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

Theorem pceulem 13182
Description: Lemma for pceu 13183. (Contributed by Mario Carneiro, 23-Feb-2014.)
Hypotheses
Ref Expression
pcval.1  |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
pcval.2  |-  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
pceu.3  |-  U  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )
pceu.4  |-  V  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  )
pceu.5  |-  ( ph  ->  P  e.  Prime )
pceu.6  |-  ( ph  ->  N  =/=  0 )
pceu.7  |-  ( ph  ->  ( x  e.  ZZ  /\  y  e.  NN ) )
pceu.8  |-  ( ph  ->  N  =  ( x  /  y ) )
pceu.9  |-  ( ph  ->  ( s  e.  ZZ  /\  t  e.  NN ) )
pceu.10  |-  ( ph  ->  N  =  ( s  /  t ) )
Assertion
Ref Expression
pceulem  |-  ( ph  ->  ( S  -  T
)  =  ( U  -  V ) )
Distinct variable groups:    n, s,
t, x, y, N    P, n, s, t, x, y    S, s, t    T, s, t
Allowed substitution hints:    ph( x, y, t, n, s)    S( x, y, n)    T( x, y, n)    U( x, y, t, n, s)    V( x, y, t, n, s)

Proof of Theorem pceulem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 pceu.7 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ZZ  /\  y  e.  NN ) )
21simprd 450 . . . . . . . . . 10  |-  ( ph  ->  y  e.  NN )
32nncnd 9980 . . . . . . . . 9  |-  ( ph  ->  y  e.  CC )
4 pceu.9 . . . . . . . . . . 11  |-  ( ph  ->  ( s  e.  ZZ  /\  t  e.  NN ) )
54simpld 446 . . . . . . . . . 10  |-  ( ph  ->  s  e.  ZZ )
65zcnd 10340 . . . . . . . . 9  |-  ( ph  ->  s  e.  CC )
73, 6mulcomd 9073 . . . . . . . 8  |-  ( ph  ->  ( y  x.  s
)  =  ( s  x.  y ) )
8 pceu.10 . . . . . . . . . 10  |-  ( ph  ->  N  =  ( s  /  t ) )
9 pceu.8 . . . . . . . . . 10  |-  ( ph  ->  N  =  ( x  /  y ) )
108, 9eqtr3d 2446 . . . . . . . . 9  |-  ( ph  ->  ( s  /  t
)  =  ( x  /  y ) )
114simprd 450 . . . . . . . . . . 11  |-  ( ph  ->  t  e.  NN )
1211nncnd 9980 . . . . . . . . . 10  |-  ( ph  ->  t  e.  CC )
131simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  x  e.  ZZ )
1413zcnd 10340 . . . . . . . . . 10  |-  ( ph  ->  x  e.  CC )
1511nnne0d 10008 . . . . . . . . . 10  |-  ( ph  ->  t  =/=  0 )
162nnne0d 10008 . . . . . . . . . 10  |-  ( ph  ->  y  =/=  0 )
176, 12, 14, 3, 15, 16divmuleqd 9800 . . . . . . . . 9  |-  ( ph  ->  ( ( s  / 
t )  =  ( x  /  y )  <-> 
( s  x.  y
)  =  ( x  x.  t ) ) )
1810, 17mpbid 202 . . . . . . . 8  |-  ( ph  ->  ( s  x.  y
)  =  ( x  x.  t ) )
197, 18eqtrd 2444 . . . . . . 7  |-  ( ph  ->  ( y  x.  s
)  =  ( x  x.  t ) )
2019breq2d 4192 . . . . . 6  |-  ( ph  ->  ( ( P ^
z )  ||  (
y  x.  s )  <-> 
( P ^ z
)  ||  ( x  x.  t ) ) )
2120rabbidv 2916 . . . . 5  |-  ( ph  ->  { z  e.  NN0  |  ( P ^ z
)  ||  ( y  x.  s ) }  =  { z  e.  NN0  |  ( P ^ z
)  ||  ( x  x.  t ) } )
22 oveq2 6056 . . . . . . 7  |-  ( n  =  z  ->  ( P ^ n )  =  ( P ^ z
) )
2322breq1d 4190 . . . . . 6  |-  ( n  =  z  ->  (
( P ^ n
)  ||  ( y  x.  s )  <->  ( P ^ z )  ||  ( y  x.  s
) ) )
2423cbvrabv 2923 . . . . 5  |-  { n  e.  NN0  |  ( P ^ n )  ||  ( y  x.  s
) }  =  {
z  e.  NN0  | 
( P ^ z
)  ||  ( y  x.  s ) }
