Theorem facp1 11190

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 9576	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		elnnuz 9975	. . . . . . 7 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
3	2	biimpi 120	. . . . . 6 |- ( N e. NN -> N e. ( ZZ>= ` 1 ) )
4		fvi 5763	. . . . . . . 8 |- ( f e. ( ZZ>= ` 1 ) -> ( _I ` f ) = f )
5		eluzelcn 9949	. . . . . . . 8 |- ( f e. ( ZZ>= ` 1 ) -> f e. CC )
6	4, 5	eqeltrd 2315	. . . . . . 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 8312	. . . . . . 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 10923	. . . . 5 |- ( N e. NN -> ( seq 1 ( x. , _I ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( _I ` ( N + 1 ) ) ) )
11		peano2nn 9325	. . . . . . 7 |- ( N e. NN -> ( N + 1 ) e. NN )
12		fvi 5763	. . . . . . 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 6104	. . . . 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 2271	. . . 4 |- ( N e. NN -> ( seq 1 ( x. , _I ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( N + 1 ) ) )
16		facnn 11187	. . . . 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 11187	. . . . 5 |- ( N e. NN -> ( ! ` N ) = ( seq 1 ( x. , _I ) ` N ) )
19	18	oveq1d 6103	. . . 4 |- ( N e. NN -> ( ( ! ` N ) x. ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( N + 1 ) ) )
20	15, 17, 19	3eqtr4d 2281	. . 3 |- ( N e. NN -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
21		0p1e1 9427	. . . . . 6 |- ( 0 + 1 ) = 1
22	21	fveq2i 5701	. . . . 5 |- ( ! ` ( 0 + 1 ) ) = ( ! ` 1 )
23		fac1 11189	. . . . 5 |- ( ! ` 1 ) = 1
24	22, 23	eqtri 2259	. . . 4 |- ( ! ` ( 0 + 1 ) ) = 1
25		fvoveq1 6111	. . . 4 |- ( N = 0 -> ( ! ` ( N + 1 ) ) = ( ! ` ( 0 + 1 ) ) )
26		fveq2 5698	. . . . . 6 |- ( N = 0 -> ( ! ` N ) = ( ! ` 0 ) )
27		oveq1 6095	. . . . . 6 |- ( N = 0 -> ( N + 1 ) = ( 0 + 1 ) )
28	26, 27	oveq12d 6106	. . . . 5 |- ( N = 0 -> ( ( ! ` N ) x. ( N + 1 ) ) = ( ( ! ` 0 ) x. ( 0 + 1 ) ) )
29		fac0 11188	. . . . . . 7 |- ( ! ` 0 ) = 1
30	29, 21	oveq12i 6100	. . . . . 6 |- ( ( ! ` 0 ) x. ( 0 + 1 ) ) = ( 1 x. 1 )
31		1t1e1 9466	. . . . . 6 |- ( 1 x. 1 ) = 1
32	30, 31	eqtri 2259	. . . . 5 |- ( ( ! ` 0 ) x. ( 0 + 1 ) ) = 1
33	28, 32	eqtrdi 2287	. . . 4 |- ( N = 0 -> ( ( ! ` N ) x. ( N + 1 ) ) = 1 )
34	24, 25, 33	3eqtr4a 2297	. . 3 |- ( N = 0 -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
35	20, 34	jaoi 728	. 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   _I cid 4435   ` cfv 5380  (class class class)co 6088   CCcc 8183   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   NNcn 9313   NN0cn0 9574   ZZ>=cuz 9937   seqcseq 10905   !cfa 11185

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-seqfrec 10906  df-fac 11186

This theorem is used by:  fac2  11191  fac3  11192  fac4  11193  facnn2  11194  faccl  11195  facdiv  11198  facwordi  11200  faclbnd  11201  faclbnd6  11204  facubnd  11205  bcm1k  11220  bcp1n  11221  4bc2eq6  11235  fprodfac  12407  efcllemp  12450  ef01bndlem  12548  eirraplem  12569  dvdsfac  12652  prmfac1  12956  pcfac  13158  2expltfac  13248  bcmono  16304  ex-fac  16947