ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pcprendvds Unicode version
Theorem pcprendvds 13098
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.
StepHypRef Expression
1 pclem.1 . . . . . . 7 |- A = { n e. NN0 | ( P ^ n ) || N }
2 pclem.2 . . . . . . 7 |- S = sup ( A , RR , < )
31, 2pcprecl 13097 . . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || N ) )
43simpld 112 . . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. NN0 )
54nn0red 9632 . . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. RR )
65ltp1d 9268 . . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S < ( S + 1 ) )
74nn0zd 9777 . . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. ZZ )
87peano2zd 9782 . . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + 1 ) e. ZZ )
9 zltnle 9701 . . . 4 |- ( ( S e. ZZ /\ ( S + 1 ) e. ZZ ) -> ( S < ( S + 1 ) <-> -. ( S + 1 ) <_ S ) )
107, 8, 9syl2anc 415 . . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S < ( S + 1 ) <-> -. ( S + 1 ) <_ S ) )
116, 10mpbid 147 . 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( S + 1 ) <_ S )
12 peano2nn0 9614 . . . 4 |- ( S e. NN0 -> ( S + 1 ) e. NN0 )
13 oveq2 6096 . . . . . . 7 |- ( x = ( S + 1 ) -> ( P ^ x ) = ( P ^ ( S + 1 ) ) )
1413breq1d 4141 . . . . . 6 |- ( x = ( S + 1 ) -> ( ( P ^ x ) || N <-> ( P ^ ( S + 1 ) ) || N ) )
15 oveq2 6096 . . . . . . . . 9 |- ( n = x -> ( P ^ n ) = ( P ^ x ) )
1615breq1d 4141 . . . . . . . 8 |- ( n = x -> ( ( P ^ n ) || N <-> ( P ^ x ) || N ) )
1716cbvrabv 2820 . . . . . . 7 |- { n e. NN0 | ( P ^ n ) || N } = { x e. NN0 | ( P ^ x ) || N }
181, 17eqtri 2259 . . . . . 6 |- A = { x e. NN0 | ( P ^ x ) || N }
1914, 18elrab2 2985 . . . . 5 |- ( ( S + 1 ) e. A <-> ( ( S + 1 ) e. NN0 /\ ( P ^ ( S + 1 ) ) || N ) )
2019simplbi2 385 . . . 4 |- ( ( S + 1 ) e. NN0 -> ( ( P ^ ( S + 1 ) ) || N -> ( S + 1 ) e. A ) )
214, 12, 203syl 17 . . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ ( S + 1 ) ) || N -> ( S + 1 ) e. A ) )
221ssrab3 3334 . . . . . . . 8 |- A C_ NN0
23 nn0ssz 9673 . . . . . . . 8 |- NN0 C_ ZZ
2422, 23sstri 3257 . . . . . . 7 |- A C_ ZZ
2524a1i 9 . . . . . 6 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> A C_ ZZ )
261pclemdc 13096 . . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> A. x e. ZZ DECID x e. A )
2726adantr 276 . . . . . 6 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> A. x e. ZZ DECID x e. A )
281pclemub 13095 . . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> E. x e. ZZ A. y e. A y <_ x )
2928adantr 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 )
3125, 27, 29, 30suprzubdc 10688 . . . . 5 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> ( S + 1 ) <_ sup ( A , RR , < ) )
3231, 2breqtrrdi 4173 . . . 4 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> ( S + 1 ) <_ S )
3332ex 115 . . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( S + 1 ) e. A -> ( S + 1 ) <_ S ) )
3421, 33syld 45 . 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ ( S + 1 ) ) || N -> ( S + 1 ) <_ S ) )
3511, 34mtod 673 1 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( S + 1 ) ) || N )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   E.wrex 2529   {crab 2532   C_ wss 3220   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   supcsup 7328   RRcr 8184   0cc0 8185   1c1 8186   + caddc 8188   < clt 8366   <_ cle 8367   2c2 9364   NN0cn0 9574   ZZcz 9655   ZZ>=cuz 9937   ^cexp 10996   || cdvds 12579
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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305
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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-sup 7330  df-inf 7331  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-dvds 12580
This theorem is used by:  pcprendvds2  13099  pczndvds  13124
  Copyright terms: Public domain W3C validator