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Theorem fprodfac 11614
Description: Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
Assertion
Ref Expression
fprodfac  |-  ( A  e.  NN0  ->  ( ! `
 A )  = 
prod_ k  e.  (
1 ... A ) k )
Distinct variable group:    A, k

Proof of Theorem fprodfac
Dummy variables  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5512 . . 3  |-  ( w  =  0  ->  ( ! `  w )  =  ( ! ` 
0 ) )
2 oveq2 5878 . . . 4  |-  ( w  =  0  ->  (
1 ... w )  =  ( 1 ... 0
) )
32prodeq1d 11563 . . 3  |-  ( w  =  0  ->  prod_ k  e.  ( 1 ... w ) k  = 
prod_ k  e.  (
1 ... 0 ) k )
41, 3eqeq12d 2192 . 2  |-  ( w  =  0  ->  (
( ! `  w
)  =  prod_ k  e.  ( 1 ... w
) k  <->  ( ! `  0 )  = 
prod_ k  e.  (
1 ... 0 ) k ) )
5 fveq2 5512 . . 3  |-  ( w  =  x  ->  ( ! `  w )  =  ( ! `  x ) )
6 oveq2 5878 . . . 4  |-  ( w  =  x  ->  (
1 ... w )  =  ( 1 ... x
) )
76prodeq1d 11563 . . 3  |-  ( w  =  x  ->  prod_ k  e.  ( 1 ... w ) k  = 
prod_ k  e.  (
1 ... x ) k )
85, 7eqeq12d 2192 . 2  |-  ( w  =  x  ->  (
( ! `  w
)  =  prod_ k  e.  ( 1 ... w
) k  <->  ( ! `  x )  =  prod_ k  e.  ( 1 ... x ) k ) )
9 fveq2 5512 . . 3  |-  ( w  =  ( x  + 
1 )  ->  ( ! `  w )  =  ( ! `  ( x  +  1
) ) )
10 oveq2 5878 . . . 4  |-  ( w  =  ( x  + 
1 )  ->  (
1 ... w )  =  ( 1 ... (
x  +  1 ) ) )
1110prodeq1d 11563 . . 3  |-  ( w  =  ( x  + 
1 )  ->  prod_ k  e.  ( 1 ... w ) k  = 
prod_ k  e.  (
1 ... ( x  + 
1 ) ) k )
129, 11eqeq12d 2192 . 2  |-  ( w  =  ( x  + 
1 )  ->  (
( ! `  w
)  =  prod_ k  e.  ( 1 ... w
) k  <->  ( ! `  ( x  +  1 ) )  =  prod_ k  e.  ( 1 ... ( x  +  1 ) ) k ) )
13 fveq2 5512 . . 3  |-  ( w  =  A  ->  ( ! `  w )  =  ( ! `  A ) )
14 oveq2 5878 . . . 4  |-  ( w  =  A  ->  (
1 ... w )  =  ( 1 ... A
) )
1514prodeq1d 11563 . . 3  |-  ( w  =  A  ->  prod_ k  e.  ( 1 ... w ) k  = 
prod_ k  e.  (
1 ... A ) k )
1613, 15eqeq12d 2192 . 2  |-  ( w  =  A  ->  (
( ! `  w
)  =  prod_ k  e.  ( 1 ... w
) k  <->  ( ! `  A )  =  prod_ k  e.  ( 1 ... A ) k ) )
17 prod0 11584 . . 3  |-  prod_ k  e.  (/)  k  =  1
18 fz10 10039 . . . 4  |-  ( 1 ... 0 )  =  (/)
1918prodeq1i 11560 . . 3  |-  prod_ k  e.  ( 1 ... 0
) k  =  prod_ k  e.  (/)  k
20 fac0 10699 . . 3  |-  ( ! `
 0 )  =  1
2117, 19, 203eqtr4ri 2209 . 2  |-  ( ! `
 0 )  = 
prod_ k  e.  (
1 ... 0 ) k
22 elnn0 9172 . . 3  |-  ( x  e.  NN0  <->  ( x  e.  NN  \/  x  =  0 ) )
23 simpr 110 . . . . . . 7  |-  ( ( x  e.  NN  /\  ( ! `  x )  =  prod_ k  e.  ( 1 ... x ) k )  ->  ( ! `  x )  =  prod_ k  e.  ( 1 ... x ) k )
2423oveq1d 5885 . . . . . 6  |-  ( ( x  e.  NN  /\  ( ! `  x )  =  prod_ k  e.  ( 1 ... x ) k )  ->  (
( ! `  x
)  x.  ( x  +  1 ) )  =  ( prod_ k  e.  ( 1 ... x
) k  x.  (
x  +  1 ) ) )
25 nnnn0 9177 . . . . . . . . 9  |-  ( x  e.  NN  ->  x  e.  NN0 )
