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Theorem pceulem 14029
Description: Lemma for pceu 14030. (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 463 . . . . . . . . . 10  |-  ( ph  ->  y  e.  NN )
32nncnd 10448 . . . . . . . . 9  |-  ( ph  ->  y  e.  CC )
4 pceu.9 . . . . . . . . . . 11  |-  ( ph  ->  ( s  e.  ZZ  /\  t  e.  NN ) )
54simpld 459 . . . . . . . . . 10  |-  ( ph  ->  s  e.  ZZ )
65zcnd 10858 . . . . . . . . 9  |-  ( ph  ->  s  e.  CC )
73, 6mulcomd 9517 . . . . . . . 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 2497 . . . . . . . . 9  |-  ( ph  ->  ( s  /  t
)  =  ( x  /  y ) )
114simprd 463 . . . . . . . . . . 11  |-  ( ph  ->  t  e.  NN )
1211nncnd 10448 . . . . . . . . . 10  |-  ( ph  ->  t  e.  CC )
131simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  x  e.  ZZ )
1413zcnd 10858 . . . . . . . . . 10  |-  ( ph  ->  x  e.  CC )
1511nnne0d 10476 . . . . . . . . . 10  |-  ( ph  ->  t  =/=  0 )
162nnne0d 10476 . . . . . . . . . 10  |-  ( ph  ->  y  =/=  0 )
176, 12, 14, 3, 15, 16divmuleqd 10263 . . . . . . . . 9  |-  ( ph  ->  ( ( s  / 
t )  =  ( x  /  y )  <-> 
( s  x.  y
)  =  ( x  x.  t ) ) )
1810, 17mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( s  x.  y
)  =  ( x  x.  t ) )
197, 18eqtrd 2495 . . . . . . 7  |-  ( ph  ->  ( y  x.  s
)  =  ( x  x.  t ) )
2019breq2d 4411 . . . . . 6  |-  ( ph  ->  ( ( P ^
z )  ||  (
y  x.  s )  <-> 
( P ^ z
)  ||  ( x  x.  t ) ) )
2120rabbidv 3068 . . . . 5  |-  ( ph  ->  { z  e.  NN0  |  ( P ^ z
)  ||  ( y  x.  s ) }  =  { z  e.  NN0  |  ( P ^ z
)  ||  ( x  x.  t ) } )
22 oveq2 6207 . . . . . . 7  |-  ( n  =  z  ->  ( P ^ n )  =  ( P ^ z
) )
2322breq1d 4409 . . . . . 6  |-  ( n  =  z  ->  (
( P ^ n
)  ||  ( y  x.  s )  <->  ( P ^ z )  ||  ( y  x.  s
) ) )
2423cbvrabv 3075 . . . . 5  |-  { n  e.  NN0  |  ( P ^ n )  ||  ( y  x.  s
) }  =  {
z  e.  NN0  | 
( P ^ z
)  ||  ( y  x.  s ) }
2522breq1d 4409 . . . . . 6  |-  ( n  =  z  ->  (
( P ^ n
)  ||  ( x  x.  t )  <->  ( P ^ z )  ||  ( x  x.  t
) ) )
2625cbvrabv 3075 . . . . 5  |-  { n  e.  NN0  |  ( P ^ n )  ||  ( x  x.  t
) }  =  {
z  e.  NN0  | 
( P ^ z
)  ||  ( x  x.  t ) }
2721, 24, 263eqtr4g 2520 . . . 4  |-  ( ph  ->  { n  e.  NN0  |  ( P ^ n
)  ||  ( y  x.  s ) }  =  { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } )
2827supeq1d 7806 . . 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 10856 . . . 4  |-  ( ph  ->  y  e.  ZZ )
31 pceu.6 . . . . 5  |-  ( ph  ->  N  =/=  0 )
3212, 15div0d 10216 . . . . . . . 8  |-  ( ph  ->  ( 0  /  t
)  =  0 )
33 oveq1 6206 . . . . . . . . 9  |-  ( s  =  0  ->  (
s  /  t )  =  ( 0  / 
t ) )
3433eqeq1d 2456 . . . . . . . 8  |-  ( s  =  0  ->  (
( s  /  t
)  =  0  <->  (
0  /  t )  =  0 ) )
3532, 34syl5ibrcom 222 . . . . . . 7  |-  ( ph  ->  ( s  =  0  ->  ( s  / 
t )  =  0 ) )
368eqeq1d 2456 . . . . . . 7  |-  ( ph  ->  ( N  =  0  <-> 
( s  /  t
)  =  0 ) )
3735, 36sylibrd 234 . . . . . 6  |-  ( ph  ->  ( s  =  0  ->  N  =  0 ) )
3837necon3d 2675 . . . . 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 2454 . . . . 5  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( y  x.  s ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( y  x.  s
) } ,  RR ,  <  )
4340, 41, 42pcpremul 14027 . . . 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 1228 . . 3  |-  ( ph  ->  ( T  +  U
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( y  x.  s ) } ,  RR ,  <  ) )
453, 16div0d 10216 . . . . . . . 8  |-  ( ph  ->  ( 0  /  y
)  =  0 )
46 oveq1 6206 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  /  y )  =  ( 0  / 
y ) )
