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Theorem prmlem0 13107
Description: Lemma for prmlem1 13109 and prmlem2 13121. (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 3298 . . . . 5  |-  ( x  e.  ( Prime  \  {
2 } )  ->  x  e.  Prime )
2 prmlem0.2 . . . . . 6  |-  ( K  e.  Prime  ->  -.  K  ||  N )
3 eleq1 2343 . . . . . . 7  |-  ( x  =  K  ->  (
x  e.  Prime  <->  K  e.  Prime ) )
4 breq1 4026 . . . . . . . 8  |-  ( x  =  K  ->  (
x  ||  N  <->  K  ||  N
) )
54notbid 285 . . . . . . 7  |-  ( x  =  K  ->  ( -.  x  ||  N  <->  -.  K  ||  N ) )
63, 5imbi12d 311 . . . . . 6  |-  ( x  =  K  ->  (
( x  e.  Prime  ->  -.  x  ||  N )  <-> 
( K  e.  Prime  ->  -.  K  ||  N ) ) )
72, 6mpbiri 224 . . . . 5  |-  ( x  =  K  ->  (
x  e.  Prime  ->  -.  x  ||  N ) )
81, 7syl5 28 . . . 4  |-  ( x  =  K  ->  (
x  e.  ( Prime  \  { 2 } )  ->  -.  x  ||  N
) )
98adantrd 454 . . 3  |-  ( x  =  K  ->  (
( x  e.  ( Prime  \  { 2 } )  /\  (
x ^ 2 )  <_  N )  ->  -.  x  ||  N ) )
109a1i 10 . 2  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  =  K  ->  ( (
x  e.  ( Prime  \  { 2 } )  /\  ( x ^
2 )  <_  N
)  ->  -.  x  ||  N ) ) )
11 uzp1 10261 . . 3  |-  ( x  e.  ( ZZ>= `  ( K  +  1 ) )  ->  ( x  =  ( K  + 
1 )  \/  x  e.  ( ZZ>= `  ( ( K  +  1 )  +  1 ) ) ) )
12 eleq1 2343 . . . . . . . 8  |-  ( x  =  ( K  + 
1 )  ->  (
x  e.  ( Prime  \  { 2 } )  <-> 
( K  +  1 )  e.  ( Prime  \  { 2 } ) ) )
1312adantl 452 . . . . . . 7  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  x  =  ( K  +  1 ) )  ->  ( x  e.  ( Prime  \  { 2 } )  <->  ( K  +  1 )  e.  ( Prime  \  { 2 } ) ) )
14 eldifsn 3749 . . . . . . . . 9  |-  ( ( K  +  1 )  e.  ( Prime  \  {
2 } )  <->  ( ( K  +  1 )  e.  Prime  /\  ( K  +  1 )  =/=  2 ) )
15 eluzel2 10235 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ZZ>= `  K
)  ->  K  e.  ZZ )
1615adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  K  e.  ZZ )
17 simpl 443 . . . . . . . . . . . . . . . 16  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  -.  2  ||  K )
18 1z 10053 . . . . . . . . . . . . . . . . 17  |-  1  e.  ZZ
19 2prm 12774 . . . . . . . . . . . . . . . . . 18  |-  2  e.  Prime
20 nprmdvds1 12790 . . . . . . . . . . . . . . . . . 18  |-  ( 2  e.  Prime  ->  -.  2  ||  1 )
2119, 20ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  -.  2  ||  1
22 opoe 12864 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  ZZ  /\ 
-.  2  ||  K
)  /\  ( 1  e.  ZZ  /\  -.  2  ||  1 ) )  ->  2  ||  ( K  +  1 ) )
2318, 21, 22mpanr12 666 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  ZZ  /\  -.  2  ||  K )  ->  2  ||  ( K  +  1 ) )
2416, 17, 23syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  2  ||  ( K  +  1 ) )
2524adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  2  ||  ( K  +  1 ) )
26 2z 10054 . . . . . . . . . . . . . . . 16  |-  2  e.  ZZ
27 uzid 10242 . . . . . . . . . . . . . . . 16  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
2826, 27mp1i 11 . . . . . . . . . . . . . . 15  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  2  e.  (
ZZ>= `  2 ) )
29 dvdsprm 12778 . . . . . . . . . . . . . . 15  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  ( K  +  1 )  e.  Prime )  ->  (
2  ||  ( K  +  1 )  <->  2  =  ( K  +  1
) ) )
3028, 29sylan 457 . . . . . . . . . . . . . 14  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  ( 2  ||  ( K  +  1
)  <->  2  =  ( K  +  1 ) ) )
3125, 30mpbid 201 . . . . . . . . . . . . 13  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  2  =  ( K  +  1 ) )
3231eqcomd 2288 . . . . . . . . . . . 12  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  ( K  + 
1 )  =  2 )
3332a1d 22 . . . . . . . . . . 11  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  ( x  ||  N  ->  ( K  + 
1 )  =  2 ) )
3433necon3ad 2482 . . . . . . . . . 10  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  ( ( K  +  1 )  =/=  2  ->  -.  x  ||  N ) )
3534expimpd 586 . . . . . . . . 9  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( ( K  +  1 )  e.  Prime  /\  ( K  +  1 )  =/=  2 )  ->  -.  x  ||  N ) )