2522breq1d 4190 . . . . . 6  |-  ( n  =  z  ->  (
( P ^ n
)  ||  ( x  x.  t )  <->  ( P ^ z )  ||  ( x  x.  t
) ) )
2625cbvrabv 2923 . . . . 5  |-  { n  e.  NN0  |  ( P ^ n )  ||  ( x  x.  t
) }  =  {
z  e.  NN0  | 
( P ^ z
)  ||  ( x  x.  t ) }
2721, 24, 263eqtr4g 2469 . . . 4  |-  ( ph  ->  { n  e.  NN0  |  ( P ^ n
)  ||  ( y  x.  s ) }  =  { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } )
2827supeq1d 7417 . . 3  |-  ( ph  ->  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( y  x.  s
) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } ,  RR ,  <  ) )
29 pceu.5 . . . 4  |-  ( ph  ->  P  e.  Prime )
302nnzd 10338 . . . 4  |-  ( ph  ->  y  e.  ZZ )
31 pceu.6 . . . . 5  |-  ( ph  ->  N  =/=  0 )
3212, 15div0d 9753 . . . . . . . 8  |-  ( ph  ->  ( 0  /  t
)  =  0 )
33 oveq1 6055 . . . . . . . . 9  |-  ( s  =  0  ->  (
s  /  t )  =  ( 0  / 
t ) )
3433eqeq1d 2420 . . . . . . . 8  |-  ( s  =  0  ->  (
( s  /  t
)  =  0  <->  (
0  /  t )  =  0 ) )
3532, 34syl5ibrcom 214 . . . . . . 7  |-  ( ph  ->  ( s  =  0  ->  ( s  / 
t )  =  0 ) )
368eqeq1d 2420 . . . . . . 7  |-  ( ph  ->  ( N  =  0  <-> 
( s  /  t
)  =  0 ) )
3735, 36sylibrd 226 . . . . . 6  |-  ( ph  ->  ( s  =  0  ->  N  =  0 ) )
3837necon3d 2613 . . . . 5  |-  ( ph  ->  ( N  =/=  0  ->  s  =/=  0 ) )
3931, 38mpd 15 . . . 4  |-  ( ph  ->  s  =/=  0 )
40 pcval.2 . . . . 5  |-  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
41 pceu.3 . . . . 5  |-  U  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )
42 eqid 2412 . . . . 5  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( y  x.  s ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( y  x.  s
) } ,  RR ,  <  )
4340, 41, 42pcpremul 13180 . . . 4  |-  ( ( P  e.  Prime  /\  (
y  e.  ZZ  /\  y  =/=  0 )  /\  ( s  e.  ZZ  /\  s  =/=  0 ) )  ->  ( T  +  U )  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( y  x.  s ) } ,  RR ,  <  ) )
4429, 30, 16, 5, 39, 43syl122anc 1193 . . 3  |-  ( ph  ->  ( T  +  U
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( y  x.  s ) } ,  RR ,  <  ) )
453, 16div0d 9753 . . . . . . . 8  |-  ( ph  ->  ( 0  /  y
)  =  0 )
46 oveq1 6055 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  /  y )  =  ( 0  / 
y ) )
4746eqeq1d 2420 . . . . . . . 8  |-  ( x  =  0  ->  (
( x  /  y
)  =  0  <->  (
0  /  y )  =  0 ) )
4845, 47syl5ibrcom 214 . . . . . . 7  |-  ( ph  ->  ( x  =  0  ->  ( x  / 
y )  =  0 ) )
499eqeq1d 2420 . . . . . . 7  |-  ( ph  ->  ( N  =  0  <-> 
( x  /  y
)  =  0 ) )
5048, 49sylibrd 226 . . . . . 6  |-  ( ph  ->  ( x  =  0  ->  N  =  0 ) )
5150necon3d 2613 . . . . 5  |-  ( ph  ->  ( N  =/=  0  ->  x  =/=  0 ) )
5231, 51mpd 15 . . . 4  |-  ( ph  ->  x  =/=  0 )
5311nnzd 10338 . . . 4  |-  ( ph  ->  t  e.  ZZ )
54 pcval.1 . . . . 5  |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
55 pceu.4 . . . . 5  |-  V  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  )
56 eqid 2412 . . . . 5  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( x  x.  t
) } ,  RR ,  <  )
