Theorem pcfaclem 12487

Description: Lemma for pcfac 12488. (Contributed by Mario Carneiro, 20-May-2014.)

Assertion

Ref	Expression
pcfaclem	|- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( |_ ` ( N / ( P ^ M ) ) ) = 0 )

Proof of Theorem pcfaclem

Step	Hyp	Ref	Expression
1		nn0ge0 9265	. . . 4 |- ( N e. NN0 -> 0 <_ N )
2	1	3ad2ant1 1020	. . 3 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 0 <_ N )
3		nn0re 9249	. . . . 5 |- ( N e. NN0 -> N e. RR )
4	3	3ad2ant1 1020	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. RR )
5		prmnn 12248	. . . . . . 7 |- ( P e. Prime -> P e. NN )
6	5	3ad2ant3 1022	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> P e. NN )
7		eluznn0 9664	. . . . . . 7 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) ) -> M e. NN0 )
8	7	3adant3 1019	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> M e. NN0 )
9	6, 8	nnexpcld 10766	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. NN )
10	9	nnred 8995	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. RR )
11	9	nngt0d 9026	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 0 < ( P ^ M ) )
12		ge0div 8890	. . . 4 |- ( ( N e. RR /\ ( P ^ M ) e. RR /\ 0 < ( P ^ M ) ) -> ( 0 <_ N <-> 0 <_ ( N / ( P ^ M ) ) ) )
13	4, 10, 11, 12	syl3anc 1249	. . 3 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( 0 <_ N <-> 0 <_ ( N / ( P ^ M ) ) ) )
14	2, 13	mpbid 147	. 2 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 0 <_ ( N / ( P ^ M ) ) )
15	8	nn0red 9294	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> M e. RR )
16		eluzle 9604	. . . . . . 7 |- ( M e. ( ZZ>= ` N ) -> N <_ M )
17	16	3ad2ant2 1021	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N <_ M )
18		prmuz2 12269	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
19	18	3ad2ant3 1022	. . . . . . 7 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> P e. ( ZZ>= ` 2 ) )
20		bernneq3 10733	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ M e. NN0 ) -> M < ( P ^ M ) )
21	19, 8, 20	syl2anc 411	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> M < ( P ^ M ) )
22	4, 15, 10, 17, 21	lelttrd 8144	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N < ( P ^ M ) )
23	9	nncnd 8996	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. CC )
24	23	mulridd 8036	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( ( P ^ M ) x. 1 ) = ( P ^ M ) )
25	22, 24	breqtrrd 4057	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N < ( ( P ^ M ) x. 1 ) )
26		1red 8034	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 1 e. RR )
27		ltdivmul 8895	. . . . 5 |- ( ( N e. RR /\ 1 e. RR /\ ( ( P ^ M ) e. RR /\ 0 < ( P ^ M ) ) ) -> ( ( N / ( P ^ M ) ) < 1 <-> N < ( ( P ^ M ) x. 1 ) ) )
28	4, 26, 10, 11, 27	syl112anc 1253	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( ( N / ( P ^ M ) ) < 1 <-> N < ( ( P ^ M ) x. 1 ) ) )
29	25, 28	mpbird 167	. . 3 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( N / ( P ^ M ) ) < 1 )
30		0p1e1 9096	. . 3 |- ( 0 + 1 ) = 1
31	29, 30	breqtrrdi 4071	. 2 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( N / ( P ^ M ) ) < ( 0 + 1 ) )
32		simp1 999	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. NN0 )
33	32	nn0zd 9437	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. ZZ )
34		znq 9689	. . . 4 |- ( ( N e. ZZ /\ ( P ^ M ) e. NN ) -> ( N / ( P ^ M ) ) e. QQ )
35	33, 9, 34	syl2anc 411	. . 3 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( N / ( P ^ M ) ) e. QQ )
36		0z 9328	. . 3 |- 0 e. ZZ
37		flqbi 10359	. . 3 |- ( ( ( N / ( P ^ M ) ) e. QQ /\ 0 e. ZZ ) -> ( ( |_ ` ( N / ( P ^ M ) ) ) = 0 <-> ( 0 <_ ( N / ( P ^ M ) ) /\ ( N / ( P ^ M ) ) < ( 0 + 1 ) ) ) )
38	35, 36, 37	sylancl 413	. 2 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( ( |_ ` ( N / ( P ^ M ) ) ) = 0 <-> ( 0 <_ ( N / ( P ^ M ) ) /\ ( N / ( P ^ M ) ) < ( 0 + 1 ) ) ) )
39	14, 31, 38	mpbir2and 946	1 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( |_ ` ( N / ( P ^ M ) ) ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   RRcr 7871   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   < clt 8054   <_ cle 8055   / cdiv 8691   NNcn 8982   2c2 9033   NN0cn0 9240   ZZcz 9317   ZZ>=cuz 9592   QQcq 9684   |_cfl 10337   ^cexp 10609   Primecprime 12245

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-en 6795  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fl 10339  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-prm 12246

This theorem is referenced by:  pcfac  12488