Theorem pcprendvds 12181

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
pcprendvds	|- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( S + 1 ) ) || N )

Distinct variable groups:   n, N   P, n

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

Proof of Theorem pcprendvds

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pclem.1	. . . . . . 7 |- A = { n e. NN0 | ( P ^ n ) || N }
2		pclem.2	. . . . . . 7 |- S = sup ( A , RR , < )
3	1, 2	pcprecl 12180	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || N ) )
4	3	simpld 111	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. NN0 )
5	4	nn0red 9150	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. RR )
6	5	ltp1d 8807	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S < ( S + 1 ) )
7	4	nn0zd 9290	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. ZZ )
8	7	peano2zd 9295	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + 1 ) e. ZZ )
9		zltnle 9219	. . . 4 |- ( ( S e. ZZ /\ ( S + 1 ) e. ZZ ) -> ( S < ( S + 1 ) <-> -. ( S + 1 ) <_ S ) )
10	7, 8, 9	syl2anc 409	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S < ( S + 1 ) <-> -. ( S + 1 ) <_ S ) )
11	6, 10	mpbid 146	. 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( S + 1 ) <_ S )
12		peano2nn0 9136	. . . 4 |- ( S e. NN0 -> ( S + 1 ) e. NN0 )
13		oveq2 5835	. . . . . . 7 |- ( x = ( S + 1 ) -> ( P ^ x ) = ( P ^ ( S + 1 ) ) )
14	13	breq1d 3977	. . . . . 6 |- ( x = ( S + 1 ) -> ( ( P ^ x ) || N <-> ( P ^ ( S + 1 ) ) || N ) )
15		oveq2 5835	. . . . . . . . 9 |- ( n = x -> ( P ^ n ) = ( P ^ x ) )
16	15	breq1d 3977	. . . . . . . 8 |- ( n = x -> ( ( P ^ n ) || N <-> ( P ^ x ) || N ) )
17	16	cbvrabv 2711	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || N } = { x e. NN0 | ( P ^ x ) || N }
18	1, 17	eqtri 2178	. . . . . 6 |- A = { x e. NN0 | ( P ^ x ) || N }
19	14, 18	elrab2 2871	. . . . 5 |- ( ( S + 1 ) e. A <-> ( ( S + 1 ) e. NN0 /\ ( P ^ ( S + 1 ) ) || N ) )
20	19	simplbi2 383	. . . 4 |- ( ( S + 1 ) e. NN0 -> ( ( P ^ ( S + 1 ) ) || N -> ( S + 1 ) e. A ) )
21	4, 12, 20	3syl 17	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ ( S + 1 ) ) || N -> ( S + 1 ) e. A ) )
22	1	ssrab3 3214	. . . . . . . 8 |- A C_ NN0
23		nn0ssz 9191	. . . . . . . 8 |- NN0 C_ ZZ
24	22, 23	sstri 3137	. . . . . . 7 |- A C_ ZZ
25	24	a1i 9	. . . . . 6 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> A C_ ZZ )
26	1	pclemdc 12179	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> A. x e. ZZ DECID x e. A )
27	26	adantr 274	. . . . . 6 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> A. x e. ZZ DECID x e. A )
28	1	pclemub 12178	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> E. x e. ZZ A. y e. A y <_ x )
29	28	adantr 274	. . . . . 6 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> E. x e. ZZ A. y e. A y <_ x )
30		simpr 109	. . . . . 6 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> ( S + 1 ) e. A )
31	25, 27, 29, 30	suprzubdc 11852	. . . . 5 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> ( S + 1 ) <_ sup ( A , RR , < ) )
32	31, 2	breqtrrdi 4009	. . . 4 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> ( S + 1 ) <_ S )
33	32	ex 114	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( S + 1 ) e. A -> ( S + 1 ) <_ S ) )
34	21, 33	syld 45	. 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ ( S + 1 ) ) || N -> ( S + 1 ) <_ S ) )
35	11, 34	mtod 653	1 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( S + 1 ) ) || N )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 820   = wceq 1335   e. wcel 2128   =/= wne 2327   A.wral 2435   E.wrex 2436   {crab 2439   C_ wss 3102   class class class wbr 3967   ` cfv 5173  (class class class)co 5827   supcsup 6929   RRcr 7734   0cc0 7735   1c1 7736   + caddc 7738   < clt 7915   <_ cle 7916   2c2 8890   NN0cn0 9096   ZZcz 9173   ZZ>=cuz 9445   ^cexp 10428   || cdvds 11695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-mulrcl 7834  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-mulass 7838  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-1rid 7842  ax-0id 7843  ax-rnegex 7844  ax-precex 7845  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-apti 7850  ax-pre-ltadd 7851  ax-pre-mulgt0 7852  ax-pre-mulext 7853  ax-arch 7854  ax-caucvg 7855

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-po 4259  df-iso 4260  df-iord 4329  df-on 4331  df-ilim 4332  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-isom 5182  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-frec 6341  df-sup 6931  df-inf 6932  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-reap 8455  df-ap 8462  df-div 8551  df-inn 8840  df-2 8898  df-3 8899  df-4 8900  df-n0 9097  df-z 9174  df-uz 9446  df-q 9536  df-rp 9568  df-fz 9920  df-fzo 10052  df-fl 10179  df-mod 10232  df-seqfrec 10355  df-exp 10429  df-cj 10754  df-re 10755  df-im 10756  df-rsqrt 10910  df-abs 10911  df-dvds 11696

This theorem is referenced by:  pcprendvds2  12182  pczndvds  12205