Theorem pcfaclem 12872

Description: Lemma for pcfac 12873. (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 9394	. . . 4 |- ( N e. NN0 -> 0 <_ N )
2	1	3ad2ant1 1042	. . 3 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 0 <_ N )
3		nn0re 9378	. . . . 5 |- ( N e. NN0 -> N e. RR )
4	3	3ad2ant1 1042	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. RR )
5		prmnn 12632	. . . . . . 7 |- ( P e. Prime -> P e. NN )
6	5	3ad2ant3 1044	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> P e. NN )
7		eluznn0 9794	. . . . . . 7 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) ) -> M e. NN0 )
8	7	3adant3 1041	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> M e. NN0 )
9	6, 8	nnexpcld 10917	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. NN )
10	9	nnred 9123	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. RR )
11	9	nngt0d 9154	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 0 < ( P ^ M ) )
12		ge0div 9018	. . . 4 |- ( ( N e. RR /\ ( P ^ M ) e. RR /\ 0 < ( P ^ M ) ) -> ( 0 <_ N <-> 0 <_ ( N / ( P ^ M ) ) ) )
13	4, 10, 11, 12	syl3anc 1271	. . 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 9423	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> M e. RR )
16		eluzle 9734	. . . . . . 7 |- ( M e. ( ZZ>= ` N ) -> N <_ M )
17	16	3ad2ant2 1043	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N <_ M )
18		prmuz2 12653	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
19	18	3ad2ant3 1044	. . . . . . 7 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> P e. ( ZZ>= ` 2 ) )
20		bernneq3 10884	. . . . . . 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 8271	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N < ( P ^ M ) )
23	9	nncnd 9124	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. CC )
24	23	mulridd 8163	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( ( P ^ M ) x. 1 ) = ( P ^ M ) )
25	22, 24	breqtrrd 4111	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N < ( ( P ^ M ) x. 1 ) )
26		1red 8161	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 1 e. RR )
27		ltdivmul 9023	. . . . 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 1275	. . . 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 9224	. . 3 |- ( 0 + 1 ) = 1
31	29, 30	breqtrrdi 4125	. 2 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( N / ( P ^ M ) ) < ( 0 + 1 ) )
32		simp1 1021	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. NN0 )
33	32	nn0zd 9567	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. ZZ )
34		znq 9819	. . . 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 9457	. . 3 |- 0 e. ZZ
37		flqbi 10510	. . 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 950	1 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( |_ ` ( N / ( P ^ M ) ) ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   RRcr 7998   0cc0 7999   1c1 8000   + caddc 8002   x. cmul 8004   < clt 8181   <_ cle 8182   / cdiv 8819   NNcn 9110   2c2 9161   NN0cn0 9369   ZZcz 9446   ZZ>=cuz 9722   QQcq 9814   |_cfl 10488   ^cexp 10760   Primecprime 12629

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-2o 6563  df-er 6680  df-en 6888  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fl 10490  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-prm 12630

This theorem is referenced by:  pcfac  12873