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Theorem pcadd2 13952
Description: The inequality of pcadd 13951 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 13927 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  A )  e. 
RR* )
61, 2, 5syl2anc 661 . . . . 5  |-  ( ph  ->  ( P  pCnt  A
)  e.  RR* )
7 pcxcl 13927 . . . . . 6  |-  ( ( P  e.  Prime  /\  B  e.  QQ )  ->  ( P  pCnt  B )  e. 
RR* )
81, 3, 7syl2anc 661 . . . . 5  |-  ( ph  ->  ( P  pCnt  B
)  e.  RR* )
9 xrltle 11126 . . . . 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 661 . . . 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 13951 . 2  |-  ( ph  ->  ( P  pCnt  A
)  <_  ( P  pCnt  ( A  +  B
) ) )
13 qaddcl 10969 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  +  B
)  e.  QQ )
142, 3, 13syl2anc 661 . . . 4  |-  ( ph  ->  ( A  +  B
)  e.  QQ )
15 qnegcl 10970 . . . . 5  |-  ( B  e.  QQ  ->  -u B  e.  QQ )
163, 15syl 16 . . . 4  |-  ( ph  -> 
-u B  e.  QQ )
17 xrltnle 9443 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( ph  ->  ( ( P  pCnt  A )  <  ( P 
pCnt  B )  <->  -.  ( P  pCnt  B )  <_ 
( P  pCnt  A
) ) )
194, 18mpbid 210 . . . . . . . 8  |-  ( ph  ->  -.  ( P  pCnt  B )  <_  ( P  pCnt  A ) )
201adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  P  e.  Prime )
2116adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  -u B  e.  QQ )
2214adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  ( A  +  B )  e.  QQ )
23 pcneg 13940 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  B  e.  QQ )  ->  ( P  pCnt  -u B )  =  ( P  pCnt  B
) )
241, 3, 23syl2anc 661 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P  pCnt  -u B
)  =  ( P 
pCnt  B ) )
2524breq1d 4302 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( P  pCnt  -u B )  <_  ( P  pCnt  ( A  +  B ) )  <->  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) ) )
2625biimpar 485 . . . . . . . . . . 11  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  ( P  pCnt  -u B )  <_  ( P  pCnt  ( A  +  B ) ) )
2720, 21, 22, 26pcadd 13951 . . . . . . . . . 10  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  ( P  pCnt  -u B )  <_  ( P  pCnt  ( -u B  +  ( A  +  B ) ) ) )
2827ex 434 . . . . . . . . 9  |-  ( ph  ->  ( ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B
) )  ->  ( P  pCnt  -u B )  <_ 
( P  pCnt  ( -u B  +  ( A  +  B ) ) ) ) )
29 qcn 10967 . . . . . . . . . . . . . . 15  |-  ( B  e.  QQ  ->  B  e.  CC )
303, 29syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  CC )
3130negcld 9706 . . . . . . . . . . . . 13  |-  ( ph  -> 
-u B  e.  CC )
32 qcn 10967 . . . . . . . . . . . . . 14  |-  ( A  e.  QQ  ->  A  e.  CC )
332, 32syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  CC )
3431, 33, 30add12d 9591 . . . . . . . . . . . 12  |-  ( ph  ->  ( -u B  +  ( A  +  B
) )  =  ( A  +  ( -u B  +  B )
) )
3531, 30addcomd 9571 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( -u B  +  B )  =  ( B  +  -u B
) )
3630negidd 9709 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( B  +  -u B )  =  0 )
3735, 36eqtrd 2475 . . . . . . . . . . . . 13  |-  ( ph  ->  ( -u B  +  B )  =  0 )
3837oveq2d 6107 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  (
-u B  +  B
) )  =  ( A  +  0 ) )
3933addid1d 9569 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  0 )  =  A )
4034, 38, 393eqtrd 2479 . . . . . . . . . . 11  |-  ( ph  ->  ( -u B  +  ( A  +  B
) )  =  A )
4140oveq2d 6107 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( -u B  +  ( A  +  B ) ) )  =  ( P 
pCnt  A ) )
4224, 41breq12d 4305 . . . . . . . . 9  |-  ( ph  ->  ( ( P  pCnt  -u B )  <_  ( P  pCnt  ( -u B  +  ( A  +  B ) ) )  <-> 
( P  pCnt  B
)  <_  ( P  pCnt  A ) ) )
4328, 42sylibd 214 . . . . . . . 8  |-  ( ph  ->  ( ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B
) )  ->  ( P  pCnt  B )  <_ 
( P  pCnt  A
) ) )
4419, 43mtod 177 . . . . . . 7  |-  ( ph  ->  -.  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B
) ) )
45 pcxcl 13927 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( A  +  B )  e.  QQ )  ->  ( P  pCnt  ( A  +  B ) )  e. 
RR* )
461, 14, 45syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  e.  RR* )
47 xrltnle 9443 . . . . . . . 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 661 . . . . . . 7  |-  ( ph  ->  ( ( P  pCnt  ( A  +  B ) )  <  ( P 
pCnt  B )  <->  -.  ( P  pCnt  B )  <_ 
( P  pCnt  ( A  +  B )
) ) )
4944, 48mpbird 232 . . . . . 6  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <  ( P  pCnt  B ) )
50 xrltle 11126 . . . . . . 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 661 . . . . . 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 4318 . . . 4  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  -u B ) )
541, 14, 16, 53pcadd 13951 . . 3  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  ( ( A  +  B )  +  -u B ) ) )
5533, 30, 31addassd 9408 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  +  -u B )  =  ( A  +  ( B  +  -u B ) ) )
5636oveq2d 6107 . . . . 5  |-  ( ph  ->  ( A  +  ( B  +  -u B
) )  =  ( A  +  0 ) )
5755, 56, 393eqtrd 2479 . . . 4  |-  ( ph  ->  ( ( A  +  B )  +  -u B )  =  A )
5857oveq2d 6107 . . 3  |-  ( ph  ->  ( P  pCnt  (
( A  +  B
)  +  -u B
) )  =  ( P  pCnt  A )
)
5954, 58breqtrd 4316 . 2  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  A ) )
60 xrletri3 11129 . . 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 661 . 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 913 1  |-  ( ph  ->  ( P  pCnt  A
)  =  ( P 
pCnt  ( A  +  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4292  (class class class)co 6091   CCcc 9280   0cc0 9282    + caddc 9285   RR*cxr 9417    < clt 9418    <_ cle 9419   -ucneg 9596   QQcq 10953   Primecprime 13763    pCnt cpc 13903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536  df-gcd 13691  df-prm 13764  df-pc 13904
This theorem is referenced by:  sylow1lem1  16097
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