MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prmlem0 Structured version   Unicode version

Theorem prmlem0 13428
Description: Lemma for prmlem1 13430 and prmlem2 13442. (Contributed by Mario Carneiro, 18-Feb-2014.)
Hypotheses
Ref Expression
prmlem0.1  |-  ( ( -.  2  ||  M  /\  x  e.  ( ZZ>=
`  M ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
prmlem0.2  |-  ( K  e.  Prime  ->  -.  K  ||  N )
prmlem0.3  |-  ( K  +  2 )  =  M
Assertion
Ref Expression
prmlem0  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
Distinct variable group:    x, N
Allowed substitution hints:    K( x)    M( x)

Proof of Theorem prmlem0
StepHypRef Expression
1 eldifi 3469 . . . . 5  |-  ( x  e.  ( Prime  \  {
2 } )  ->  x  e.  Prime )
2 prmlem0.2 . . . . . 6  |-  ( K  e.  Prime  ->  -.  K  ||  N )
3 eleq1 2496 . . . . . . 7  |-  ( x  =  K  ->  (
x  e.  Prime  <->  K  e.  Prime ) )
4 breq1 4215 . . . . . . . 8  |-  ( x  =  K  ->  (
x  ||  N  <->  K  ||  N
) )
54notbid 286 . . . . . . 7  |-  ( x  =  K  ->  ( -.  x  ||  N  <->  -.  K  ||  N ) )
63, 5imbi12d 312 . . . . . 6  |-  ( x  =  K  ->  (
( x  e.  Prime  ->  -.  x  ||  N )  <-> 
( K  e.  Prime  ->  -.  K  ||  N ) ) )
72, 6mpbiri 225 . . . . 5  |-  ( x  =  K  ->  (
x  e.  Prime  ->  -.  x  ||  N ) )
81, 7syl5 30 . . . 4  |-  ( x  =  K  ->  (
x  e.  ( Prime  \  { 2 } )  ->  -.  x  ||  N
) )
98adantrd 455 . . 3  |-  ( x  =  K  ->  (
( x  e.  ( Prime  \  { 2 } )  /\  (
x ^ 2 )  <_  N )  ->  -.  x  ||  N ) )
109a1i 11 . 2  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  =  K  ->  ( (
x  e.  ( Prime  \  { 2 } )  /\  ( x ^
2 )  <_  N
)  ->  -.  x  ||  N ) ) )
11 uzp1 10519 . . 3  |-  ( x  e.  ( ZZ>= `  ( K  +  1 ) )  ->  ( x  =  ( K  + 
1 )  \/  x  e.  ( ZZ>= `  ( ( K  +  1 )  +  1 ) ) ) )
12 eleq1 2496 . . . . . . . 8  |-  ( x  =  ( K  + 
1 )  ->  (
x  e.  ( Prime  \  { 2 } )  <-> 
( K  +  1 )  e.  ( Prime  \  { 2 } ) ) )
1312adantl 453 . . . . . . 7  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  x  =  ( K  +  1 ) )  ->  ( x  e.  ( Prime  \  { 2 } )  <->  ( K  +  1 )  e.  ( Prime  \  { 2 } ) ) )
14 eldifsn 3927 . . . . . . . . 9  |-  ( ( K  +  1 )  e.  ( Prime  \  {
2 } )  <->  ( ( K  +  1 )  e.  Prime  /\  ( K  +  1 )  =/=  2 ) )
15 eluzel2 10493 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ZZ>= `  K
)  ->  K  e.  ZZ )
1615adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  K  e.  ZZ )
17 simpl 444 . . . . . . . . . . . . . . . 16  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  -.  2  ||  K )
18 1z 10311 . . . . . . . . . . . . . . . . 17  |-  1  e.  ZZ
19 2prm 13095 . . . . . . . . . . . . . . . . . 18  |-  2  e.  Prime
20 nprmdvds1 13111 . . . . . . . . . . . . . . . . . 18  |-  ( 2  e.  Prime  ->  -.  2  ||  1 )
2119, 20ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  -.  2  ||  1
22 opoe 13185 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  ZZ  /\ 
-.  2  ||  K
)  /\  ( 1  e.  ZZ  /\  -.  2  ||  1 ) )  ->  2  ||  ( K  +  1 ) )
2318, 21, 22mpanr12 667 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  ZZ  /\  -.  2  ||  K )  ->  2  ||  ( K  +  1 ) )
2416, 17, 23syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  2  ||  ( K  +  1 ) )
2524adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  2  ||  ( K  +  1 ) )
26 2z 10312 . . . . . . . . . . . . . . . 16  |-  2  e.  ZZ
27 uzid 10500 . . . . . . . . . . . . . . . 16  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
2826, 27mp1i 12 . . . . . . . . . . . . . . 15  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  2  e.  (
ZZ>= `  2 ) )
29 dvdsprm 13099 . . . . . . . . . . . . . . 15  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  ( K  +  1 )  e.  Prime )  ->  (
2  ||  ( K  +  1 )  <->  2  =  ( K  +  1
) ) )
3028, 29sylan 458 . . . . . . . . . . . . . 14  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  ( 2  ||  ( K  +  1
)  <->  2  =  ( K  +  1 ) ) )
3125, 30mpbid 202 . . . . . . . . . . . . 13  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  2  =  ( K  +  1 ) )
3231eqcomd 2441 . . . . . . . . . . . 12  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  ( K  + 
1 )  =  2 )
3332a1d 23 . . . . . . . . . . 11  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  ( x  ||  N  ->  ( K  + 
1 )  =  2 ) )
3433necon3ad 2637 . . . . . . . . . 10  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  ( ( K  +  1 )  =/=  2  ->  -.  x  ||  N ) )
