Theorem facp1 10804

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 9245	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		elnnuz 9632	. . . . . . 7 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
3	2	biimpi 120	. . . . . 6 |- ( N e. NN -> N e. ( ZZ>= ` 1 ) )
4		fvi 5615	. . . . . . . 8 |- ( f e. ( ZZ>= ` 1 ) -> ( _I ` f ) = f )
5		eluzelcn 9606	. . . . . . . 8 |- ( f e. ( ZZ>= ` 1 ) -> f e. CC )
6	4, 5	eqeltrd 2270	. . . . . . 7 |- ( f e. ( ZZ>= ` 1 ) -> ( _I ` f ) e. CC )
7	6	adantl 277	. . . . . 6 |- ( ( N e. NN /\ f e. ( ZZ>= ` 1 ) ) -> ( _I ` f ) e. CC )
8		mulcl 8001	. . . . . . 7 |- ( ( f e. CC /\ g e. CC ) -> ( f x. g ) e. CC )
9	8	adantl 277	. . . . . 6 |- ( ( N e. NN /\ ( f e. CC /\ g e. CC ) ) -> ( f x. g ) e. CC )
10	3, 7, 9	seq3p1 10539	. . . . 5 |- ( N e. NN -> ( seq 1 ( x. , _I ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( _I ` ( N + 1 ) ) ) )
11		peano2nn 8996	. . . . . . 7 |- ( N e. NN -> ( N + 1 ) e. NN )
12		fvi 5615	. . . . . . 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 5935	. . . . 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 2226	. . . 4 |- ( N e. NN -> ( seq 1 ( x. , _I ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( N + 1 ) ) )
16		facnn 10801	. . . . 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 10801	. . . . 5 |- ( N e. NN -> ( ! ` N ) = ( seq 1 ( x. , _I ) ` N ) )
19	18	oveq1d 5934	. . . 4 |- ( N e. NN -> ( ( ! ` N ) x. ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( N + 1 ) ) )
20	15, 17, 19	3eqtr4d 2236	. . 3 |- ( N e. NN -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
21		0p1e1 9098	. . . . . 6 |- ( 0 + 1 ) = 1
22	21	fveq2i 5558	. . . . 5 |- ( ! ` ( 0 + 1 ) ) = ( ! ` 1 )
23		fac1 10803	. . . . 5 |- ( ! ` 1 ) = 1
24	22, 23	eqtri 2214	. . . 4 |- ( ! ` ( 0 + 1 ) ) = 1
25		fvoveq1 5942	. . . 4 |- ( N = 0 -> ( ! ` ( N + 1 ) ) = ( ! ` ( 0 + 1 ) ) )
26		fveq2 5555	. . . . . 6 |- ( N = 0 -> ( ! ` N ) = ( ! ` 0 ) )
27		oveq1 5926	. . . . . 6 |- ( N = 0 -> ( N + 1 ) = ( 0 + 1 ) )
28	26, 27	oveq12d 5937	. . . . 5 |- ( N = 0 -> ( ( ! ` N ) x. ( N + 1 ) ) = ( ( ! ` 0 ) x. ( 0 + 1 ) ) )
29		fac0 10802	. . . . . . 7 |- ( ! ` 0 ) = 1
30	29, 21	oveq12i 5931	. . . . . 6 |- ( ( ! ` 0 ) x. ( 0 + 1 ) ) = ( 1 x. 1 )
31		1t1e1 9137	. . . . . 6 |- ( 1 x. 1 ) = 1
32	30, 31	eqtri 2214	. . . . 5 |- ( ( ! ` 0 ) x. ( 0 + 1 ) ) = 1
33	28, 32	eqtrdi 2242	. . . 4 |- ( N = 0 -> ( ( ! ` N ) x. ( N + 1 ) ) = 1 )
34	24, 25, 33	3eqtr4a 2252	. . 3 |- ( N = 0 -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
35	20, 34	jaoi 717	. 2 |- ( ( N e. NN \/ N = 0 ) -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
36	1, 35	sylbi 121	1 |- ( N e. NN0 -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2164   _I cid 4320   ` cfv 5255  (class class class)co 5919   CCcc 7872   0cc0 7874   1c1 7875   + caddc 7877   x. cmul 7879   NNcn 8984   NN0cn0 9243   ZZ>=cuz 9595   seqcseq 10521   !cfa 10799

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-seqfrec 10522  df-fac 10800

This theorem is referenced by:  fac2  10805  fac3  10806  fac4  10807  facnn2  10808  faccl  10809  facdiv  10812  facwordi  10814  faclbnd  10815  faclbnd6  10818  facubnd  10819  bcm1k  10834  bcp1n  10835  4bc2eq6  10848  fprodfac  11761  efcllemp  11804  ef01bndlem  11902  eirraplem  11923  dvdsfac  12005  prmfac1  12293  pcfac  12491  ex-fac  15290