Theorem pcprendvds 12992

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 12991	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || N ) )
4	3	simpld 112	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. NN0 )
5	4	nn0red 9556	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. RR )
6	5	ltp1d 9206	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S < ( S + 1 ) )
7	4	nn0zd 9701	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. ZZ )
8	7	peano2zd 9706	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + 1 ) e. ZZ )
9		zltnle 9625	. . . 4 |- ( ( S e. ZZ /\ ( S + 1 ) e. ZZ ) -> ( S < ( S + 1 ) <-> -. ( S + 1 ) <_ S ) )
10	7, 8, 9	syl2anc 411	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S < ( S + 1 ) <-> -. ( S + 1 ) <_ S ) )
11	6, 10	mpbid 147	. 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( S + 1 ) <_ S )
12		peano2nn0 9538	. . . 4 |- ( S e. NN0 -> ( S + 1 ) e. NN0 )
13		oveq2 6060	. . . . . . 7 |- ( x = ( S + 1 ) -> ( P ^ x ) = ( P ^ ( S + 1 ) ) )
14	13	breq1d 4121	. . . . . 6 |- ( x = ( S + 1 ) -> ( ( P ^ x ) || N <-> ( P ^ ( S + 1 ) ) || N ) )
15		oveq2 6060	. . . . . . . . 9 |- ( n = x -> ( P ^ n ) = ( P ^ x ) )
16	15	breq1d 4121	. . . . . . . 8 |- ( n = x -> ( ( P ^ n ) || N <-> ( P ^ x ) || N ) )
17	16	cbvrabv 2814	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || N } = { x e. NN0 | ( P ^ x ) || N }
18	1, 17	eqtri 2255	. . . . . 6 |- A = { x e. NN0 | ( P ^ x ) || N }
19	14, 18	elrab2 2978	. . . . 5 |- ( ( S + 1 ) e. A <-> ( ( S + 1 ) e. NN0 /\ ( P ^ ( S + 1 ) ) || N ) )
20	19	simplbi2 385	. . . 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 3326	. . . . . . . 8 |- A C_ NN0
23		nn0ssz 9597	. . . . . . . 8 |- NN0 C_ ZZ
24	22, 23	sstri 3249	. . . . . . 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 12990	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> A. x e. ZZ DECID x e. A )
27	26	adantr 276	. . . . . 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 12989	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> E. x e. ZZ A. y e. A y <_ x )
29	28	adantr 276	. . . . . 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 110	. . . . . 6 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> ( S + 1 ) e. A )
31	25, 27, 29, 30	suprzubdc 10600	. . . . 5 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> ( S + 1 ) <_ sup ( A , RR , < ) )
32	31, 2	breqtrrdi 4153	. . . 4 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> ( S + 1 ) <_ S )
33	32	ex 115	. . 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 669	1 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( S + 1 ) ) || N )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   E.wrex 2523   {crab 2526   C_ wss 3213   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   supcsup 7275   RRcr 8128   0cc0 8129   1c1 8130   + caddc 8132   < clt 8310   <_ cle 8311   2c2 9290   NN0cn0 9498   ZZcz 9579   ZZ>=cuz 9856   ^cexp 10904   || cdvds 12477

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-sup 7277  df-inf 7278  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-fz 10346  df-fzo 10481  df-fl 10634  df-mod 10689  df-seqfrec 10814  df-exp 10905  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-dvds 12478

This theorem is referenced by:  pcprendvds2  12993  pczndvds  13018