4746eqeq1d 2456 . . . . . . . 8  |-  ( x  =  0  ->  (
( x  /  y
)  =  0  <->  (
0  /  y )  =  0 ) )
4845, 47syl5ibrcom 222 . . . . . . 7  |-  ( ph  ->  ( x  =  0  ->  ( x  / 
y )  =  0 ) )
499eqeq1d 2456 . . . . . . 7  |-  ( ph  ->  ( N  =  0  <-> 
( x  /  y
)  =  0 ) )
5048, 49sylibrd 234 . . . . . 6  |-  ( ph  ->  ( x  =  0  ->  N  =  0 ) )
5150necon3d 2675 . . . . 5  |-  ( ph  ->  ( N  =/=  0  ->  x  =/=  0 ) )
5231, 51mpd 15 . . . 4  |-  ( ph  ->  x  =/=  0 )
5311nnzd 10856 . . . 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 2454 . . . . 5  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( x  x.  t
) } ,  RR ,  <  )
5754, 55, 56pcpremul 14027 . . . 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 1228 . . 3  |-  ( ph  ->  ( S  +  V
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } ,  RR ,  <  ) )
5928, 44, 583eqtr4d 2505 . 2  |-  ( ph  ->  ( T  +  U
)  =  ( S  +  V ) )
60 prmuz2 13898 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
6129, 60syl 16 . . . . 5  |-  ( ph  ->  P  e.  ( ZZ>= ` 
2 ) )
62 eqid 2454 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  y }
6362, 40pcprecl 14023 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  ( T  e. 
NN0  /\  ( P ^ T )  ||  y
) )
6463simpld 459 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  T  e.  NN0 )
6561, 30, 16, 64syl12anc 1217 . . . 4  |-  ( ph  ->  T  e.  NN0 )
6665nn0cnd 10748 . . 3  |-  ( ph  ->  T  e.  CC )
67 eqid 2454 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  s }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  s }
6867, 41pcprecl 14023 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
s  e.  ZZ  /\  s  =/=  0 ) )  ->  ( U  e. 
NN0  /\  ( P ^ U )  ||  s
) )
6968simpld 459 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
s  e.  ZZ  /\  s  =/=  0 ) )  ->  U  e.  NN0 )
7061, 5, 39, 69syl12anc 1217 . . . 4  |-  ( ph  ->  U  e.  NN0 )
7170nn0cnd 10748 . . 3  |-  ( ph  ->  U  e.  CC )
72 eqid 2454 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  x }
7372, 54pcprecl 14023 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( S  e. 
NN0  /\  ( P ^ S )  ||  x
) )
7473simpld 459 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  S  e.  NN0 )
7561, 13, 52, 74syl12anc 1217 . . . 4  |-  ( ph  ->  S  e.  NN0 )
7675nn0cnd 10748 . . 3  |-  ( ph  ->  S  e.  CC )
77 eqid 2454 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  t }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  t }
7877, 55pcprecl 14023 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
t  e.  ZZ  /\  t  =/=  0 ) )  ->  ( V  e. 
NN0  /\  ( P ^ V )  ||  t
) )
7978simpld 459 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  (
t  e.  ZZ  /\  t  =/=  0 ) )  ->  V  e.  NN0 )
8061, 53, 15, 79syl12anc 1217 . . . 4  |-  ( ph  ->  V  e.  NN0 )
8180nn0cnd 10748 . . 3  |-  ( ph  ->  V  e.  CC )
8266, 71, 76, 81addsubeq4d 9880 . 2  |-  ( ph  ->  ( ( T  +  U )  =  ( S  +  V )  <-> 
( S  -  T
)  =  ( U  -  V ) ) )
8359, 82mpbid 210 1  |-  ( ph  ->  ( S  -  T
)  =  ( U  -  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2647   {crab 2802   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   supcsup 7800   RRcr 9391   0cc0 9392    + caddc 9395    x. cmul 9397    < clt 9528    - cmin 9705    / cdiv 10103   NNcn 10432   2c2 10481   NN0cn0 10689   ZZcz 10756   ZZ>=cuz 10971   ^cexp 11981    || cdivides 13652   Primecprime 13880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-rp 11102  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-dvds 13653  df-gcd 13808  df-prm 13881
This theorem is referenced by:  pceu  14030
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