Theorem pcprendvds2 12854

Description: Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Hypotheses

Ref	Expression
pclem.1	|- A = { n e. NN0 | ( P ^ n ) || N }
pclem.2	|- S = sup ( A , RR , < )

Assertion

Ref	Expression
pcprendvds2	|- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( N / ( P ^ S ) ) )

Distinct variable groups:   n, N   P, n

Allowed substitution hints:   A( n)   S( n)

Proof of Theorem pcprendvds2

Step	Hyp	Ref	Expression
1		pclem.1	. . 3 |- A = { n e. NN0 | ( P ^ n ) || N }
2		pclem.2	. . 3 |- S = sup ( A , RR , < )
3	1, 2	pcprendvds 12853	. 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( S + 1 ) ) || N )
4		eluz2nn 9790	. . . . . 6 |- ( P e. ( ZZ>= ` 2 ) -> P e. NN )
5	4	adantr 276	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. NN )
6	5	nnzd 9591	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. ZZ )
7	1, 2	pcprecl 12852	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || N ) )
8	7	simprd 114	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) || N )
9	7	simpld 112	. . . . . . . 8 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. NN0 )
10	5, 9	nnexpcld 10947	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. NN )
11	10	nnzd 9591	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. ZZ )
12	10	nnne0d 9178	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) =/= 0 )
13		simprl 529	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. ZZ )
14		dvdsval2 12341	. . . . . 6 |- ( ( ( P ^ S ) e. ZZ /\ ( P ^ S ) =/= 0 /\ N e. ZZ ) -> ( ( P ^ S ) || N <-> ( N / ( P ^ S ) ) e. ZZ ) )
15	11, 12, 13, 14	syl3anc 1271	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ S ) || N <-> ( N / ( P ^ S ) ) e. ZZ ) )
16	8, 15	mpbid 147	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( N / ( P ^ S ) ) e. ZZ )
17		dvdscmul 12369	. . . 4 |- ( ( P e. ZZ /\ ( N / ( P ^ S ) ) e. ZZ /\ ( P ^ S ) e. ZZ ) -> ( P || ( N / ( P ^ S ) ) -> ( ( P ^ S ) x. P ) || ( ( P ^ S ) x. ( N / ( P ^ S ) ) ) ) )
18	6, 16, 11, 17	syl3anc 1271	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P || ( N / ( P ^ S ) ) -> ( ( P ^ S ) x. P ) || ( ( P ^ S ) x. ( N / ( P ^ S ) ) ) ) )
19	5	nncnd 9147	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. CC )
20	19, 9	expp1d 10926	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + 1 ) ) = ( ( P ^ S ) x. P ) )
21	20	eqcomd 2235	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ S ) x. P ) = ( P ^ ( S + 1 ) ) )
22		zcn 9474	. . . . . 6 |- ( N e. ZZ -> N e. CC )
23	22	ad2antrl 490	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. CC )
24	10	nncnd 9147	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. CC )
25	10	nnap0d 9179	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) # 0 )
26	23, 24, 25	divcanap2d 8962	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ S ) x. ( N / ( P ^ S ) ) ) = N )
27	21, 26	breq12d 4099	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( ( P ^ S ) x. P ) || ( ( P ^ S ) x. ( N / ( P ^ S ) ) ) <-> ( P ^ ( S + 1 ) ) || N ) )
28	18, 27	sylibd 149	. 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P || ( N / ( P ^ S ) ) -> ( P ^ ( S + 1 ) ) || N ) )
29	3, 28	mtod 667	1 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( N / ( P ^ S ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   =/= wne 2400   {crab 2512   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   supcsup 7172   CCcc 8020   RRcr 8021   0cc0 8022   1c1 8023   + caddc 8025   x. cmul 8027   < clt 8204   / cdiv 8842   NNcn 9133   2c2 9184   NN0cn0 9392   ZZcz 9469   ZZ>=cuz 9745   ^cexp 10790   || cdvds 12338

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-dvds 12339

This theorem is referenced by:  pcpremul  12856  pczndvds2  12881