Theorem pcprendvds 12828

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 12827	. . . . . 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 9434	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. RR )
6	5	ltp1d 9088	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S < ( S + 1 ) )
7	4	nn0zd 9578	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. ZZ )
8	7	peano2zd 9583	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + 1 ) e. ZZ )
9		zltnle 9503	. . . 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 9420	. . . 4 |- ( S e. NN0 -> ( S + 1 ) e. NN0 )
13		oveq2 6015	. . . . . . 7 |- ( x = ( S + 1 ) -> ( P ^ x ) = ( P ^ ( S + 1 ) ) )
14	13	breq1d 4093	. . . . . 6 |- ( x = ( S + 1 ) -> ( ( P ^ x ) || N <-> ( P ^ ( S + 1 ) ) || N ) )
15		oveq2 6015	. . . . . . . . 9 |- ( n = x -> ( P ^ n ) = ( P ^ x ) )
16	15	breq1d 4093	. . . . . . . 8 |- ( n = x -> ( ( P ^ n ) || N <-> ( P ^ x ) || N ) )
17	16	cbvrabv 2798	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || N } = { x e. NN0 | ( P ^ x ) || N }
18	1, 17	eqtri 2250	. . . . . 6 |- A = { x e. NN0 | ( P ^ x ) || N }
19	14, 18	elrab2 2962	. . . . 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 3310	. . . . . . . 8 |- A C_ NN0
23		nn0ssz 9475	. . . . . . . 8 |- NN0 C_ ZZ
24	22, 23	sstri 3233	. . . . . . 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 12826	. . . . . . 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 12825	. . . . . . 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 10468	. . . . 5 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> ( S + 1 ) <_ sup ( A , RR , < ) )
32	31, 2	breqtrrdi 4125	. . . 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 667	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 839   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   E.wrex 2509   {crab 2512   C_ wss 3197   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   supcsup 7160   RRcr 8009   0cc0 8010   1c1 8011   + caddc 8013   < clt 8192   <_ cle 8193   2c2 9172   NN0cn0 9380   ZZcz 9457   ZZ>=cuz 9733   ^cexp 10772   || cdvds 12313

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-cj 11368  df-re 11369  df-im 11370  df-rsqrt 11524  df-abs 11525  df-dvds 12314

This theorem is referenced by:  pcprendvds2  12829  pczndvds  12854