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Theorem facp1 9754
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.
StepHypRef Expression
1 elnn0 8357 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 elnnuz 8736 . . . . . . 7  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
32biimpi 118 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  ( ZZ>= `  1 )
)
4 fvi 5262 . . . . . . . 8  |-  ( f  e.  ( ZZ>= `  1
)  ->  (  _I  `  f )  =  f )
5 eluzelcn 8711 . . . . . . . 8  |-  ( f  e.  ( ZZ>= `  1
)  ->  f  e.  CC )
64, 5eqeltrd 2156 . . . . . . 7  |-  ( f  e.  ( ZZ>= `  1
)  ->  (  _I  `  f )  e.  CC )
76adantl 271 . . . . . 6  |-  ( ( N  e.  NN  /\  f  e.  ( ZZ>= ` 
1 ) )  -> 
(  _I  `  f
)  e.  CC )
8 mulcl 7162 . . . . . . 7  |-  ( ( f  e.  CC  /\  g  e.  CC )  ->  ( f  x.  g
)  e.  CC )
98adantl 271 . . . . . 6  |-  ( ( N  e.  NN  /\  ( f  e.  CC  /\  g  e.  CC ) )  ->  ( f  x.  g )  e.  CC )
103, 7, 9iseqp1 9538 . . . . 5  |-  ( N  e.  NN  ->  (  seq 1 (  x.  ,  _I  ,  CC ) `  ( N  +  1
) )  =  ( (  seq 1 (  x.  ,  _I  ,  CC ) `  N )  x.  (  _I  `  ( N  +  1
) ) ) )
11 peano2nn 8118 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
12 fvi 5262 . . . . . . 7  |-  ( ( N  +  1 )  e.  NN  ->  (  _I  `  ( N  + 
1 ) )  =  ( N  +  1 ) )
1311, 12syl 14 . . . . . 6  |-  ( N  e.  NN  ->  (  _I  `  ( N  + 
1 ) )  =  ( N  +  1 ) )
1413oveq2d 5559 . . . . 5  |-  ( N  e.  NN  ->  (
(  seq 1 (  x.  ,  _I  ,  CC ) `  N )  x.  (  _I  `  ( N  +  1 ) ) )  =  ( (  seq 1 (  x.  ,  _I  ,  CC ) `  N )  x.  ( N  + 
1 ) ) )
1510, 14eqtrd 2114 . . . 4  |-  ( N  e.  NN  ->  (  seq 1 (  x.  ,  _I  ,  CC ) `  ( N  +  1
) )  =  ( (  seq 1 (  x.  ,  _I  ,  CC ) `  N )  x.  ( N  + 
1 ) ) )
16 facnn 9751 . . . . 5  |-  ( ( N  +  1 )  e.  NN  ->  ( ! `  ( N  +  1 ) )  =  (  seq 1
(  x.  ,  _I  ,  CC ) `  ( N  +  1 ) ) )
1711, 16syl 14 . . . 4  |-  ( N  e.  NN  ->  ( ! `  ( N  +  1 ) )  =  (  seq 1
(  x.  ,  _I  ,  CC ) `  ( N  +  1 ) ) )
18 facnn 9751 . . . . 5  |-  ( N  e.  NN  ->  ( ! `  N )  =  (  seq 1
(  x.  ,  _I  ,  CC ) `  N
) )
1918oveq1d 5558 . . . 4  |-  ( N  e.  NN  ->  (
( ! `  N
)  x.  ( N  +  1 ) )  =  ( (  seq 1 (  x.  ,  _I  ,  CC ) `  N )  x.  ( N  +  1 ) ) )
2015, 17, 193eqtr4d 2124 . . 3  |-  ( N  e.  NN  ->  ( ! `  ( N  +  1 ) )  =  ( ( ! `
 N )  x.  ( N  +  1 ) ) )
21 0p1e1 8220 . . . . . 6  |-  ( 0  +  1 )  =  1
2221fveq2i 5212 . . . . 5  |-  ( ! `
 ( 0  +  1 ) )  =  ( ! `  1
)
23 fac1 9753 . . . . 5  |-  ( ! `
 1 )  =  1
2422, 23eqtri 2102 . . . 4  |-  ( ! `
 ( 0  +  1 ) )  =  1
25 oveq1 5550 . . . . 5  |-  ( N  =  0  ->  ( N  +  1 )  =  ( 0  +  1 ) )
2625fveq2d 5213 . . . 4  |-  ( N  =  0  ->  ( ! `  ( N  +  1 ) )  =  ( ! `  ( 0  +  1 ) ) )
27 fveq2 5209 . . . . . 6  |-  ( N  =  0  ->  ( ! `  N )  =  ( ! ` 
0 ) )
2827, 25oveq12d 5561 . . . . 5  |-  ( N  =  0  ->  (
( ! `  N
)  x.  ( N  +  1 ) )  =  ( ( ! `
 0 )  x.  ( 0  +  1 ) ) )
29 fac0 9752 . . . . . . 7  |-  ( ! `
 0 )  =  1
3029, 21oveq12i 5555 . . . . . 6  |-  ( ( ! `  0 )  x.  ( 0  +  1 ) )  =  ( 1  x.  1 )
31 1t1e1 8251 . . . . . 6  |-  ( 1  x.  1 )  =  1
3230, 31eqtri 2102 . . . . 5  |-  ( ( ! `  0 )  x.  ( 0  +  1 ) )  =  1
3328, 32syl6eq 2130 . . . 4  |-  ( N  =  0  ->  (
( ! `  N
)  x.  ( N  +  1 ) )  =  1 )
3424, 26, 333eqtr4a 2140 . . 3  |-  ( N  =  0  ->  ( ! `  ( N  +  1 ) )  =  ( ( ! `
 N )  x.  ( N  +  1 ) ) )
3520, 34jaoi 669 . 2  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  ( ! `  ( N  +  1
) )  =  ( ( ! `  N
)  x.  ( N  +  1 ) ) )
361, 35sylbi 119 1  |-  ( N  e.  NN0  ->  ( ! `
 ( N  + 
1 ) )  =  ( ( ! `  N )  x.  ( N  +  1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662    = wceq 1285    e. wcel 1434    _I cid 4051   ` cfv 4932  (class class class)co 5543   CCcc 7041   0cc0 7043   1c1 7044    + caddc 7046    x. cmul 7048   NNcn 8106   NN0cn0 8355   ZZ>=cuz 8700    seqcseq 9521   !cfa 9749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-iseq 9522  df-fac 9750
This theorem is referenced by:  fac2  9755  fac3  9756  fac4  9757  facnn2  9758  faccl  9759  facdiv  9762  facwordi  9764  faclbnd  9765  faclbnd6  9768  facubnd  9769  bcm1k  9784  bcp1n  9785  4bc2eq6  9798  dvdsfac  10405  prmfac1  10675  ex-fac  10716
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