MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pcadd2 Structured version   Unicode version

Theorem pcadd2 13259
Description: The inequality of pcadd 13258 becomes an equality when one of the factors has prime count strictly less than the other. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
pcadd2.1  |-  ( ph  ->  P  e.  Prime )
pcadd2.2  |-  ( ph  ->  A  e.  QQ )
pcadd2.3  |-  ( ph  ->  B  e.  QQ )
pcadd2.4  |-  ( ph  ->  ( P  pCnt  A
)  <  ( P  pCnt  B ) )
Assertion
Ref Expression
pcadd2  |-  ( ph  ->  ( P  pCnt  A
)  =  ( P 
pCnt  ( A  +  B ) ) )

Proof of Theorem pcadd2
StepHypRef Expression
1 pcadd2.1 . . 3  |-  ( ph  ->  P  e.  Prime )
2 pcadd2.2 . . 3  |-  ( ph  ->  A  e.  QQ )
3 pcadd2.3 . . 3  |-  ( ph  ->  B  e.  QQ )
4 pcadd2.4 . . . 4  |-  ( ph  ->  ( P  pCnt  A
)  <  ( P  pCnt  B ) )
5 pcxcl 13234 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  A )  e. 
RR* )
61, 2, 5syl2anc 643 . . . . 5  |-  ( ph  ->  ( P  pCnt  A
)  e.  RR* )
7 pcxcl 13234 . . . . . 6  |-  ( ( P  e.  Prime  /\  B  e.  QQ )  ->  ( P  pCnt  B )  e. 
RR* )
81, 3, 7syl2anc 643 . . . . 5  |-  ( ph  ->  ( P  pCnt  B
)  e.  RR* )
9 xrltle 10742 . . . . 5  |-  ( ( ( P  pCnt  A
)  e.  RR*  /\  ( P  pCnt  B )  e. 
RR* )  ->  (
( P  pCnt  A
)  <  ( P  pCnt  B )  ->  ( P  pCnt  A )  <_ 
( P  pCnt  B
) ) )
106, 8, 9syl2anc 643 . . . 4  |-  ( ph  ->  ( ( P  pCnt  A )  <  ( P 
pCnt  B )  ->  ( P  pCnt  A )  <_ 
( P  pCnt  B
) ) )
114, 10mpd 15 . . 3  |-  ( ph  ->  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )
121, 2, 3, 11pcadd 13258 . 2  |-  ( ph  ->  ( P  pCnt  A
)  <_  ( P  pCnt  ( A  +  B
) ) )
13 qaddcl 10590 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  +  B
)  e.  QQ )
142, 3, 13syl2anc 643 . . . 4  |-  ( ph  ->  ( A  +  B
)  e.  QQ )
15 qnegcl 10591 . . . . 5  |-  ( B  e.  QQ  ->  -u B  e.  QQ )
163, 15syl 16 . . . 4  |-  ( ph  -> 
-u B  e.  QQ )
17 xrltnle 9144 . . . . . . . . . 10  |-  ( ( ( P  pCnt  A
)  e.  RR*  /\  ( P  pCnt  B )  e. 
RR* )  ->  (
( P  pCnt  A
)  <  ( P  pCnt  B )  <->  -.  ( P  pCnt  B )  <_ 
( P  pCnt  A
) ) )
186, 8, 17syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( ( P  pCnt  A )  <  ( P 
pCnt  B )  <->  -.  ( P  pCnt  B )  <_ 
( P  pCnt  A
) ) )
194, 18mpbid 202 . . . . . . . 8  |-  ( ph  ->  -.  ( P  pCnt  B )  <_  ( P  pCnt  A ) )
201adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  P  e.  Prime )
2116adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  -u B  e.  QQ )
2214adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  ( A  +  B )  e.  QQ )
23 pcneg 13247 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  B  e.  QQ )  ->  ( P  pCnt  -u B )  =  ( P  pCnt  B
) )
241, 3, 23syl2anc 643 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P  pCnt  -u B
)  =  ( P 
pCnt  B ) )
2524breq1d 4222 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( P  pCnt  -u B )  <_  ( P  pCnt  ( A  +  B ) )  <->  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) ) )
2625biimpar 472 . . . . . . . . . . 11  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  ( P  pCnt  -u B )  <_  ( P  pCnt  ( A  +  B ) ) )
2720, 21, 22, 26pcadd 13258 . . . . . . . . . 10  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  ( P  pCnt  -u B )  <_  ( P  pCnt  ( -u B  +  ( A  +  B ) ) ) )
2827ex 424 . . . . . . . . 9  |-  ( ph  ->  ( ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B
) )  ->  ( P  pCnt  -u B )  <_ 
( P  pCnt  ( -u B  +  ( A  +  B ) ) ) ) )
29 qcn 10588 . . . . . . . . . . . . . . 15  |-  ( B  e.  QQ  ->  B  e.  CC )
303, 29syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  CC )
3130negcld 9398 . . . . . . . . . . . . 13  |-  ( ph  -> 
-u B  e.  CC )
32 qcn 10588 . . . . . . . . . . . . . 14  |-  ( A  e.  QQ  ->  A  e.  CC )
332, 32syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  CC )
3431, 33, 30add12d 9287 . . . . . . . . . . . 12  |-  ( ph  ->  ( -u B  +  ( A  +  B
) )  =  ( A  +  ( -u B  +  B )
) )
3531, 30addcomd 9268 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( -u B  +  B )  =  ( B  +  -u B
) )
3630negidd 9401 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( B  +  -u B )  =  0 )
3735, 36eqtrd 2468 . . . . . . . . . . . . 13  |-  ( ph  ->  ( -u B  +  B )  =  0 )
3837oveq2d 6097 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  (
-u B  +  B
) )  =  ( A  +  0 ) )
3933addid1d 9266 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  0 )  =  A )
4034, 38, 393eqtrd 2472 . . . . . . . . . . 11  |-  ( ph  ->  ( -u B  +  ( A  +  B
) )  =  A )
4140oveq2d 6097 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( -u B  +  ( A  +  B ) ) )  =  ( P 
pCnt  A ) )
4224, 41breq12d 4225 . . . . . . . . 9  |-  ( ph  ->  ( ( P  pCnt  -u B )  <_  ( P  pCnt  ( -u B  +  ( A  +  B ) ) )  <-> 
( P  pCnt  B
)  <_  ( P  pCnt  A ) ) )
4328, 42sylibd 206 . . . . . . . 8  |-  ( ph  ->  ( ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B
) )  ->  ( P  pCnt  B )  <_ 
( P  pCnt  A
) ) )
4419, 43mtod 170 . . . . . . 7  |-  ( ph  ->  -.  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B
) ) )
45 pcxcl 13234 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( A  +  B )  e.  QQ )  ->  ( P  pCnt  ( A  +  B ) )  e. 
