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Theorem prmlem0 14593
Description: Lemma for prmlem1 14595 and prmlem2 14607. (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 3540 . . . . 5  |-  ( x  e.  ( Prime  \  {
2 } )  ->  x  e.  Prime )
2 prmlem0.2 . . . . . 6  |-  ( K  e.  Prime  ->  -.  K  ||  N )
3 eleq1 2454 . . . . . . 7  |-  ( x  =  K  ->  (
x  e.  Prime  <->  K  e.  Prime ) )
4 breq1 4370 . . . . . . . 8  |-  ( x  =  K  ->  (
x  ||  N  <->  K  ||  N
) )
54notbid 292 . . . . . . 7  |-  ( x  =  K  ->  ( -.  x  ||  N  <->  -.  K  ||  N ) )
63, 5imbi12d 318 . . . . . 6  |-  ( x  =  K  ->  (
( x  e.  Prime  ->  -.  x  ||  N )  <-> 
( K  e.  Prime  ->  -.  K  ||  N ) ) )
72, 6mpbiri 233 . . . . 5  |-  ( x  =  K  ->  (
x  e.  Prime  ->  -.  x  ||  N ) )
81, 7syl5 32 . . . 4  |-  ( x  =  K  ->  (
x  e.  ( Prime  \  { 2 } )  ->  -.  x  ||  N
) )
98adantrd 466 . . 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 11034 . . 3  |-  ( x  e.  ( ZZ>= `  ( K  +  1 ) )  ->  ( x  =  ( K  + 
1 )  \/  x  e.  ( ZZ>= `  ( ( K  +  1 )  +  1 ) ) ) )
12 eleq1 2454 . . . . . . . 8  |-  ( x  =  ( K  + 
1 )  ->  (
x  e.  ( Prime  \  { 2 } )  <-> 
( K  +  1 )  e.  ( Prime  \  { 2 } ) ) )
1312adantl 464 . . . . . . 7  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  x  =  ( K  +  1 ) )  ->  ( x  e.  ( Prime  \  { 2 } )  <->  ( K  +  1 )  e.  ( Prime  \  { 2 } ) ) )
14 eldifsn 4069 . . . . . . . . 9  |-  ( ( K  +  1 )  e.  ( Prime  \  {
2 } )  <->  ( ( K  +  1 )  e.  Prime  /\  ( K  +  1 )  =/=  2 ) )
15 eluzel2 11006 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ZZ>= `  K
)  ->  K  e.  ZZ )
1615adantl 464 . . . . . . . . . . . . . . . 16  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  K  e.  ZZ )
17 simpl 455 . . . . . . . . . . . . . . . 16  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  -.  2  ||  K )
18 1z 10811 . . . . . . . . . . . . . . . . 17  |-  1  e.  ZZ
19 n2dvds1 14037 . . . . . . . . . . . . . . . . 17  |-  -.  2  ||  1
20 opoe 14337 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  ZZ  /\ 
-.  2  ||  K
)  /\  ( 1  e.  ZZ  /\  -.  2  ||  1 ) )  ->  2  ||  ( K  +  1 ) )
2118, 19, 20mpanr12 683 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  ZZ  /\  -.  2  ||  K )  ->  2  ||  ( K  +  1 ) )
2216, 17, 21syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  2  ||  ( K  +  1 ) )
2322adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  2  ||  ( K  +  1 ) )
24 2z 10813 . . . . . . . . . . . . . . . 16  |-  2  e.  ZZ
25 uzid 11015 . . . . . . . . . . . . . . . 16  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
2624, 25mp1i 12 . . . . . . . . . . . . . . 15  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  2  e.  (
ZZ>= `  2 ) )
27 dvdsprm 14242 . . . . . . . . . . . . . . 15  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  ( K  +  1 )  e.  Prime )  ->  (
2  ||  ( K  +  1 )  <->  2  =  ( K  +  1
) ) )
2826, 27sylan 469 . . . . . . . . . . . . . 14  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  ( 2  ||  ( K  +  1
)  <->  2  =  ( K  +  1 ) ) )
2923, 28mpbid 210 . . . . . . . . . . . . 13  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  2  =  ( K  +  1 ) )
3029eqcomd 2390 . . . . . . . . . . . 12  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  ( K  + 
1 )  =  2 )
3130a1d 25 . . . . . . . . . . 11  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  ( x  ||  N  ->  ( K  + 
1 )  =  2 ) )
3231necon3ad 2592 . . . . . . . . . 10  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  ( K  + 
1 )  e.  Prime )  ->  ( ( K  +  1 )  =/=  2  ->  -.  x  ||  N ) )
3332expimpd 601 . . . . . . . . 9  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( ( K  +  1 )  e.  Prime  /\  ( K  +  1 )  =/=  2 )  ->  -.  x  ||  N ) )
3414, 33syl5bi 217 . . . . . . . 8  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( K  +  1 )  e.  ( Prime  \  { 2 } )  ->  -.  x  ||  N ) )
3534adantr 463 . . . . . . 7  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  x  =  ( K  +  1 ) )  ->  ( ( K  +  1 )  e.  ( Prime  \  {
