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Theorem dvdsfac 12627
Description: A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
Assertion
Ref Expression
dvdsfac |- ( ( K e. NN /\ N e. ( ZZ>= ` K ) ) -> K || ( ! ` N ) )
Proof of Theorem dvdsfac
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5695 . . . . 5 |- ( x = K -> ( ! ` x ) = ( ! ` K ) )
21breq2d 4142 . . . 4 |- ( x = K -> ( K || ( ! ` x ) <-> K || ( ! ` K ) ) )
32imbi2d 230 . . 3 |- ( x = K -> ( ( K e. NN -> K || ( ! ` x ) ) <-> ( K e. NN -> K || ( ! ` K ) ) ) )
4 fveq2 5695 . . . . 5 |- ( x = y -> ( ! ` x ) = ( ! ` y ) )
54breq2d 4142 . . . 4 |- ( x = y -> ( K || ( ! ` x ) <-> K || ( ! ` y ) ) )
65imbi2d 230 . . 3 |- ( x = y -> ( ( K e. NN -> K || ( ! ` x ) ) <-> ( K e. NN -> K || ( ! ` y ) ) ) )
7 fveq2 5695 . . . . 5 |- ( x = ( y + 1 ) -> ( ! ` x ) = ( ! ` ( y + 1 ) ) )
87breq2d 4142 . . . 4 |- ( x = ( y + 1 ) -> ( K || ( ! ` x ) <-> K || ( ! ` ( y + 1 ) ) ) )
98imbi2d 230 . . 3 |- ( x = ( y + 1 ) -> ( ( K e. NN -> K || ( ! ` x ) ) <-> ( K e. NN -> K || ( ! ` ( y + 1 ) ) ) ) )
10 fveq2 5695 . . . . 5 |- ( x = N -> ( ! ` x ) = ( ! ` N ) )
1110breq2d 4142 . . . 4 |- ( x = N -> ( K || ( ! ` x ) <-> K || ( ! ` N ) ) )
1211imbi2d 230 . . 3 |- ( x = N -> ( ( K e. NN -> K || ( ! ` x ) ) <-> ( K e. NN -> K || ( ! ` N ) ) ) )
13 nnm1nn0 9604 . . . . . . . 8 |- ( K e. NN -> ( K - 1 ) e. NN0 )
14 faccl 11173 . . . . . . . 8 |- ( ( K - 1 ) e. NN0 -> ( ! ` ( K - 1 ) ) e. NN )
1513, 14syl 14 . . . . . . 7 |- ( K e. NN -> ( ! ` ( K - 1 ) ) e. NN )
1615nnzd 9767 . . . . . 6 |- ( K e. NN -> ( ! ` ( K - 1 ) ) e. ZZ )
17 nnz 9663 . . . . . 6 |- ( K e. NN -> K e. ZZ )
18 dvdsmul2 12581 . . . . . 6 |- ( ( ( ! ` ( K - 1 ) ) e. ZZ /\ K e. ZZ ) -> K || ( ( ! ` ( K - 1 ) ) x. K ) )
1916, 17, 18syl2anc 415 . . . . 5 |- ( K e. NN -> K || ( ( ! ` ( K - 1 ) ) x. K ) )
20 facnn2 11172 . . . . 5 |- ( K e. NN -> ( ! ` K ) = ( ( ! ` ( K - 1 ) ) x. K ) )
2119, 20breqtrrd 4158 . . . 4 |- ( K e. NN -> K || ( ! ` K ) )
2221a1i 9 . . 3 |- ( K e. ZZ -> ( K e. NN -> K || ( ! ` K ) ) )
2317adantl 277 . . . . . . 7 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> K e. ZZ )
24 elnnuz 9959 . . . . . . . . . . . 12 |- ( K e. NN <-> K e. ( ZZ>= ` 1 ) )
25 uztrn 9939 . . . . . . . . . . . 12 |- ( ( y e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` 1 ) ) -> y e. ( ZZ>= ` 1 ) )
2624, 25sylan2b 287 . . . . . . . . . . 11 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> y e. ( ZZ>= ` 1 ) )
27 elnnuz 9959 . . . . . . . . . . 11 |- ( y e. NN <-> y e. ( ZZ>= ` 1 ) )
2826, 27sylibr 134 . . . . . . . . . 10 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> y e. NN )
2928nnnn0d 9620 . . . . . . . . 9 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> y e. NN0 )
30 faccl 11173 . . . . . . . . 9 |- ( y e. NN0 -> ( ! ` y ) e. NN )
3129, 30syl 14 . . . . . . . 8 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( ! ` y ) e. NN )
3231nnzd 9767 . . . . . . 7 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( ! ` y ) e. ZZ )
3328nnzd 9767 . . . . . . . 8 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> y e. ZZ )
3433peano2zd 9771 . . . . . . 7 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( y + 1 ) e. ZZ )
35 dvdsmultr1 12598 . . . . . . 7 |- ( ( K e. ZZ /\ ( ! ` y ) e. ZZ /\ ( y + 1 ) e. ZZ ) -> ( K || ( ! ` y ) -> K || ( ( ! ` y ) x. ( y + 1 ) ) ) )
3623, 32, 34, 35syl3anc 1278 . . . . . 6 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( K || ( ! ` y ) -> K || ( ( ! ` y ) x. ( y + 1 ) ) ) )
37 facp1 11168 . . . . . . . 8 |- ( y e. NN0 -> ( ! ` ( y + 1 ) ) = ( ( ! ` y ) x. ( y + 1 ) ) )
3829, 37syl 14 . . . . . . 7 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( ! ` ( y + 1 ) ) = ( ( ! ` y ) x. ( y + 1 ) ) )
3938breq2d 4142 . . . . . 6 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( K || ( ! ` ( y + 1 ) ) <-> K || ( ( ! ` y ) x. ( y + 1 ) ) ) )
4036, 39sylibrd 169 . . . . 5 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( K || ( ! ` y ) -> K || ( ! ` ( y + 1 ) ) ) )
4140ex 115 . . . 4 |- ( y e. ( ZZ>= ` K ) -> ( K e. NN -> ( K || ( ! ` y ) -> K || ( ! ` ( y + 1 ) ) ) ) )
4241a2d 26 . . 3 |- ( y e. ( ZZ>= ` K ) -> ( ( K e. NN -> K || ( ! ` y ) ) -> ( K e. NN -> K || ( ! ` ( y + 1 ) ) ) ) )
433, 6, 9, 12, 22, 42uzind4 9988 . 2 |- ( N e. ( ZZ>= ` K ) -> ( K e. NN -> K || ( ! ` N ) ) )
4443impcom 125 1 |- ( ( K e. NN /\ N e. ( ZZ>= ` K ) ) -> K || ( ! ` N ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   1c1 8180   + caddc 8182   x. cmul 8184   - cmin 8497   NNcn 9304   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   !cfa 11163   || cdvds 12554
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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295
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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-seqfrec 10885  df-fac 11164  df-dvds 12555
This theorem is used by:  prmunb  13141
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