26 facp1 10701 . . . . . . . . 9  |-  ( x  e.  NN0  ->  ( ! `
 ( x  + 
1 ) )  =  ( ( ! `  x )  x.  (
x  +  1 ) ) )
2725, 26syl 14 . . . . . . . 8  |-  ( x  e.  NN  ->  ( ! `  ( x  +  1 ) )  =  ( ( ! `
 x )  x.  ( x  +  1 ) ) )
28 elnnuz 9558 . . . . . . . . . 10  |-  ( x  e.  NN  <->  x  e.  ( ZZ>= `  1 )
)
2928biimpi 120 . . . . . . . . 9  |-  ( x  e.  NN  ->  x  e.  ( ZZ>= `  1 )
)
30 elfzelz 10018 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... ( x  +  1 ) )  ->  k  e.  ZZ )
3130zcnd 9370 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( x  +  1 ) )  ->  k  e.  CC )
3231adantl 277 . . . . . . . . 9  |-  ( ( x  e.  NN  /\  k  e.  ( 1 ... ( x  + 
1 ) ) )  ->  k  e.  CC )
33 id 19 . . . . . . . . 9  |-  ( k  =  ( x  + 
1 )  ->  k  =  ( x  + 
1 ) )
3429, 32, 33fprodp1 11599 . . . . . . . 8  |-  ( x  e.  NN  ->  prod_ k  e.  ( 1 ... ( x  +  1 ) ) k  =  ( prod_ k  e.  ( 1 ... x ) k  x.  ( x  +  1 ) ) )
3527, 34eqeq12d 2192 . . . . . . 7  |-  ( x  e.  NN  ->  (
( ! `  (
x  +  1 ) )  =  prod_ k  e.  ( 1 ... (
x  +  1 ) ) k  <->  ( ( ! `  x )  x.  ( x  +  1 ) )  =  (
prod_ k  e.  (
1 ... x ) k  x.  ( x  + 
1 ) ) ) )
3635adantr 276 . . . . . 6  |-  ( ( x  e.  NN  /\  ( ! `  x )  =  prod_ k  e.  ( 1 ... x ) k )  ->  (
( ! `  (
x  +  1 ) )  =  prod_ k  e.  ( 1 ... (
x  +  1 ) ) k  <->  ( ( ! `  x )  x.  ( x  +  1 ) )  =  (
prod_ k  e.  (
1 ... x ) k  x.  ( x  + 
1 ) ) ) )
3724, 36mpbird 167 . . . . 5  |-  ( ( x  e.  NN  /\  ( ! `  x )  =  prod_ k  e.  ( 1 ... x ) k )  ->  ( ! `  ( x  +  1 ) )  =  prod_ k  e.  ( 1 ... ( x  +  1 ) ) k )
3837ex 115 . . . 4  |-  ( x  e.  NN  ->  (
( ! `  x
)  =  prod_ k  e.  ( 1 ... x
) k  ->  ( ! `  ( x  +  1 ) )  =  prod_ k  e.  ( 1 ... ( x  +  1 ) ) k ) )
39 1zzd 9274 . . . . . . 7  |-  ( x  =  0  ->  1  e.  ZZ )
40 1cnd 7968 . . . . . . 7  |-  ( x  =  0  ->  1  e.  CC )
41 id 19 . . . . . . . 8  |-  ( k  =  1  ->  k  =  1 )
4241fprod1 11593 . . . . . . 7  |-  ( ( 1  e.  ZZ  /\  1  e.  CC )  ->  prod_ k  e.  ( 1 ... 1 ) k  =  1 )
4339, 40, 42syl2anc 411 . . . . . 6  |-  ( x  =  0  ->  prod_ k  e.  ( 1 ... 1 ) k  =  1 )
44 oveq1 5877 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  +  1 )  =  ( 0  +  1 ) )
45 0p1e1 9027 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4644, 45eqtrdi 2226 . . . . . . . 8  |-  ( x  =  0  ->  (
x  +  1 )  =  1 )
4746oveq2d 5886 . . . . . . 7  |-  ( x  =  0  ->  (
1 ... ( x  + 
1 ) )  =  ( 1 ... 1
) )
4847prodeq1d 11563 . . . . . 6  |-  ( x  =  0  ->  prod_ k  e.  ( 1 ... ( x  +  1 ) ) k  = 
prod_ k  e.  (
1 ... 1 ) k )
49 fv0p1e1 9028 . . . . . . 7  |-  ( x  =  0  ->  ( ! `  ( x  +  1 ) )  =  ( ! ` 
1 ) )
50 fac1 10700 . . . . . . 7  |-  ( ! `
 1 )  =  1
5149, 50eqtrdi 2226 . . . . . 6  |-  ( x  =  0  ->  ( ! `  ( x  +  1 ) )  =  1 )
