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Theorem pcadd2 14421
Description: The inequality of pcadd 14420 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 14396 . . . . . 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 14396 . . . . . 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 11380 . . . . 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 14420 . 2  |-  ( ph  ->  ( P  pCnt  A
)  <_  ( P  pCnt  ( A  +  B
) ) )
13 qaddcl 11223 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  +  B
)  e.  QQ )
142, 3, 13syl2anc 661 . . . 4  |-  ( ph  ->  ( A  +  B
)  e.  QQ )
15 qnegcl 11224 . . . . 5  |-  ( B  e.  QQ  ->  -u B  e.  QQ )
163, 15syl 16 . . . 4  |-  ( ph  -> 
-u B  e.  QQ )
17 xrltnle 9670 . . . . . . . . . 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 14409 . . . . . . . . . . . . . 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 4466 . . . . . . . . . . . 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 14420 . . . . . . . . . 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 11221 . . . . . . . . . . . . . . 15  |-  ( B  e.  QQ  ->  B  e.  CC )
303, 29syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  CC )
3130negcld 9937 . . . . . . . . . . . . 13  |-  ( ph  -> 
-u B  e.  CC )
32 qcn 11221 . . . . . . . . . . . . . 14  |-  ( A  e.  QQ  ->  A  e.  CC )
332, 32syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  CC )
3431, 33, 30add12d 9820 . . . . . . . . . . . 12  |-  ( ph  ->  ( -u B  +  ( A  +  B
) )  =  ( A  +  ( -u B  +  B )
) )
3531, 30addcomd 9799 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( -u B  +  B )  =  ( B  +  -u B
) )
3630negidd 9940 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( B  +  -u B )  =  0 )
3735, 36eqtrd 2498 . . . . . . . . . . . . 13  |-  ( ph  ->  ( -u B  +  B )  =  0 )
3837oveq2d 6312 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  (
-u B  +  B
) )  =  ( A  +  0 ) )
3933addid1d 9797 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  0 )  =  A )
4034, 38, 393eqtrd 2502 . . . . . . . . . . 11  |-  ( ph  ->  ( -u B  +  ( A  +  B
) )  =  A )
4140oveq2d 6312 . . . . . . . . . 10  |-  ( ph  ->  ( P  pCnt  ( -u B  +  ( A  +  B ) ) )  =  ( P 
pCnt  A ) )
4224, 41breq12d 4469 . . . . . . . . 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 14396 . . . . . . . . 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 9670 . . . . . . . 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 11380 . . . . . . 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 4482 . . . 4  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  -u B ) )
541, 14, 16, 53pcadd 14420 . . 3  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  ( ( A  +  B )  +  -u B ) ) )
5533, 30, 31addassd 9635 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  +  -u B )  =  ( A  +  ( B  +  -u B ) ) )
5636oveq2d 6312 . . . . 5  |-  ( ph  ->  ( A  +  ( B  +  -u B
) )  =  ( A  +  0 ) )
5755, 56, 393eqtrd 2502 . . . 4  |-  ( ph  ->  ( ( A  +  B )  +  -u B )  =  A )
5857oveq2d 6312 . . 3  |-  ( ph  ->  ( P  pCnt  (
( A  +  B
)  +  -u B
) )  =  ( P  pCnt  A )
)
5954, 58breqtrd 4480 . 2  |-  ( ph  ->  ( P  pCnt  ( A  +  B )
)  <_  ( P  pCnt  A ) )
60 xrletri3 11383 . . 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 922 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 1395    e. wcel 1819   class class class wbr 4456  (class class class)co 6296   CCcc 9507   0cc0 9509    + caddc 9512   RR*cxr 9644    < clt 9645    <_ cle 9646   -ucneg 9825   QQcq 11207   Primecprime 14229    pCnt cpc 14372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-prm 14230  df-pc 14373
This theorem is referenced by:  sylow1lem1  16745
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