Theorem pcprendvds2 13070

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 13069	. 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( S + 1 ) ) || N )
4		eluz2nn 9966	. . . . . 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 9767	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. ZZ )
7	1, 2	pcprecl 13068	. . . . . 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 11133	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. NN )
11	10	nnzd 9767	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. ZZ )
12	10	nnne0d 9349	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) =/= 0 )
13		simprl 535	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. ZZ )
14		dvdsval2 12557	. . . . . 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 1278	. . . . 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 12585	. . . 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 1278	. . 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 9318	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. CC )
20	19, 9	expp1d 11112	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + 1 ) ) = ( ( P ^ S ) x. P ) )
21	20	eqcomd 2244	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ S ) x. P ) = ( P ^ ( S + 1 ) ) )
22		zcn 9649	. . . . . 6 |- ( N e. ZZ -> N e. CC )
23	22	ad2antrl 494	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. CC )
24	10	nncnd 9318	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. CC )
25	10	nnap0d 9350	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) # 0 )
26	23, 24, 25	divcanap2d 9122	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ S ) x. ( N / ( P ^ S ) ) ) = N )
27	21, 26	breq12d 4143	. . 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 673	1 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( N / ( P ^ S ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   supcsup 7322   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   / cdiv 9002   NNcn 9304   2c2 9355   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   ^cexp 10975   || cdvds 12554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555

This theorem is used by:  pcpremul  13072  pczndvds2  13097