Theorem pcprendvds2 12614

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 12613	. 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( S + 1 ) ) || N )
4		eluz2nn 9687	. . . . . 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 9494	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. ZZ )
7	1, 2	pcprecl 12612	. . . . . 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 10840	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. NN )
11	10	nnzd 9494	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. ZZ )
12	10	nnne0d 9081	. . . . . 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 12101	. . . . . 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 1250	. . . . 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 12129	. . . 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 1250	. . 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 9050	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. CC )
20	19, 9	expp1d 10819	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + 1 ) ) = ( ( P ^ S ) x. P ) )
21	20	eqcomd 2211	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ S ) x. P ) = ( P ^ ( S + 1 ) ) )
22		zcn 9377	. . . . . 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 9050	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. CC )
25	10	nnap0d 9082	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) # 0 )
26	23, 24, 25	divcanap2d 8865	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ S ) x. ( N / ( P ^ S ) ) ) = N )
27	21, 26	breq12d 4057	. . 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 665	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 1373   e. wcel 2176   =/= wne 2376   {crab 2488   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   supcsup 7084   CCcc 7923   RRcr 7924   0cc0 7925   1c1 7926   + caddc 7928   x. cmul 7930   < clt 8107   / cdiv 8745   NNcn 9036   2c2 9087   NN0cn0 9295   ZZcz 9372   ZZ>=cuz 9648   ^cexp 10683   || cdvds 12098

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-sup 7086  df-inf 7087  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-fz 10131  df-fzo 10265  df-fl 10413  df-mod 10468  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-dvds 12099

This theorem is referenced by:  pcpremul  12616  pczndvds2  12641