Theorem facp1 10710

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 9178	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		elnnuz 9564	. . . . . . 7 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
3	2	biimpi 120	. . . . . 6 |- ( N e. NN -> N e. ( ZZ>= ` 1 ) )
4		fvi 5574	. . . . . . . 8 |- ( f e. ( ZZ>= ` 1 ) -> ( _I ` f ) = f )
5		eluzelcn 9539	. . . . . . . 8 |- ( f e. ( ZZ>= ` 1 ) -> f e. CC )
6	4, 5	eqeltrd 2254	. . . . . . 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 7938	. . . . . . 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 10462	. . . . 5 |- ( N e. NN -> ( seq 1 ( x. , _I ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( _I ` ( N + 1 ) ) ) )
11		peano2nn 8931	. . . . . . 7 |- ( N e. NN -> ( N + 1 ) e. NN )
12		fvi 5574	. . . . . . 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 5891	. . . . 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 2210	. . . 4 |- ( N e. NN -> ( seq 1 ( x. , _I ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( N + 1 ) ) )
16		facnn 10707	. . . . 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 10707	. . . . 5 |- ( N e. NN -> ( ! ` N ) = ( seq 1 ( x. , _I ) ` N ) )
19	18	oveq1d 5890	. . . 4 |- ( N e. NN -> ( ( ! ` N ) x. ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( N + 1 ) ) )
20	15, 17, 19	3eqtr4d 2220	. . 3 |- ( N e. NN -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
21		0p1e1 9033	. . . . . 6 |- ( 0 + 1 ) = 1
22	21	fveq2i 5519	. . . . 5 |- ( ! ` ( 0 + 1 ) ) = ( ! ` 1 )
23		fac1 10709	. . . . 5 |- ( ! ` 1 ) = 1
24	22, 23	eqtri 2198	. . . 4 |- ( ! ` ( 0 + 1 ) ) = 1
25		fvoveq1 5898	. . . 4 |- ( N = 0 -> ( ! ` ( N + 1 ) ) = ( ! ` ( 0 + 1 ) ) )
26		fveq2 5516	. . . . . 6 |- ( N = 0 -> ( ! ` N ) = ( ! ` 0 ) )
27		oveq1 5882	. . . . . 6 |- ( N = 0 -> ( N + 1 ) = ( 0 + 1 ) )
28	26, 27	oveq12d 5893	. . . . 5 |- ( N = 0 -> ( ( ! ` N ) x. ( N + 1 ) ) = ( ( ! ` 0 ) x. ( 0 + 1 ) ) )
29		fac0 10708	. . . . . . 7 |- ( ! ` 0 ) = 1
30	29, 21	oveq12i 5887	. . . . . 6 |- ( ( ! ` 0 ) x. ( 0 + 1 ) ) = ( 1 x. 1 )
31		1t1e1 9071	. . . . . 6 |- ( 1 x. 1 ) = 1
32	30, 31	eqtri 2198	. . . . 5 |- ( ( ! ` 0 ) x. ( 0 + 1 ) ) = 1
33	28, 32	eqtrdi 2226	. . . 4 |- ( N = 0 -> ( ( ! ` N ) x. ( N + 1 ) ) = 1 )
34	24, 25, 33	3eqtr4a 2236	. . 3 |- ( N = 0 -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
35	20, 34	jaoi 716	. 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 708   = wceq 1353   e. wcel 2148   _I cid 4289   ` cfv 5217  (class class class)co 5875   CCcc 7809   0cc0 7811   1c1 7812   + caddc 7814   x. cmul 7816   NNcn 8919   NN0cn0 9176   ZZ>=cuz 9528   seqcseq 10445   !cfa 10705

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-seqfrec 10446  df-fac 10706

This theorem is referenced by:  fac2  10711  fac3  10712  fac4  10713  facnn2  10714  faccl  10715  facdiv  10718  facwordi  10720  faclbnd  10721  faclbnd6  10724  facubnd  10725  bcm1k  10740  bcp1n  10741  4bc2eq6  10754  fprodfac  11623  efcllemp  11666  ef01bndlem  11764  eirraplem  11784  dvdsfac  11866  prmfac1  12152  pcfac  12348  ex-fac  14483