Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rexzrexnn0 Structured version   Unicode version

Theorem rexzrexnn0 35099
Description: Rewrite a quantification over integers into a quantification over naturals. (Contributed by Stefan O'Rear, 11-Oct-2014.)
Hypotheses
Ref Expression
rexzrexnn0.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
rexzrexnn0.2  |-  ( x  =  -u y  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
rexzrexnn0  |-  ( E. x  e.  ZZ  ph  <->  E. y  e.  NN0  ( ps  \/  ch ) )
Distinct variable groups:    ph, y    ps, x    ch, x    x, y
Allowed substitution hints:    ph( x)    ps( y)    ch( y)

Proof of Theorem rexzrexnn0
StepHypRef Expression
1 elznn0 10920 . . . . . . 7  |-  ( x  e.  ZZ  <->  ( x  e.  RR  /\  ( x  e.  NN0  \/  -u x  e.  NN0 ) ) )
21simprbi 462 . . . . . 6  |-  ( x  e.  ZZ  ->  (
x  e.  NN0  \/  -u x  e.  NN0 )
)
32adantr 463 . . . . 5  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( x  e.  NN0  \/  -u x  e.  NN0 ) )
4 simpr 459 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\ 
ph )  /\  x  e.  NN0 )  ->  x  e.  NN0 )
5 simplr 754 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\ 
ph )  /\  x  e.  NN0 )  ->  ph )
6 rexzrexnn0.1 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
76equcoms 1819 . . . . . . . . . 10  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
87bicomd 201 . . . . . . . . 9  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
98rspcev 3160 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  ph )  ->  E. y  e.  NN0  ps )
104, 5, 9syl2anc 659 . . . . . . 7  |-  ( ( ( x  e.  ZZ  /\ 
ph )  /\  x  e.  NN0 )  ->  E. y  e.  NN0  ps )
1110ex 432 . . . . . 6  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( x  e.  NN0  ->  E. y  e.  NN0  ps ) )
12 simpr 459 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  -> 
-u x  e.  NN0 )
13 zcn 10910 . . . . . . . . . . . . . . 15  |-  ( x  e.  ZZ  ->  x  e.  CC )
1413negnegd 9958 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  ->  -u -u x  =  x )
1514eqcomd 2410 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  x  =  -u -u x )
16 negeq 9848 . . . . . . . . . . . . . 14  |-  ( y  =  -u x  ->  -u y  =  -u -u x )
1716eqeq2d 2416 . . . . . . . . . . . . 13  |-  ( y  =  -u x  ->  (
x  =  -u y  <->  x  =  -u -u x ) )
1815, 17syl5ibrcom 222 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  (
y  =  -u x  ->  x  =  -u y
) )
1918imp 427 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  x  =  -u y )
20 rexzrexnn0.2 . . . . . . . . . . 11  |-  ( x  =  -u y  ->  ( ph 
<->  ch ) )
2119, 20syl 17 . . . . . . . . . 10  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  ( ph  <->  ch )
)
2221bicomd 201 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  ( ch  <->  ph ) )
2322adantlr 713 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  /\  y  =  -u x )  ->  ( ch 
<-> 
ph ) )
2412, 23rspcedv 3164 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  ->  ( ph  ->  E. y  e.  NN0  ch ) )
2524impancom 438 . . . . . 6  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( -u x  e.  NN0  ->  E. y  e.  NN0  ch ) )
2611, 25orim12d 839 . . . . 5  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( (
x  e.  NN0  \/  -u x  e.  NN0 )  ->  ( E. y  e. 
NN0  ps  \/  E. y  e.  NN0  ch ) ) )
273, 26mpd 15 . . . 4  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( E. y  e.  NN0  ps  \/  E. y  e.  NN0  ch ) )
28 r19.43 2963 . . . 4  |-  ( E. y  e.  NN0  ( ps  \/  ch )  <->  ( E. y  e.  NN0  ps  \/  E. y  e.  NN0  ch ) )
2927, 28sylibr 212 . . 3  |-  ( ( x  e.  ZZ  /\  ph )  ->  E. y  e.  NN0  ( ps  \/  ch ) )
3029rexlimiva 2892 . 2  |-  ( E. x  e.  ZZ  ph  ->  E. y  e.  NN0  ( ps  \/  ch ) )
31 nn0z 10928 . . . . 5  |-  ( y  e.  NN0  ->  y  e.  ZZ )
326rspcev 3160 . . . . 5  |-  ( ( y  e.  ZZ  /\  ps )  ->  E. x  e.  ZZ  ph )
3331, 32sylan 469 . . . 4  |-  ( ( y  e.  NN0  /\  ps )  ->  E. x  e.  ZZ  ph )
34 nn0negz 10943 . . . . 5  |-  ( y  e.  NN0  ->  -u y  e.  ZZ )
3520rspcev 3160 . . . . 5  |-  ( (
-u y  e.  ZZ  /\ 
ch )  ->  E. x  e.  ZZ  ph )
3634, 35sylan 469 . . . 4  |-  ( ( y  e.  NN0  /\  ch )  ->  E. x  e.  ZZ  ph )
3733, 36jaodan 786 . . 3  |-  ( ( y  e.  NN0  /\  ( ps  \/  ch ) )  ->  E. x  e.  ZZ  ph )
3837rexlimiva 2892 . 2  |-  ( E. y  e.  NN0  ( ps  \/  ch )  ->  E. x  e.  ZZ  ph )
3930, 38impbii 187 1  |-  ( E. x  e.  ZZ  ph  <->  E. y  e.  NN0  ( ps  \/  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2755   RRcr 9521   -ucneg 9842   NN0cn0 10836   ZZcz 10905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-n0 10837  df-z 10906
This theorem is referenced by:  dvdsrabdioph  35105
  Copyright terms: Public domain W3C validator