfa 9652

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432  df-fac 9653

This theorem is referenced by:  fac2  9658  fac3  9659  fac4  9660  facnn2  9661  faccl  9662  facdiv  9665  facwordi  9667  faclbnd  9668  faclbnd6  9671  facubnd  9672  bcm1k  9687  bcp1n  9688  4bc2eq6  9701  dvdsfac  10260  prmfac1  10531  ex-fac  10565