Theorem facp1 10664

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 9137	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		elnnuz 9523	. . . . . . 7 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
3	2	biimpi 119	. . . . . 6 |- ( N e. NN -> N e. ( ZZ>= ` 1 ) )
4		fvi 5553	. . . . . . . 8 |- ( f e. ( ZZ>= ` 1 ) -> ( _I ` f ) = f )
5		eluzelcn 9498	. . . . . . . 8 |- ( f e. ( ZZ>= ` 1 ) -> f e. CC )
6	4, 5	eqeltrd 2247	. . . . . . 7 |- ( f e. ( ZZ>= ` 1 ) -> ( _I ` f ) e. CC )
7	6	adantl 275	. . . . . 6 |- ( ( N e. NN /\ f e. ( ZZ>= ` 1 ) ) -> ( _I ` f ) e. CC )
8		mulcl 7901	. . . . . . 7 |- ( ( f e. CC /\ g e. CC ) -> ( f x. g ) e. CC )
9	8	adantl 275	. . . . . 6 |- ( ( N e. NN /\ ( f e. CC /\ g e. CC ) ) -> ( f x. g ) e. CC )
10	3, 7, 9	seq3p1 10418	. . . . 5 |- ( N e. NN -> ( seq 1 ( x. , _I ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( _I ` ( N + 1 ) ) ) )
11		peano2nn 8890	. . . . . . 7 |- ( N e. NN -> ( N + 1 ) e. NN )
12		fvi 5553	. . . . . . 7 |- ( ( N + 1 ) e. NN -> ( _I ` ( N + 1 ) ) = ( N + 1 ) )
13	11, 12	syl 14	. . . . . 6 |- ( N e. NN -> ( _I ` ( N + 1 ) ) = ( N + 1 ) )
14	13	oveq2d 5869	. . . . 5 |- ( N e. NN -> ( ( seq 1 ( x. , _I ) ` N ) x. ( _I ` ( N + 1 ) ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( N + 1 ) ) )
15	10, 14	eqtrd 2203	. . . 4 |- ( N e. NN -> ( seq 1 ( x. , _I ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( N + 1 ) ) )
16		facnn 10661	. . . . 5 |- ( ( N + 1 ) e. NN -> ( ! ` ( N + 1 ) ) = ( seq 1 ( x. , _I ) ` ( N + 1 ) ) )
17	11, 16	syl 14	. . . 4 |- ( N e. NN -> ( ! ` ( N + 1 ) ) = ( seq 1 ( x. , _I ) ` ( N + 1 ) ) )
18		facnn 10661	. . . . 5 |- ( N e. NN -> ( ! ` N ) = ( seq 1 ( x. , _I ) ` N ) )
19	18	oveq1d 5868	. . . 4 |- ( N e. NN -> ( ( ! ` N ) x. ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( N + 1 ) ) )
20	15, 17, 19	3eqtr4d 2213	. . 3 |- ( N e. NN -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
21		0p1e1 8992	. . . . . 6 |- ( 0 + 1 ) = 1
22	21	fveq2i 5499	. . . . 5 |- ( ! ` ( 0 + 1 ) ) = ( ! ` 1 )
23		fac1 10663	. . . . 5 |- ( ! ` 1 ) = 1
24	22, 23	eqtri 2191	. . . 4 |- ( ! ` ( 0 + 1 ) ) = 1
25		fvoveq1 5876	. . . 4 |- ( N = 0 -> ( ! ` ( N + 1 ) ) = ( ! ` ( 0 + 1 ) ) )
26		fveq2 5496	. . . . . 6 |- ( N = 0 -> ( ! ` N ) = ( ! ` 0 ) )
27		oveq1 5860	. . . . . 6 |- ( N = 0 -> ( N + 1 ) = ( 0 + 1 ) )
28	26, 27	oveq12d 5871	. . . . 5 |- ( N = 0 -> ( ( ! ` N ) x. ( N + 1 ) ) = ( ( ! ` 0 ) x. ( 0 + 1 ) ) )
29		fac0 10662	. . . . . . 7 |- ( ! ` 0 ) = 1
30	29, 21	oveq12i 5865	. . . . . 6 |- ( ( ! ` 0 ) x. ( 0 + 1 ) ) = ( 1 x. 1 )
31		1t1e1 9030	. . . . . 6 |- ( 1 x. 1 ) = 1
32	30, 31	eqtri 2191	. . . . 5 |- ( ( ! ` 0 ) x. ( 0 + 1 ) ) = 1
33	28, 32	eqtrdi 2219	. . . 4 |- ( N = 0 -> ( ( ! ` N ) x. ( N + 1 ) ) = 1 )
34	24, 25, 33	3eqtr4a 2229	. . 3 |- ( N = 0 -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
35	20, 34	jaoi 711	. 2 |- ( ( N e. NN \/ N = 0 ) -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
36	1, 35	sylbi 120	1 |- ( N e. NN0 -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 703   = wceq 1348   e. wcel 2141   _I cid 4273   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   NNcn 8878   NN0cn0 9135   ZZ>=cuz 9487   seqcseq 10401   !cfa 10659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402  df-fac 10660

This theorem is referenced by:  fac2  10665  fac3  10666  fac4  10667  facnn2  10668  faccl  10669  facdiv  10672  facwordi  10674  faclbnd  10675  faclbnd6  10678  facubnd  10679  bcm1k  10694  bcp1n  10695  4bc2eq6  10708  fprodfac  11578  efcllemp  11621  ef01bndlem  11719  eirraplem  11739  dvdsfac  11820  prmfac1  12106  pcfac  12302  ex-fac  13763