5754, 55, 56pcpremul 13180 . . . 4  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  ( t  e.  ZZ  /\  t  =/=  0 ) )  ->  ( S  +  V )  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } ,  RR ,  <  ) )
5829, 13, 52, 53, 15, 57syl122anc 1193 . . 3  |-  ( ph  ->  ( S  +  V
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } ,  RR ,  <  ) )
5928, 44, 583eqtr4d 2454 . 2  |-  ( ph  ->  ( T  +  U
)  =  ( S  +  V ) )
60 prmuz2 13060 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
6129, 60syl 16 . . . . 5  |-  ( ph  ->  P  e.  ( ZZ>= ` 
2 ) )
62 eqid 2412 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  y }
6362, 40pcprecl 13176 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  ( T  e. 
NN0  /\  ( P ^ T )  ||  y
) )
6463simpld 446 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  T  e.  NN0 )
6561, 30, 16, 64syl12anc 1182 . . . 4  |-  ( ph  ->  T  e.  NN0 )
6665nn0cnd 10240 . . 3  |-  ( ph  ->  T  e.  CC )
67 eqid 2412 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  s }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  s }
6867, 41pcprecl 13176 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
s  e.  ZZ  /\  s  =/=  0 ) )  ->  ( U  e. 
NN0  /\  ( P ^ U )  ||  s
) )
6968simpld 446 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
s  e.  ZZ  /\  s  =/=  0 ) )  ->  U  e.  NN0 )
7061, 5, 39, 69syl12anc 1182 . . . 4  |-  ( ph  ->  U  e.  NN0 )
7170nn0cnd 10240 . . 3  |-  ( ph  ->  U  e.  CC )
72 eqid 2412 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  x }
7372, 54pcprecl 13176 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( S  e. 
NN0  /\  ( P ^ S )  ||  x
) )
7473simpld 446 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  S  e.  NN0 )
7561, 13, 52, 74syl12anc 1182 . . . 4  |-  ( ph  ->  S  e.  NN0 )
7675nn0cnd 10240 . . 3  |-  ( ph  ->  S  e.  CC )
77 eqid 2412 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  t }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  t }
7877, 55pcprecl 13176 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
t  e.  ZZ  /\  t  =/=  0 ) )  ->  ( V  e. 
NN0  /\  ( P ^ V )  ||  t
) )
7978simpld 446 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
t  e.  ZZ  /\  t  =/=  0 ) )  ->  V  e.  NN0 )
8061, 53, 15, 79syl12anc 1182 . . . 4  |-  ( ph  ->  V  e.  NN0 )
8180nn0cnd 10240 . . 3  |-  ( ph  ->  V  e.  CC )
8266, 71, 76, 81addsubeq4d 9426 . 2  |-  ( ph  ->  ( ( T  +  U )  =  ( S  +  V )  <-> 
( S  -  T
)  =  ( U  -  V ) ) )
8359, 82mpbid 202 1  |-  ( ph  ->  ( S  -  T
)  =  ( U  -  V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   {crab 2678   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   supcsup 7411   RRcr 8953   0cc0 8954    + caddc 8957    x. cmul 8959    < clt 9084    - cmin 9255    / cdiv 9641   NNcn 9964   2c2 10013   NN0cn0 10185   ZZcz 10246   ZZ>=cuz 10452   ^cexp 11345    || cdivides 12815   Primecprime 13042
This theorem is referenced by:  pceu  13183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-dvds 12816  df-gcd 12970  df-prm 13043
  Copyright terms: Public domain W3C validator