Theorem pcprendvds2 12283

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 12282	. 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( S + 1 ) ) || N )
4		eluz2nn 9562	. . . . . 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 9370	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. ZZ )
7	1, 2	pcprecl 12281	. . . . . 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 10670	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. NN )
11	10	nnzd 9370	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. ZZ )
12	10	nnne0d 8960	. . . . . 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 11790	. . . . . 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 1238	. . . . 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 11818	. . . 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 1238	. . 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 8929	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. CC )
20	19, 9	expp1d 10649	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + 1 ) ) = ( ( P ^ S ) x. P ) )
21	20	eqcomd 2183	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ S ) x. P ) = ( P ^ ( S + 1 ) ) )
22		zcn 9254	. . . . . 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 8929	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. CC )
25	10	nnap0d 8961	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) # 0 )
26	23, 24, 25	divcanap2d 8745	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ S ) x. ( N / ( P ^ S ) ) ) = N )
27	21, 26	breq12d 4015	. . 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 663	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 1353   e. wcel 2148   =/= wne 2347   {crab 2459   class class class wbr 4002   ` cfv 5215  (class class class)co 5872   supcsup 6978   CCcc 7806   RRcr 7807   0cc0 7808   1c1 7809   + caddc 7811   x. cmul 7813   < clt 7988   / cdiv 8625   NNcn 8915   2c2 8966   NN0cn0 9172   ZZcz 9249   ZZ>=cuz 9524   ^cexp 10514   || cdvds 11787

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-mulrcl 7907  ax-addcom 7908  ax-mulcom 7909  ax-addass 7910  ax-mulass 7911  ax-distr 7912  ax-i2m1 7913  ax-0lt1 7914  ax-1rid 7915  ax-0id 7916  ax-rnegex 7917  ax-precex 7918  ax-cnre 7919  ax-pre-ltirr 7920  ax-pre-ltwlin 7921  ax-pre-lttrn 7922  ax-pre-apti 7923  ax-pre-ltadd 7924  ax-pre-mulgt0 7925  ax-pre-mulext 7926  ax-arch 7927  ax-caucvg 7928

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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-isom 5224  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-recs 6303  df-frec 6389  df-sup 6980  df-inf 6981  df-pnf 7990  df-mnf 7991  df-xr 7992  df-ltxr 7993  df-le 7994  df-sub 8126  df-neg 8127  df-reap 8528  df-ap 8535  df-div 8626  df-inn 8916  df-2 8974  df-3 8975  df-4 8976  df-n0 9173  df-z 9250  df-uz 9525  df-q 9616  df-rp 9650  df-fz 10005  df-fzo 10138  df-fl 10265  df-mod 10318  df-seqfrec 10441  df-exp 10515  df-cj 10844  df-re 10845  df-im 10846  df-rsqrt 11000  df-abs 11001  df-dvds 11788

This theorem is referenced by:  pcpremul  12285  pczndvds2  12309