Theorem facp1 10651

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 9124	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		elnnuz 9510	. . . . . . 7 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
3	2	biimpi 119	. . . . . 6 |- ( N e. NN -> N e. ( ZZ>= ` 1 ) )
4		fvi 5551	. . . . . . . 8 |- ( f e. ( ZZ>= ` 1 ) -> ( _I ` f ) = f )
5		eluzelcn 9485	. . . . . . . 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 7888	. . . . . . 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 10405	. . . . 5 |- ( N e. NN -> ( seq 1 ( x. , _I ) ` ( N + 1 ) ) = ( ( seq 1 ( x. , _I ) ` N ) x. ( _I ` ( N + 1 ) ) ) )
11		peano2nn 8877	. . . . . . 7 |- ( N e. NN -> ( N + 1 ) e. NN )
12		fvi 5551	. . . . . . 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 5866	. . . . 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 10648	. . . . 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 10648	. . . . 5 |- ( N e. NN -> ( ! ` N ) = ( seq 1 ( x. , _I ) ` N ) )
19	18	oveq1d 5865	. . . 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 8979	. . . . . 6 |- ( 0 + 1 ) = 1
22	21	fveq2i 5497	. . . . 5 |- ( ! ` ( 0 + 1 ) ) = ( ! ` 1 )
23		fac1 10650	. . . . 5 |- ( ! ` 1 ) = 1
24	22, 23	eqtri 2191	. . . 4 |- ( ! ` ( 0 + 1 ) ) = 1
25		fvoveq1 5873	. . . 4 |- ( N = 0 -> ( ! ` ( N + 1 ) ) = ( ! ` ( 0 + 1 ) ) )
26		fveq2 5494	. . . . . 6 |- ( N = 0 -> ( ! ` N ) = ( ! ` 0 ) )
27		oveq1 5857	. . . . . 6 |- ( N = 0 -> ( N + 1 ) = ( 0 + 1 ) )
28	26, 27	oveq12d 5868	. . . . 5 |- ( N = 0 -> ( ( ! ` N ) x. ( N + 1 ) ) = ( ( ! ` 0 ) x. ( 0 + 1 ) ) )
29		fac0 10649	. . . . . . 7 |- ( ! ` 0 ) = 1
30	29, 21	oveq12i 5862	. . . . . 6 |- ( ( ! ` 0 ) x. ( 0 + 1 ) ) = ( 1 x. 1 )
31		1t1e1 9017	. . . . . 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 4271   ` cfv 5196  (class class class)co 5850   CCcc 7759   0cc0 7761   1c1 7762   + caddc 7764   x. cmul 7766   NNcn 8865   NN0cn0 9122   ZZ>=cuz 9474   seqcseq 10388   !cfa 10646

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 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-ltadd 7877

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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475  df-seqfrec 10389  df-fac 10647

This theorem is referenced by:  fac2  10652  fac3  10653  fac4  10654  facnn2  10655  faccl  10656  facdiv  10659  facwordi  10661  faclbnd  10662  faclbnd6  10665  facubnd  10666  bcm1k  10681  bcp1n  10682  4bc2eq6  10695  fprodfac  11565  efcllemp  11608  ef01bndlem  11706  eirraplem  11726  dvdsfac  11807  prmfac1  12093  pcfac  12289  ex-fac  13684