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Theorem pcadd2 12954
Description: The inequality of pcadd 12953 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 12929 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  A )  e. 
RR* )
61, 2, 5syl2anc 642 . . . . 5  |-  ( ph  ->  ( P  pCnt  A
)  e.  RR* )
7 pcxcl 12929 . . . . . 6  |-  ( ( P  e.  Prime  /\  B  e.  QQ )  ->  ( P  pCnt  B )  e. 
RR* )
81, 3, 7syl2anc 642 . . . . 5  |-  ( ph  ->  ( P  pCnt  B
)  e.  RR* )
9 xrltle 10499 . . . . 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 642 . . . 4  |-  ( ph  ->  ( ( P  pCnt  A )  <  ( P 
pCnt  B )  ->  ( P  pCnt  A )  <_ 
( P  pCnt  B
) ) )
114, 10mpd 14 . . 3  |-  ( ph  ->  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )
121, 2, 3, 11pcadd 12953 . 2  |-  ( ph  ->  ( P  pCnt  A
)  <_  ( P  pCnt  ( A  +  B
) ) )
13 qaddcl 10348 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  +  B
)  e.  QQ )
142, 3, 13syl2anc 642 . . . 4  |-  ( ph  ->  ( A  +  B
)  e.  QQ )
15 qnegcl 10349 . . . . 5  |-  ( B  e.  QQ  ->  -u B  e.  QQ )
163, 15syl 15 . . . 4  |-  ( ph  -> 
-u B  e.  QQ )
17 xrltnle 8907 . . . . . . . . . 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 642 . . . . . . . . 9  |-  ( ph  ->  ( ( P  pCnt  A )  <  ( P 
pCnt  B )  <->  -.  ( P  pCnt  B )  <_ 
( P  pCnt  A
) ) )
194, 18mpbid 201 . . . . . . . 8  |-  ( ph  ->  -.  ( P  pCnt  B )  <_  ( P  pCnt  A ) )
201adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  P  e.  Prime )
2116adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  -u B  e.  QQ )
2214adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  ( A  +  B )  e.  QQ )
23 pcneg 12942 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  B  e.  QQ )  ->  ( P  pCnt  -u B )  =  ( P  pCnt  B
) )
241, 3, 23syl2anc 642 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P  pCnt  -u B
)  =  ( P 
pCnt  B ) )
2524breq1d 4049 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( P  pCnt  -u B )  <_  ( P  pCnt  ( A  +  B ) )  <->  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) ) )
2625biimpar 471 . . . . . . . . . . 11  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  ( P  pCnt  -u B )  <_  ( P  pCnt  ( A  +  B ) ) )
2720, 21, 22, 26pcadd 12953 . . . . . . . . . 10  |-  ( (
ph  /\  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B ) ) )  ->  ( P  pCnt  -u B )  <_  ( P  pCnt  ( -u B  +  ( A  +  B ) ) ) )
2827ex 423 . . . . . . . . 9  |-  ( ph  ->  ( ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B
) )  ->  ( P  pCnt  -u B )  <_ 
( P  pCnt  ( -u B  +  ( A  +  B ) ) ) ) )
29 qcn 10346 . . . . . . . . . . . . . . 15  |-  ( B  e.  QQ  ->  B  e.  CC )
303, 29syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  CC )
3130negcld 9160 . . . . . . . . . . . . 13  |-  ( ph  -> 
-u B  e.  CC )
32 qcn 10346 . . . . . . . . . . . . . 14  |-  ( A  e.  QQ  ->  A  e.  CC )
332, 32syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  CC )
3431, 33, 30add12d 9049 . . . . . . . . . . . 12  |-  ( ph  ->  ( -u B  +  ( A  +  B
) )  =  ( A  +  ( -u B  +  B )
) )
3531, 30addcomd 9030 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( -u B  +  B )  =  ( B  +  -u B
) )
3630negidd 9163 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( B  +  -u B )  =  0 )
3735, 36eqtrd 2328 . . . . . . . . . . . . 13  |-  ( ph  ->  ( -u B  +  B )  =  0 )
3837oveq2d 5890 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  (
-u B  +  B
) )  =  ( A  +  0 ) )
3933addid1d 9028 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  0 )  =  A )
4034, 38, 393eqtrd 2332 . . . . . . . . . . 11  |-  ( ph  ->  ( -u B  +  ( A  +  B
) )  =  A )
4140oveq2d 5890 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( -u B  +  ( A  +  B ) ) )  =  ( P 
pCnt  A ) )
4224, 41breq12d 4052 . . . . . . . . 9  |-  ( ph  ->  ( ( P  pCnt  -u B )  <_  ( P  pCnt  ( -u B  +  ( A  +  B ) ) )  <-> 
( P  pCnt  B
)  <_  ( P  pCnt  A ) ) )
4328, 42sylibd 205 . . . . . . . 8  |-  ( ph  ->  ( ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B
) )  ->  ( P  pCnt  B )  <_ 
( P  pCnt  A
) ) )
4419, 43mtod 168 . . . . . . 7  |-  ( ph  ->  -.  ( P  pCnt  B )  <_  ( P  pCnt  ( A  +  B
) ) )
45 pcxcl 12929 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( A  +  B )  e.  QQ )  ->  ( P  pCnt  ( A  +  B ) )  e. 
RR* )
461, 14, 45syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  e.  RR* )
47 xrltnle 8907 . . . . . . . 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 642 . . . . . . 7  |-  ( ph  ->  ( ( P  pCnt  ( A  +  B ) )  <  ( P 
pCnt  B )  <->  -.  ( P  pCnt  B )  <_ 
( P  pCnt  ( A  +  B )
) ) )
4944, 48mpbird 223 . . . . . 6  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <  ( P  pCnt  B ) )
50 xrltle 10499 . . . . . . 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 642 . . . . . 6  |-  ( ph  ->  ( ( P  pCnt  ( A  +  B ) )  <  ( P 
pCnt  B )  ->  ( P  pCnt  ( A  +  B ) )  <_ 
( P  pCnt  B
) ) )
5249, 51mpd 14 . . . . 5  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  B ) )
5352, 24breqtrrd 4065 . . . 4  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  -u B ) )
541, 14, 16, 53pcadd 12953 . . 3  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  ( ( A  +  B )  +  -u B ) ) )
5533, 30, 31addassd 8873 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  +  -u B )  =  ( A  +  ( B  +  -u B ) ) )
5636oveq2d 5890 . . . . 5  |-  ( ph  ->  ( A  +  ( B  +  -u B
) )  =  ( A  +  0 ) )
5755, 56, 393eqtrd 2332 . . . 4  |-  ( ph  ->  ( ( A  +  B )  +  -u B )  =  A )
5857oveq2d 5890 . . 3  |-  ( ph  ->  ( P  pCnt  (
( A  +  B
)  +  -u B
) )  =  ( P  pCnt  A )
)
5954, 58breqtrd 4063 . 2  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  A ) )
60 xrletri3 10502 . . 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 642 . 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 888 1  |-  ( ph  ->  ( P  pCnt  A
)  =  ( P 
pCnt  ( A  +  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   CCcc 8751   0cc0 8753    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884   -ucneg 9054   QQcq 10332   Primecprime 12774    pCnt cpc 12905
This theorem is referenced by:  sylow1lem1  14925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906
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