RR* )
461, 14, 45syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  e.  RR* )
47 xrltnle 9144 . . . . . . . 8  |-  ( ( ( P  pCnt  ( A  +  B )
)  e.  RR*  /\  ( P  pCnt  B )  e. 
RR* )  ->  (
( P  pCnt  ( A  +  B )
)  <  ( P  pCnt  B )  <->  -.  ( P  pCnt  B )  <_ 
( P  pCnt  ( A  +  B )
) ) )
4846, 8, 47syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( P  pCnt  ( A  +  B ) )  <  ( P 
pCnt  B )  <->  -.  ( P  pCnt  B )  <_ 
( P  pCnt  ( A  +  B )
) ) )
4944, 48mpbird 224 . . . . . 6  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <  ( P  pCnt  B ) )
50 xrltle 10742 . . . . . . 7  |-  ( ( ( P  pCnt  ( A  +  B )
)  e.  RR*  /\  ( P  pCnt  B )  e. 
RR* )  ->  (
( P  pCnt  ( A  +  B )
)  <  ( P  pCnt  B )  ->  ( P  pCnt  ( A  +  B ) )  <_ 
( P  pCnt  B
) ) )
5146, 8, 50syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( P  pCnt  ( A  +  B ) )  <  ( P 
pCnt  B )  ->  ( P  pCnt  ( A  +  B ) )  <_ 
( P  pCnt  B
) ) )
5249, 51mpd 15 . . . . 5  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  B ) )
5352, 24breqtrrd 4238 . . . 4  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  -u B ) )
541, 14, 16, 53pcadd 13258 . . 3  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  ( ( A  +  B )  +  -u B ) ) )
5533, 30, 31addassd 9110 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  +  -u B )  =  ( A  +  ( B  +  -u B ) ) )
5636oveq2d 6097 . . . . 5  |-  ( ph  ->  ( A  +  ( B  +  -u B
) )  =  ( A  +  0 ) )
5755, 56, 393eqtrd 2472 . . . 4  |-  ( ph  ->  ( ( A  +  B )  +  -u B )  =  A )
5857oveq2d 6097 . . 3  |-  ( ph  ->  ( P  pCnt  (
( A  +  B
)  +  -u B
) )  =  ( P  pCnt  A )
)
5954, 58breqtrd 4236 . 2  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  A ) )
60 xrletri3 10745 . . 3  |-  ( ( ( P  pCnt  A
)  e.  RR*  /\  ( P  pCnt  ( A  +  B ) )  e. 
RR* )  ->  (
( P  pCnt  A
)  =  ( P 
pCnt  ( A  +  B ) )  <->  ( ( P  pCnt  A )  <_ 
( P  pCnt  ( A  +  B )
)  /\  ( P  pCnt  ( A  +  B
) )  <_  ( P  pCnt  A ) ) ) )
616, 46, 60syl2anc 643 . 2  |-  ( ph  ->  ( ( P  pCnt  A )  =  ( P 
pCnt  ( A  +  B ) )  <->  ( ( P  pCnt  A )  <_ 
( P  pCnt  ( A  +  B )
)  /\  ( P  pCnt  ( A  +  B
) )  <_  ( P  pCnt  A ) ) ) )
6212, 59, 61mpbir2and 889 1  |-  ( ph  ->  ( P  pCnt  A
)  =  ( P 
pCnt  ( A  +  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   CCcc 8988   0cc0 8990    + caddc 8993   RR*cxr 9119    < clt 9120    <_ cle 9121   -ucneg 9292   QQcq 10574   Primecprime 13079    pCnt cpc 13210
This theorem is referenced by:  sylow1lem1  15232
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  ax-pre-sup 9068
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-sup 7446  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-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211
  Copyright terms: Public domain W3C validator