3534expimpd 587 . . . . . . . . 9  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( ( K  +  1 )  e.  Prime  /\  ( K  +  1 )  =/=  2 )  ->  -.  x  ||  N ) )
3614, 35syl5bi 209 . . . . . . . 8  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( K  +  1 )  e.  ( Prime  \  { 2 } )  ->  -.  x  ||  N ) )
3736adantr 452 . . . . . . 7  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  x  =  ( K  +  1 ) )  ->  ( ( K  +  1 )  e.  ( Prime  \  {
2 } )  ->  -.  x  ||  N ) )
3813, 37sylbid 207 . . . . . 6  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  x  =  ( K  +  1 ) )  ->  ( x  e.  ( Prime  \  { 2 } )  ->  -.  x  ||  N ) )
3938adantrd 455 . . . . 5  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  x  =  ( K  +  1 ) )  ->  ( (
x  e.  ( Prime  \  { 2 } )  /\  ( x ^
2 )  <_  N
)  ->  -.  x  ||  N ) )
4039ex 424 . . . 4  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  =  ( K  +  1 )  ->  ( (
x  e.  ( Prime  \  { 2 } )  /\  ( x ^
2 )  <_  N
)  ->  -.  x  ||  N ) ) )
4116zcnd 10376 . . . . . . . . 9  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  K  e.  CC )
42 ax-1cn 9048 . . . . . . . . . 10  |-  1  e.  CC
43 addass 9077 . . . . . . . . . 10  |-  ( ( K  e.  CC  /\  1  e.  CC  /\  1  e.  CC )  ->  (
( K  +  1 )  +  1 )  =  ( K  +  ( 1  +  1 ) ) )
4442, 42, 43mp3an23 1271 . . . . . . . . 9  |-  ( K  e.  CC  ->  (
( K  +  1 )  +  1 )  =  ( K  +  ( 1  +  1 ) ) )
4541, 44syl 16 . . . . . . . 8  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( K  +  1 )  +  1 )  =  ( K  +  ( 1  +  1 ) ) )
46 1p1e2 10094 . . . . . . . . . 10  |-  ( 1  +  1 )  =  2
4746oveq2i 6092 . . . . . . . . 9  |-  ( K  +  ( 1  +  1 ) )  =  ( K  +  2 )
48 prmlem0.3 . . . . . . . . 9  |-  ( K  +  2 )  =  M
4947, 48eqtri 2456 . . . . . . . 8  |-  ( K  +  ( 1  +  1 ) )  =  M
5045, 49syl6eq 2484 . . . . . . 7  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( K  +  1 )  +  1 )  =  M )
5150fveq2d 5732 . . . . . 6  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ZZ>= `  (
( K  +  1 )  +  1 ) )  =  ( ZZ>= `  M ) )
5251eleq2d 2503 . . . . 5  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  e.  ( ZZ>= `  ( ( K  +  1 )  +  1 ) )  <-> 
x  e.  ( ZZ>= `  M ) ) )
53 dvdsaddr 12889 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  K  e.  ZZ )  ->  ( 2  ||  K  <->  2 
||  ( K  + 
2 ) ) )
5426, 16, 53sylancr 645 . . . . . . . 8  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( 2  ||  K 
<->  2  ||  ( K  +  2 ) ) )
5548breq2i 4220 . . . . . . . 8  |-  ( 2 
||  ( K  + 
2 )  <->  2  ||  M )
5654, 55syl6bb 253 . . . . . . 7  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( 2  ||  K 
<->  2  ||  M ) )
5717, 56mtbid 292 . . . . . 6  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  -.  2  ||  M )
58 prmlem0.1 . . . . . . 7  |-  ( ( -.  2  ||  M  /\  x  e.  ( ZZ>=
`  M ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
5958ex 424 . . . . . 6  |-  ( -.  2  ||  M  -> 
( x  e.  (
ZZ>= `  M )  -> 
( ( x  e.  ( Prime  \  { 2 } )  /\  (
x ^ 2 )  <_  N )  ->  -.  x  ||  N ) ) )
6057, 59syl 16 . . . . 5  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  e.  ( ZZ>= `  M )  ->  ( ( x  e.  ( Prime  \  { 2 } )  /\  (
x ^ 2 )  <_  N )  ->  -.  x  ||  N ) ) )
6152, 60sylbid 207 . . . 4  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  e.  ( ZZ>= `  ( ( K  +  1 )  +  1 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) ) )
6240, 61jaod 370 . . 3  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( x  =  ( K  + 
1 )  \/  x  e.  ( ZZ>= `  ( ( K  +  1 )  +  1 ) ) )  ->  ( (
x  e.  ( Prime  \  { 2 } )  /\  ( x ^
2 )  <_  N
)  ->  -.  x  ||  N ) ) )
6311, 62syl5 30 . 2  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  e.  ( ZZ>= `  ( K  +  1 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) ) )
64 uzp1 10519 . . 3  |-  ( x  e.  ( ZZ>= `  K
)  ->  ( x  =  K  \/  x  e.  ( ZZ>= `  ( K  +  1 ) ) ) )
6564adantl 453 . 2  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  =  K  \/  x  e.  ( ZZ>= `  ( K  +  1 ) ) ) )
6610, 63, 65mpjaod 371 1  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317   {csn 3814   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   1c1 8991    + caddc 8993    <_ cle 9121   2c2 10049   ZZcz 10282   ZZ>=cuz 10488   ^cexp 11382    || cdivides 12852   Primecprime 13079
This theorem is referenced by:  prmlem1a  13429  prmlem2  13442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-dvds 12853  df-prm 13080
  Copyright terms: Public domain W3C validator