Theorem pcfaclem 12314

Description: Lemma for pcfac 12315. (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 9174	. . . 4 |- ( N e. NN0 -> 0 <_ N )
2	1	3ad2ant1 1018	. . 3 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 0 <_ N )
3		nn0re 9158	. . . . 5 |- ( N e. NN0 -> N e. RR )
4	3	3ad2ant1 1018	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. RR )
5		prmnn 12077	. . . . . . 7 |- ( P e. Prime -> P e. NN )
6	5	3ad2ant3 1020	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> P e. NN )
7		eluznn0 9572	. . . . . . 7 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) ) -> M e. NN0 )
8	7	3adant3 1017	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> M e. NN0 )
9	6, 8	nnexpcld 10645	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. NN )
10	9	nnred 8905	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. RR )
11	9	nngt0d 8936	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 0 < ( P ^ M ) )
12		ge0div 8801	. . . 4 |- ( ( N e. RR /\ ( P ^ M ) e. RR /\ 0 < ( P ^ M ) ) -> ( 0 <_ N <-> 0 <_ ( N / ( P ^ M ) ) ) )
13	4, 10, 11, 12	syl3anc 1238	. . 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 9203	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> M e. RR )
16		eluzle 9513	. . . . . . 7 |- ( M e. ( ZZ>= ` N ) -> N <_ M )
17	16	3ad2ant2 1019	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N <_ M )
18		prmuz2 12098	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
19	18	3ad2ant3 1020	. . . . . . 7 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> P e. ( ZZ>= ` 2 ) )
20		bernneq3 10612	. . . . . . 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 8056	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N < ( P ^ M ) )
23	9	nncnd 8906	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. CC )
24	23	mulid1d 7949	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( ( P ^ M ) x. 1 ) = ( P ^ M ) )
25	22, 24	breqtrrd 4026	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N < ( ( P ^ M ) x. 1 ) )
26		1red 7947	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 1 e. RR )
27		ltdivmul 8806	. . . . 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 1242	. . . 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 9006	. . 3 |- ( 0 + 1 ) = 1
31	29, 30	breqtrrdi 4040	. 2 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( N / ( P ^ M ) ) < ( 0 + 1 ) )
32		simp1 997	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. NN0 )
33	32	nn0zd 9346	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. ZZ )
34		znq 9597	. . . 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 9237	. . 3 |- 0 e. ZZ
37		flqbi 10260	. . 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 944	1 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( |_ ` ( N / ( P ^ M ) ) ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2146   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   RRcr 7785   0cc0 7786   1c1 7787   + caddc 7789   x. cmul 7791   < clt 7966   <_ cle 7967   / cdiv 8602   NNcn 8892   2c2 8943   NN0cn0 9149   ZZcz 9226   ZZ>=cuz 9501   QQcq 9592   |_cfl 10238   ^cexp 10489   Primecprime 12074

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906

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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-1o 6407  df-2o 6408  df-er 6525  df-en 6731  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-3 8952  df-4 8953  df-n0 9150  df-z 9227  df-uz 9502  df-q 9593  df-rp 9625  df-fl 10240  df-seqfrec 10416  df-exp 10490  df-cj 10819  df-re 10820  df-im 10821  df-rsqrt 10975  df-abs 10976  df-dvds 11763  df-prm 12075

This theorem is referenced by:  pcfac  12315