5243, 48, 513eqtr4rd 2221 . . . . 5  |-  ( x  =  0  ->  ( ! `  ( x  +  1 ) )  =  prod_ k  e.  ( 1 ... ( x  +  1 ) ) k )
5352a1d 22 . . . 4  |-  ( x  =  0  ->  (
( ! `  x
)  =  prod_ k  e.  ( 1 ... x
) k  ->  ( ! `  ( x  +  1 ) )  =  prod_ k  e.  ( 1 ... ( x  +  1 ) ) k ) )
5438, 53jaoi 716 . . 3  |-  ( ( x  e.  NN  \/  x  =  0 )  ->  ( ( ! `
 x )  = 
prod_ k  e.  (
1 ... x ) k  ->  ( ! `  ( x  +  1
) )  =  prod_ k  e.  ( 1 ... ( x  +  1 ) ) k ) )
5522, 54sylbi 121 . 2  |-  ( x  e.  NN0  ->  ( ( ! `  x )  =  prod_ k  e.  ( 1 ... x ) k  ->  ( ! `  ( x  +  1 ) )  =  prod_ k  e.  ( 1 ... ( x  +  1 ) ) k ) )
564, 8, 12, 16, 21, 55nn0ind 9361 1  |-  ( A  e.  NN0  ->  ( ! `
 A )  = 
prod_ k  e.  (
1 ... A ) k )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    = wceq 1353    e. wcel 2148   (/)c0 3422   ` cfv 5213  (class class class)co 5870   CCcc 7804   0cc0 7806   1c1 7807    + caddc 7809    x. cmul 7811   NNcn 8913   NN0cn0 9170   ZZcz 9247   ZZ>=cuz 9522   ...cfz 10002   !cfa 10696   prod_cprod 11549
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 4116  ax-sep 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-iinf 4585  ax-cnex 7897  ax-resscn 7898  ax-1cn 7899  ax-1re 7900  ax-icn 7901  ax-addcl 7902  ax-addrcl 7903  ax-mulcl 7904  ax-mulrcl 7905  ax-addcom 7906  ax-mulcom 7907  ax-addass 7908  ax-mulass 7909  ax-distr 7910  ax-i2m1 7911  ax-0lt1 7912  ax-1rid 7913  ax-0id 7914  ax-rnegex 7915  ax-precex 7916  ax-cnre 7917  ax-pre-ltirr 7918  ax-pre-ltwlin 7919  ax-pre-lttrn 7920  ax-pre-apti 7921  ax-pre-ltadd 7922  ax-pre-mulgt0 7923  ax-pre-mulext 7924  ax-arch 7925  ax-caucvg 7926
This theorem depends on definitions:  df-bi 117  df-dc 835  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-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-iun 3887  df-br 4002  df-opab 4063  df-mpt 4064  df-tr 4100  df-id 4291  df-po 4294  df-iso 4295  df-iord 4364  df-on 4366  df-ilim 4367  df-suc 4369  df-iom 4588  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-isom 5222  df-riota 5826  df-ov 5873  df-oprab 5874  df-mpo 5875  df-1st 6136  df-2nd 6137  df-recs 6301  df-irdg 6366  df-frec 6387  df-1o 6412  df-oadd 6416  df-er 6530  df-en 6736  df-dom 6737  df-fin 6738  df-pnf 7988  df-mnf 7989  df-xr 7990  df-ltxr 7991  df-le 7992  df-sub 8124  df-neg 8125  df-reap 8526  df-ap 8533  df-div 8624  df-inn 8914  df-2 8972  df-3 8973  df-4 8974  df-n0 9171  df-z 9248  df-uz 9523  df-q 9614  df-rp 9648  df-fz 10003  df-fzo 10136  df-seqfrec 10439  df-exp 10513  df-fac 10697  df-ihash 10747  df-cj 10842  df-re 10843  df-im 10844  df-rsqrt 10998  df-abs 10999  df-clim 11278  df-proddc 11550
This theorem is referenced by: (None)
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