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Theorem pceulem 14381
Description: Lemma for pceu 14382. (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 10572 . . . . . . . . 9  |-  ( ph  ->  y  e.  CC )
4 pceu.9 . . . . . . . . . . 11  |-  ( ph  ->  ( s  e.  ZZ  /\  t  e.  NN ) )
54simpld 459 . . . . . . . . . 10  |-  ( ph  ->  s  e.  ZZ )
65zcnd 10991 . . . . . . . . 9  |-  ( ph  ->  s  e.  CC )
73, 6mulcomd 9634 . . . . . . . 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 2500 . . . . . . . . 9  |-  ( ph  ->  ( s  /  t
)  =  ( x  /  y ) )
114simprd 463 . . . . . . . . . . 11  |-  ( ph  ->  t  e.  NN )
1211nncnd 10572 . . . . . . . . . 10  |-  ( ph  ->  t  e.  CC )
131simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  x  e.  ZZ )
1413zcnd 10991 . . . . . . . . . 10  |-  ( ph  ->  x  e.  CC )
1511nnne0d 10601 . . . . . . . . . 10  |-  ( ph  ->  t  =/=  0 )
162nnne0d 10601 . . . . . . . . . 10  |-  ( ph  ->  y  =/=  0 )
176, 12, 14, 3, 15, 16divmuleqd 10387 . . . . . . . . 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 2498 . . . . . . 7  |-  ( ph  ->  ( y  x.  s
)  =  ( x  x.  t ) )
2019breq2d 4468 . . . . . 6  |-  ( ph  ->  ( ( P ^
z )  ||  (
y  x.  s )  <-> 
( P ^ z
)  ||  ( x  x.  t ) ) )
2120rabbidv 3101 . . . . 5  |-  ( ph  ->  { z  e.  NN0  |  ( P ^ z
)  ||  ( y  x.  s ) }  =  { z  e.  NN0  |  ( P ^ z
)  ||  ( x  x.  t ) } )
22 oveq2 6304 . . . . . . 7  |-  ( n  =  z  ->  ( P ^ n )  =  ( P ^ z
) )
2322breq1d 4466 . . . . . 6  |-  ( n  =  z  ->  (
( P ^ n
)  ||  ( y  x.  s )  <->  ( P ^ z )  ||  ( y  x.  s
) ) )
2423cbvrabv 3108 . . . . 5  |-  { n  e.  NN0  |  ( P ^ n )  ||  ( y  x.  s
) }  =  {
z  e.  NN0  | 
( P ^ z
)  ||  ( y  x.  s ) }
2522breq1d 4466 . . . . . 6  |-  ( n  =  z  ->  (
( P ^ n
)  ||  ( x  x.  t )  <->  ( P ^ z )  ||  ( x  x.  t
) ) )
2625cbvrabv 3108 . . . . 5  |-  { n  e.  NN0  |  ( P ^ n )  ||  ( x  x.  t
) }  =  {
z  e.  NN0  | 
( P ^ z
)  ||  ( x  x.  t ) }
2721, 24, 263eqtr4g 2523 . . . 4  |-  ( ph  ->  { n  e.  NN0  |  ( P ^ n
)  ||  ( y  x.  s ) }  =  { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } )
2827supeq1d 7923 . . 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 10989 . . . 4  |-  ( ph  ->  y  e.  ZZ )
31 pceu.6 . . . . 5  |-  ( ph  ->  N  =/=  0 )
3212, 15div0d 10340 . . . . . . . 8  |-  ( ph  ->  ( 0  /  t
)  =  0 )
33 oveq1 6303 . . . . . . . . 9  |-  ( s  =  0  ->  (
s  /  t )  =  ( 0  / 
t ) )
3433eqeq1d 2459 . . . . . . . 8  |-  ( s  =  0  ->  (
( s  /  t
)  =  0  <->  (
0  /  t )  =  0 ) )
3532, 34syl5ibrcom 222 . . . . . . 7  |-  ( ph  ->  ( s  =  0  ->  ( s  / 
t )  =  0 ) )
368eqeq1d 2459 . . . . . . 7  |-  ( ph  ->  ( N  =  0  <-> 
( s  /  t
)  =  0 ) )
3735, 36sylibrd 234 . . . . . 6  |-  ( ph  ->  ( s  =  0  ->  N  =  0 ) )
3837necon3d 2681 . . . . 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 2457 . . . . 5  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( y  x.  s ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( y  x.  s
) } ,  RR ,  <  )
4340, 41, 42pcpremul 14379 . . . 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 1237 . . 3  |-  ( ph  ->  ( T  +  U