2 } )  ->  -.  x  ||  N ) )
3613, 35sylbid 215 . . . . . 6  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  x  =  ( K  +  1 ) )  ->  ( x  e.  ( Prime  \  { 2 } )  ->  -.  x  ||  N ) )
3736adantrd 466 . . . . 5  |-  ( ( ( -.  2  ||  K  /\  x  e.  (
ZZ>= `  K ) )  /\  x  =  ( K  +  1 ) )  ->  ( (
x  e.  ( Prime  \  { 2 } )  /\  ( x ^
2 )  <_  N
)  ->  -.  x  ||  N ) )
3837ex 432 . . . 4  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  =  ( K  +  1 )  ->  ( (
x  e.  ( Prime  \  { 2 } )  /\  ( x ^
2 )  <_  N
)  ->  -.  x  ||  N ) ) )
3916zcnd 10885 . . . . . . . . 9  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  K  e.  CC )
40 ax-1cn 9461 . . . . . . . . . 10  |-  1  e.  CC
41 addass 9490 . . . . . . . . . 10  |-  ( ( K  e.  CC  /\  1  e.  CC  /\  1  e.  CC )  ->  (
( K  +  1 )  +  1 )  =  ( K  +  ( 1  +  1 ) ) )
4240, 40, 41mp3an23 1314 . . . . . . . . 9  |-  ( K  e.  CC  ->  (
( K  +  1 )  +  1 )  =  ( K  +  ( 1  +  1 ) ) )
4339, 42syl 16 . . . . . . . 8  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( K  +  1 )  +  1 )  =  ( K  +  ( 1  +  1 ) ) )
44 1p1e2 10566 . . . . . . . . . 10  |-  ( 1  +  1 )  =  2
4544oveq2i 6207 . . . . . . . . 9  |-  ( K  +  ( 1  +  1 ) )  =  ( K  +  2 )
46 prmlem0.3 . . . . . . . . 9  |-  ( K  +  2 )  =  M
4745, 46eqtri 2411 . . . . . . . 8  |-  ( K  +  ( 1  +  1 ) )  =  M
4843, 47syl6eq 2439 . . . . . . 7  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( K  +  1 )  +  1 )  =  M )
4948fveq2d 5778 . . . . . 6  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ZZ>= `  (
( K  +  1 )  +  1 ) )  =  ( ZZ>= `  M ) )
5049eleq2d 2452 . . . . 5  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  e.  ( ZZ>= `  ( ( K  +  1 )  +  1 ) )  <-> 
x  e.  ( ZZ>= `  M ) ) )
51 dvdsaddr 14027 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  K  e.  ZZ )  ->  ( 2  ||  K  <->  2 
||  ( K  + 
2 ) ) )
5224, 16, 51sylancr 661 . . . . . . . 8  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( 2  ||  K 
<->  2  ||  ( K  +  2 ) ) )
5346breq2i 4375 . . . . . . . 8  |-  ( 2 
||  ( K  + 
2 )  <->  2  ||  M )
5452, 53syl6bb 261 . . . . . . 7  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( 2  ||  K 
<->  2  ||  M ) )
5517, 54mtbid 298 . . . . . 6  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  -.  2  ||  M )
56 prmlem0.1 . . . . . . 7  |-  ( ( -.  2  ||  M  /\  x  e.  ( ZZ>=
`  M ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
5756ex 432 . . . . . 6  |-  ( -.  2  ||  M  -> 
( x  e.  (
ZZ>= `  M )  -> 
( ( x  e.  ( Prime  \  { 2 } )  /\  (
x ^ 2 )  <_  N )  ->  -.  x  ||  N ) ) )
5855, 57syl 16 . . . . 5  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  e.  ( ZZ>= `  M )  ->  ( ( x  e.  ( Prime  \  { 2 } )  /\  (
x ^ 2 )  <_  N )  ->  -.  x  ||  N ) ) )
5950, 58sylbid 215 . . . 4  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  e.  ( ZZ>= `  ( ( K  +  1 )  +  1 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) ) )
6038, 59jaod 378 . . 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 ) ) )
6111, 60syl5 32 . 2  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  e.  ( ZZ>= `  ( K  +  1 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) ) )
62 uzp1 11034 . . 3  |-  ( x  e.  ( ZZ>= `  K
)  ->  ( x  =  K  \/  x  e.  ( ZZ>= `  ( K  +  1 ) ) ) )
6362adantl 464 . 2  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( x  =  K  \/  x  e.  ( ZZ>= `  ( K  +  1 ) ) ) )
6410, 61, 63mpjaod 379 1  |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>=
`  K ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577    \ cdif 3386   {csn 3944   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   CCcc 9401   1c1 9404    + caddc 9406    <_ cle 9540   2c2 10502   ZZcz 10781   ZZ>=cuz 11001   ^cexp 12069    || cdvds 13988   Primecprime 14219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-dvds 13989  df-prm 14220
This theorem is referenced by:  prmlem1a  14594  prmlem2  14607
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