Theorem dvdsfac 12546

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.

Step	Hyp	Ref	Expression
1		fveq2 5670	. . . . 5 |- ( x = K -> ( ! ` x ) = ( ! ` K ) )
2	1	breq2d 4121	. . . 4 |- ( x = K -> ( K || ( ! ` x ) <-> K || ( ! ` K ) ) )
3	2	imbi2d 230	. . 3 |- ( x = K -> ( ( K e. NN -> K || ( ! ` x ) ) <-> ( K e. NN -> K || ( ! ` K ) ) ) )
4		fveq2 5670	. . . . 5 |- ( x = y -> ( ! ` x ) = ( ! ` y ) )
5	4	breq2d 4121	. . . 4 |- ( x = y -> ( K || ( ! ` x ) <-> K || ( ! ` y ) ) )
6	5	imbi2d 230	. . 3 |- ( x = y -> ( ( K e. NN -> K || ( ! ` x ) ) <-> ( K e. NN -> K || ( ! ` y ) ) ) )
7		fveq2 5670	. . . . 5 |- ( x = ( y + 1 ) -> ( ! ` x ) = ( ! ` ( y + 1 ) ) )
8	7	breq2d 4121	. . . 4 |- ( x = ( y + 1 ) -> ( K || ( ! ` x ) <-> K || ( ! ` ( y + 1 ) ) ) )
9	8	imbi2d 230	. . 3 |- ( x = ( y + 1 ) -> ( ( K e. NN -> K || ( ! ` x ) ) <-> ( K e. NN -> K || ( ! ` ( y + 1 ) ) ) ) )
10		fveq2 5670	. . . . 5 |- ( x = N -> ( ! ` x ) = ( ! ` N ) )
11	10	breq2d 4121	. . . 4 |- ( x = N -> ( K || ( ! ` x ) <-> K || ( ! ` N ) ) )
12	11	imbi2d 230	. . 3 |- ( x = N -> ( ( K e. NN -> K || ( ! ` x ) ) <-> ( K e. NN -> K || ( ! ` N ) ) ) )
13		nnm1nn0 9537	. . . . . . . 8 |- ( K e. NN -> ( K - 1 ) e. NN0 )
14		faccl 11097	. . . . . . . 8 |- ( ( K - 1 ) e. NN0 -> ( ! ` ( K - 1 ) ) e. NN )
15	13, 14	syl 14	. . . . . . 7 |- ( K e. NN -> ( ! ` ( K - 1 ) ) e. NN )
16	15	nnzd 9699	. . . . . 6 |- ( K e. NN -> ( ! ` ( K - 1 ) ) e. ZZ )
17		nnz 9596	. . . . . 6 |- ( K e. NN -> K e. ZZ )
18		dvdsmul2 12500	. . . . . 6 |- ( ( ( ! ` ( K - 1 ) ) e. ZZ /\ K e. ZZ ) -> K || ( ( ! ` ( K - 1 ) ) x. K ) )
19	16, 17, 18	syl2anc 411	. . . . 5 |- ( K e. NN -> K || ( ( ! ` ( K - 1 ) ) x. K ) )
20		facnn2 11096	. . . . 5 |- ( K e. NN -> ( ! ` K ) = ( ( ! ` ( K - 1 ) ) x. K ) )
21	19, 20	breqtrrd 4137	. . . 4 |- ( K e. NN -> K || ( ! ` K ) )
22	21	a1i 9	. . 3 |- ( K e. ZZ -> ( K e. NN -> K || ( ! ` K ) ) )
23	17	adantl 277	. . . . . . 7 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> K e. ZZ )
24		elnnuz 9891	. . . . . . . . . . . 12 |- ( K e. NN <-> K e. ( ZZ>= ` 1 ) )
25		uztrn 9871	. . . . . . . . . . . 12 |- ( ( y e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` 1 ) ) -> y e. ( ZZ>= ` 1 ) )
26	24, 25	sylan2b 287	. . . . . . . . . . 11 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> y e. ( ZZ>= ` 1 ) )
27		elnnuz 9891	. . . . . . . . . . 11 |- ( y e. NN <-> y e. ( ZZ>= ` 1 ) )
28	26, 27	sylibr 134	. . . . . . . . . 10 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> y e. NN )
29	28	nnnn0d 9553	. . . . . . . . 9 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> y e. NN0 )
30		faccl 11097	. . . . . . . . 9 |- ( y e. NN0 -> ( ! ` y ) e. NN )
31	29, 30	syl 14	. . . . . . . 8 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( ! ` y ) e. NN )
32	31	nnzd 9699	. . . . . . 7 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( ! ` y ) e. ZZ )
33	28	nnzd 9699	. . . . . . . 8 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> y e. ZZ )
34	33	peano2zd 9703	. . . . . . 7 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( y + 1 ) e. ZZ )
35		dvdsmultr1 12517	. . . . . . 7 |- ( ( K e. ZZ /\ ( ! ` y ) e. ZZ /\ ( y + 1 ) e. ZZ ) -> ( K || ( ! ` y ) -> K || ( ( ! ` y ) x. ( y + 1 ) ) ) )
36	23, 32, 34, 35	syl3anc 1274	. . . . . 6 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( K || ( ! ` y ) -> K || ( ( ! ` y ) x. ( y + 1 ) ) ) )
37		facp1 11092	. . . . . . . 8 |- ( y e. NN0 -> ( ! ` ( y + 1 ) ) = ( ( ! ` y ) x. ( y + 1 ) ) )
38	29, 37	syl 14	. . . . . . 7 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( ! ` ( y + 1 ) ) = ( ( ! ` y ) x. ( y + 1 ) ) )
39	38	breq2d 4121	. . . . . 6 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( K || ( ! ` ( y + 1 ) ) <-> K || ( ( ! ` y ) x. ( y + 1 ) ) ) )
40	36, 39	sylibrd 169	. . . . 5 |- ( ( y e. ( ZZ>= ` K ) /\ K e. NN ) -> ( K || ( ! ` y ) -> K || ( ! ` ( y + 1 ) ) ) )
41	40	ex 115	. . . 4 |- ( y e. ( ZZ>= ` K ) -> ( K e. NN -> ( K || ( ! ` y ) -> K || ( ! ` ( y + 1 ) ) ) ) )
42	41	a2d 26	. . 3 |- ( y e. ( ZZ>= ` K ) -> ( ( K e. NN -> K || ( ! ` y ) ) -> ( K e. NN -> K || ( ! ` ( y + 1 ) ) ) ) )
43	3, 6, 9, 12, 22, 42	uzind4 9920	. 2 |- ( N e. ( ZZ>= ` K ) -> ( K e. NN -> K || ( ! ` N ) ) )
44	43	impcom 125	1 |- ( ( K e. NN /\ N e. ( ZZ>= ` K ) ) -> K || ( ! ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   1c1 8128   + caddc 8130   x. cmul 8132   - cmin 8444   NNcn 9237   NN0cn0 9496   ZZcz 9577   ZZ>=cuz 9853   !cfa 11087   || cdvds 12473

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-seqfrec 10810  df-fac 11088  df-dvds 12474

This theorem is referenced by:  prmunb  13060