Theorem pcfaclem 13002

Description: Lemma for pcfac 13003. (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 9486	. . . 4 |- ( N e. NN0 -> 0 <_ N )
2	1	3ad2ant1 1045	. . 3 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 0 <_ N )
3		nn0re 9470	. . . . 5 |- ( N e. NN0 -> N e. RR )
4	3	3ad2ant1 1045	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. RR )
5		prmnn 12762	. . . . . . 7 |- ( P e. Prime -> P e. NN )
6	5	3ad2ant3 1047	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> P e. NN )
7		eluznn0 9894	. . . . . . 7 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) ) -> M e. NN0 )
8	7	3adant3 1044	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> M e. NN0 )
9	6, 8	nnexpcld 11020	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. NN )
10	9	nnred 9215	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. RR )
11	9	nngt0d 9246	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 0 < ( P ^ M ) )
12		ge0div 9110	. . . 4 |- ( ( N e. RR /\ ( P ^ M ) e. RR /\ 0 < ( P ^ M ) ) -> ( 0 <_ N <-> 0 <_ ( N / ( P ^ M ) ) ) )
13	4, 10, 11, 12	syl3anc 1274	. . 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 9517	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> M e. RR )
16		eluzle 9829	. . . . . . 7 |- ( M e. ( ZZ>= ` N ) -> N <_ M )
17	16	3ad2ant2 1046	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N <_ M )
18		prmuz2 12783	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
19	18	3ad2ant3 1047	. . . . . . 7 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> P e. ( ZZ>= ` 2 ) )
20		bernneq3 10987	. . . . . . 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 8363	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N < ( P ^ M ) )
23	9	nncnd 9216	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. CC )
24	23	mulridd 8256	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( ( P ^ M ) x. 1 ) = ( P ^ M ) )
25	22, 24	breqtrrd 4121	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N < ( ( P ^ M ) x. 1 ) )
26		1red 8254	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 1 e. RR )
27		ltdivmul 9115	. . . . 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 1278	. . . 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 9316	. . 3 |- ( 0 + 1 ) = 1
31	29, 30	breqtrrdi 4135	. 2 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( N / ( P ^ M ) ) < ( 0 + 1 ) )
32		simp1 1024	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. NN0 )
33	32	nn0zd 9661	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. ZZ )
34		znq 9919	. . . 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 9551	. . 3 |- 0 e. ZZ
37		flqbi 10613	. . 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 953	1 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( |_ ` ( N / ( P ^ M ) ) ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   RRcr 8091   0cc0 8092   1c1 8093   + caddc 8095   x. cmul 8097   < clt 8273   <_ cle 8274   / cdiv 8911   NNcn 9202   2c2 9253   NN0cn0 9461   ZZcz 9540   ZZ>=cuz 9816   QQcq 9914   |_cfl 10591   ^cexp 10863   Primecprime 12759

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fl 10593  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-dvds 12429  df-prm 12760

This theorem is referenced by:  pcfac  13003