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Theorem pceulem 13174
Description: Lemma for pceu 13175. (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 9972 . . . . . . . . 9  |-  ( ph  ->  y  e.  CC )
4 pceu.9 . . . . . . . . . . 11  |-  ( ph  ->  ( s  e.  ZZ  /\  t  e.  NN ) )
54simpld 446 . . . . . . . . . 10  |-  ( ph  ->  s  e.  ZZ )
65zcnd 10332 . . . . . . . . 9  |-  ( ph  ->  s  e.  CC )
73, 6mulcomd 9065 . . . . . . . 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 2438 . . . . . . . . 9  |-  ( ph  ->  ( s  /  t
)  =  ( x  /  y ) )
114simprd 450 . . . . . . . . . . 11  |-  ( ph  ->  t  e.  NN )
1211nncnd 9972 . . . . . . . . . 10  |-  ( ph  ->  t  e.  CC )
131simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  x  e.  ZZ )
1413zcnd 10332 . . . . . . . . . 10  |-  ( ph  ->  x  e.  CC )
1511nnne0d 10000 . . . . . . . . . 10  |-  ( ph  ->  t  =/=  0 )
162nnne0d 10000 . . . . . . . . . 10  |-  ( ph  ->  y  =/=  0 )
176, 12, 14, 3, 15, 16divmuleqd 9792 . . . . . . . . 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 2436 . . . . . . 7  |-  ( ph  ->  ( y  x.  s
)  =  ( x  x.  t ) )
2019breq2d 4184 . . . . . 6  |-  ( ph  ->  ( ( P ^
z )  ||  (
y  x.  s )  <-> 
( P ^ z
)  ||  ( x  x.  t ) ) )
2120rabbidv 2908 . . . . 5  |-  ( ph  ->  { z  e.  NN0  |  ( P ^ z
)  ||  ( y  x.  s ) }  =  { z  e.  NN0  |  ( P ^ z
)  ||  ( x  x.  t ) } )
22 oveq2 6048 . . . . . . 7  |-  ( n  =  z  ->  ( P ^ n )  =  ( P ^ z
) )
2322breq1d 4182 . . . . . 6  |-  ( n  =  z  ->  (
( P ^ n
)  ||  ( y  x.  s )  <->  ( P ^ z )  ||  ( y  x.  s
) ) )
2423cbvrabv 2915 . . . . 5  |-  { n  e.  NN0  |  ( P ^ n )  ||  ( y  x.  s
) }  =  {
z  e.  NN0  | 
( P ^ z
)  ||  ( y  x.  s ) }
2522breq1d 4182 . . . . . 6  |-  ( n  =  z  ->  (
( P ^ n
)  ||  ( x  x.  t )  <->  ( P ^ z )  ||  ( x  x.  t
) ) )
2625cbvrabv 2915 . . . . 5  |-  { n  e.  NN0  |  ( P ^ n )  ||  ( x  x.  t
) }  =  {
z  e.  NN0  | 
( P ^ z
)  ||  ( x  x.  t ) }
2721, 24, 263eqtr4g 2461 . . . 4  |-  ( ph  ->  { n  e.  NN0  |  ( P ^ n
)  ||  ( y  x.  s ) }  =  { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } )
2827supeq1d 7409 . . 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 10330 . . . 4  |-  ( ph  ->  y  e.  ZZ )
31 pceu.6 . . . . 5  |-  ( ph  ->  N  =/=  0 )
3212, 15div0d 9745 . . . . . . . 8  |-  ( ph  ->  ( 0  /  t
)  =  0 )
33 oveq1 6047 . . . . . . . . 9  |-  ( s  =  0  ->  (
s  /  t )  =  ( 0  / 
t ) )
3433eqeq1d 2412 . . . . . . . 8  |-  ( s  =  0  ->  (
( s  /  t
)  =  0  <->  (
0  /  t )  =  0 ) )
3532, 34syl5ibrcom 214 . . . . . . 7  |-  ( ph  ->  ( s  =  0  ->  ( s  / 
t )  =  0 ) )
368eqeq1d 2412 . . . . . . 7  |-  ( ph  ->  ( N  =  0  <-> 
( s  /  t
)  =  0 ) )
3735, 36sylibrd 226 . . . . . 6  |-  ( ph  ->  ( s  =  0  ->  N  =  0 ) )
3837necon3d 2605 . . . . 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 2404 . . . . 5  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( y  x.  s ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( y  x.  s
) } ,  RR ,  <  )
4340, 41, 42pcpremul 13172 . . . 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 9745 . . . . . . . 8  |-  ( ph  ->  ( 0  /  y
)  =  0 )
46 oveq1 6047 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  /  y )  =  ( 0  / 
y ) )
4746eqeq1d 2412 . . . . . . . 8  |-  ( x  =  0  ->  (
( x  /  y
)  =  0  <->  (
0  /  y )  =  0 ) )
4845, 47syl5ibrcom 214 . . . . . . 7  |-  ( ph  ->  ( x  =  0  ->  ( x  / 
y )  =  0 ) )
499eqeq1d 2412 . . . . . . 7  |-  ( ph  ->  ( N  =  0  <-> 
( x  /  y
)  =  0 ) )
5048, 49sylibrd 226 . . . . . 6  |-  ( ph  ->  ( x  =  0  ->  N  =  0 ) )
5150necon3d 2605 . . . . 5  |-  ( ph  ->  ( N  =/=  0  ->  x  =/=  0 ) )
5231, 51mpd 15 . . . 4  |-  ( ph  ->  x  =/=  0 )
5311nnzd 10330 . . . 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 2404 . . . . 5  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( x  x.  t
) } ,  RR ,  <  )
5754, 55, 56pcpremul 13172 . . . 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 2446 . 2  |-  ( ph  ->  ( T  +  U
)  =  ( S  +  V ) )
60 prmuz2 13052 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
6129, 60syl 16 . . . . 5  |-  ( ph  ->  P  e.  ( ZZ>= ` 
2 ) )
62 eqid 2404 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  y }
6362, 40pcprecl 13168 . . . . . 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 10232 . . 3  |-  ( ph  ->  T  e.  CC )
67 eqid 2404 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  s }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  s }
6867, 41pcprecl 13168 . . . . . 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 10232 . . 3  |-  ( ph  ->  U  e.  CC )
72 eqid 2404 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  x }
7372, 54pcprecl 13168 . . . . . 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 10232 . . 3  |-  ( ph  ->  S  e.  CC )
77 eqid 2404 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  t }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  t }
7877, 55pcprecl 13168 . . . . . 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 10232 . . 3  |-  ( ph  ->  V  e.  CC )
8266, 71, 76, 81addsubeq4d 9418 . 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 2567   {crab 2670   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   supcsup 7403   RRcr 8945   0cc0 8946    + caddc 8949    x. cmul 8951    < clt 9076    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ^cexp 11337    || cdivides 12807   Primecprime 13034
This theorem is referenced by:  pceu  13175
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035
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