Theorem pcfaclem 12787

Description: Lemma for pcfac 12788. (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 9355	. . . 4 |- ( N e. NN0 -> 0 <_ N )
2	1	3ad2ant1 1021	. . 3 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 0 <_ N )
3		nn0re 9339	. . . . 5 |- ( N e. NN0 -> N e. RR )
4	3	3ad2ant1 1021	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. RR )
5		prmnn 12547	. . . . . . 7 |- ( P e. Prime -> P e. NN )
6	5	3ad2ant3 1023	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> P e. NN )
7		eluznn0 9755	. . . . . . 7 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) ) -> M e. NN0 )
8	7	3adant3 1020	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> M e. NN0 )
9	6, 8	nnexpcld 10877	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. NN )
10	9	nnred 9084	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. RR )
11	9	nngt0d 9115	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 0 < ( P ^ M ) )
12		ge0div 8979	. . . 4 |- ( ( N e. RR /\ ( P ^ M ) e. RR /\ 0 < ( P ^ M ) ) -> ( 0 <_ N <-> 0 <_ ( N / ( P ^ M ) ) ) )
13	4, 10, 11, 12	syl3anc 1250	. . 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 9384	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> M e. RR )
16		eluzle 9695	. . . . . . 7 |- ( M e. ( ZZ>= ` N ) -> N <_ M )
17	16	3ad2ant2 1022	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N <_ M )
18		prmuz2 12568	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
19	18	3ad2ant3 1023	. . . . . . 7 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> P e. ( ZZ>= ` 2 ) )
20		bernneq3 10844	. . . . . . 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 8232	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N < ( P ^ M ) )
23	9	nncnd 9085	. . . . . 6 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( P ^ M ) e. CC )
24	23	mulridd 8124	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( ( P ^ M ) x. 1 ) = ( P ^ M ) )
25	22, 24	breqtrrd 4087	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N < ( ( P ^ M ) x. 1 ) )
26		1red 8122	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> 1 e. RR )
27		ltdivmul 8984	. . . . 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 1254	. . . 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 9185	. . 3 |- ( 0 + 1 ) = 1
31	29, 30	breqtrrdi 4101	. 2 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( N / ( P ^ M ) ) < ( 0 + 1 ) )
32		simp1 1000	. . . . 5 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. NN0 )
33	32	nn0zd 9528	. . . 4 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> N e. ZZ )
34		znq 9780	. . . 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 9418	. . 3 |- 0 e. ZZ
37		flqbi 10470	. . 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 947	1 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ P e. Prime ) -> ( |_ ` ( N / ( P ^ M ) ) ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   x. cmul 7965   < clt 8142   <_ cle 8143   / cdiv 8780   NNcn 9071   2c2 9122   NN0cn0 9330   ZZcz 9407   ZZ>=cuz 9683   QQcq 9775   |_cfl 10448   ^cexp 10720   Primecprime 12544

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-2o 6526  df-er 6643  df-en 6851  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fl 10450  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-prm 12545

This theorem is referenced by:  pcfac  12788