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( y  x.  s ) } ,  RR ,  <  ) )
453, 16div0d 10340 . . . . . . . 8  |-  ( ph  ->  ( 0  /  y
)  =  0 )
46 oveq1 6303 . . . . . . . . 9  |-  ( x  =  0  ->  (
x  /  y )  =  ( 0  / 
y ) )
4746eqeq1d 2459 . . . . . . . 8  |-  ( x  =  0  ->  (
( x  /  y
)  =  0  <->  (
0  /  y )  =  0 ) )
4845, 47syl5ibrcom 222 . . . . . . 7  |-  ( ph  ->  ( x  =  0  ->  ( x  / 
y )  =  0 ) )
499eqeq1d 2459 . . . . . . 7  |-  ( ph  ->  ( N  =  0  <-> 
( x  /  y
)  =  0 ) )
5048, 49sylibrd 234 . . . . . 6  |-  ( ph  ->  ( x  =  0  ->  N  =  0 ) )
5150necon3d 2681 . . . . 5  |-  ( ph  ->  ( N  =/=  0  ->  x  =/=  0 ) )
5231, 51mpd 15 . . . 4  |-  ( ph  ->  x  =/=  0 )
5311nnzd 10989 . . . 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 2457 . . . . 5  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( x  x.  t
) } ,  RR ,  <  )
5754, 55, 56pcpremul 14379 . . . 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 1237 . . 3  |-  ( ph  ->  ( S  +  V
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( x  x.  t ) } ,  RR ,  <  ) )
5928, 44, 583eqtr4d 2508 . 2  |-  ( ph  ->  ( T  +  U
)  =  ( S  +  V ) )
60 prmuz2 14247 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
6129, 60syl 16 . . . . 5  |-  ( ph  ->  P  e.  ( ZZ>= ` 
2 ) )
62 eqid 2457 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  y }
6362, 40pcprecl 14375 . . . . . 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 1226 . . . 4  |-  ( ph  ->  T  e.  NN0 )
6665nn0cnd 10875 . . 3  |-  ( ph  ->  T  e.  CC )
67 eqid 2457 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  s }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  s }
6867, 41pcprecl 14375 . . . . . 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 1226 . . . 4  |-  ( ph  ->  U  e.  NN0 )
7170nn0cnd 10875 . . 3  |-  ( ph  ->  U  e.  CC )
72 eqid 2457 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  x }
7372, 54pcprecl 14375 . . . . . 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 1226 . . . 4  |-  ( ph  ->  S  e.  NN0 )
7675nn0cnd 10875 . . 3  |-  ( ph  ->  S  e.  CC )
77 eqid 2457 . . . . . . 7  |-  { n  e.  NN0  |  ( P ^ n )  ||  t }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  t }
7877, 55pcprecl 14375 . . . . . 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 1226 . . . 4  |-  ( ph  ->  V  e.  NN0 )
8180nn0cnd 10875 . . 3  |-  ( ph  ->  V  e.  CC )
8266, 71, 76, 81addsubeq4d 10001 . 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 1395    e. wcel 1819    =/= wne 2652   {crab 2811   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   supcsup 7918   RRcr 9508   0cc0 9509    + caddc 9512    x. cmul 9514    < clt 9645    - cmin 9824    / cdiv 10227   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   ^cexp 12169    || cdvds 13998   Primecprime 14229
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-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-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
This theorem is referenced by:  pceu  14382
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