3614, 35syl5bi 208 . . . . . . . 8  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( K  +  1 )  e.  ( Prime  \  { 2 } )  ->  -.  x  ||  N ) )
3736adantr 451 . . . . . . 7  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  x  =  ( K  +  1 ) )  ->  ( ( K  +  1 )  e.  ( Prime  \  {
2 } )  ->  -.  x  ||  N ) )
3813, 37sylbid 206 . . . . . 6  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  x  =  ( K  +  1 ) )  ->  ( x  e.  ( Prime  \  { 2 } )  ->  -.  x  ||  N ) )
3938adantrd 454 . . . . 5  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  x  =  ( K  +  1 ) )  ->  ( (
x  e.  ( Prime  \  { 2 } )  /\  ( x ^
2 )  <_  N
)  ->  -.  x  ||  N ) )
4039ex 423 . . . 4  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  =  ( K  +  1 )  ->  ( (
x  e.  ( Prime  \  { 2 } )  /\  ( x ^
2 )  <_  N
)  ->  -.  x  ||  N ) ) )
4116zcnd 10118 . . . . . . . . 9  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  K  e.  CC )
42 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
43 addass 8824 . . . . . . . . . 10  |-  ( ( K  e.  CC  /\  1  e.  CC  /\  1  e.  CC )  ->  (
( K  +  1 )  +  1 )  =  ( K  +  ( 1  +  1 ) ) )
4442, 42, 43mp3an23 1269 . . . . . . . . 9  |-  ( K  e.  CC  ->  (
( K  +  1 )  +  1 )  =  ( K  +  ( 1  +  1 ) ) )
4541, 44syl 15 . . . . . . . 8  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( K  +  1 )  +  1 )  =  ( K  +  ( 1  +  1 ) ) )
46 1p1e2 9840 . . . . . . . . . 10  |-  ( 1  +  1 )  =  2
4746oveq2i 5869 . . . . . . . . 9  |-  ( K  +  ( 1  +  1 ) )  =  ( K  +  2 )
48 prmlem0.3 . . . . . . . . 9  |-  ( K  +  2 )  =  M
4947, 48eqtri 2303 . . . . . . . 8  |-  ( K  +  ( 1  +  1 ) )  =  M
5045, 49syl6eq 2331 . . . . . . 7  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( K  +  1 )  +  1 )  =  M )
5150fveq2d 5529 . . . . . 6  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ZZ>= `  (
( K  +  1 )  +  1 ) )  =  ( ZZ>= `  M ) )
5251eleq2d 2350 . . . . 5  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  e.  ( ZZ>= `  ( ( K  +  1 )  +  1 ) )  <-> 
x  e.  ( ZZ>= `  M ) ) )
53 dvdsaddr 12568 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  K  e.  ZZ )  ->  ( 2  ||  K  <->  2 
||  ( K  + 
2 ) ) )
5426, 16, 53sylancr 644 . . . . . . . 8  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( 2  ||  K 
<->  2  ||  ( K  +  2 ) ) )
5548breq2i 4031 . . . . . . . 8  |-  ( 2 
||  ( K  + 
2 )  <->  2  ||  M )
5654, 55syl6bb 252 . . . . . . 7  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( 2  ||  K 
<->  2  ||  M ) )
5717, 56mtbid 291 . . . . . 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 423 . . . . . 6  |-  ( -.  2  ||  M  -> 
( x  e.  (
ZZ>= `  M )  -> 
( ( x  e.  ( Prime  \  { 2 } )  /\  (
x ^ 2 )  <_  N )  ->  -.  x  ||  N ) ) )
6057, 59syl 15 . . . . 5  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  e.  ( ZZ>= `  M )  ->  ( ( x  e.  ( Prime  \  { 2 } )  /\  (
x ^ 2 )  <_  N )  ->  -.  x  ||  N ) ) )
6152, 60sylbid 206 . . . 4  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  e.  ( ZZ>= `  ( ( K  +  1 )  +  1 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) ) )
6240, 61jaod 369 . . 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 28 . 2  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  e.  ( ZZ>= `  ( K  +  1 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) ) )
64 uzp1 10261 . . 3  |-  ( x  e.  ( ZZ>= `  K
)  ->  ( x  =  K  \/  x  e.  ( ZZ>= `  ( K  +  1 ) ) ) )
6564adantl 452 . 2  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  =  K  \/  x  e.  ( ZZ>= `  ( K  +  1 ) ) ) )
6610, 63, 65mpjaod 370 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 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    <_ cle 8868   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   ^cexp 11104    || cdivides 12531   Primecprime 12758
This theorem is referenced by:  prmlem1a  13108  prmlem2  13121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-dvds 12532  df-prm 12759
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