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Theorem rexzrexnn0 26754
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 10252 . . . . . . 7  |-  ( x  e.  ZZ  <->  ( x  e.  RR  /\  ( x  e.  NN0  \/  -u x  e.  NN0 ) ) )
21simprbi 451 . . . . . 6  |-  ( x  e.  ZZ  ->  (
x  e.  NN0  \/  -u x  e.  NN0 )
)
32adantr 452 . . . . 5  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( x  e.  NN0  \/  -u x  e.  NN0 ) )
4 simpr 448 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\ 
ph )  /\  x  e.  NN0 )  ->  x  e.  NN0 )
5 simplr 732 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\ 
ph )  /\  x  e.  NN0 )  ->  ph )
6 rexzrexnn0.1 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
76equcoms 1689 . . . . . . . . . 10  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
87bicomd 193 . . . . . . . . 9  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
98rspcev 3012 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  ph )  ->  E. y  e.  NN0  ps )
104, 5, 9syl2anc 643 . . . . . . 7  |-  ( ( ( x  e.  ZZ  /\ 
ph )  /\  x  e.  NN0 )  ->  E. y  e.  NN0  ps )
1110ex 424 . . . . . 6  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( x  e.  NN0  ->  E. y  e.  NN0  ps ) )
12 simpr 448 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  -> 
-u x  e.  NN0 )
13 zcn 10243 . . . . . . . . . . . . . . 15  |-  ( x  e.  ZZ  ->  x  e.  CC )
1413negnegd 9358 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  ->  -u -u x  =  x )
1514eqcomd 2409 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  x  =  -u -u x )
16 negeq 9254 . . . . . . . . . . . . . 14  |-  ( y  =  -u x  ->  -u y  =  -u -u x )
1716eqeq2d 2415 . . . . . . . . . . . . 13  |-  ( y  =  -u x  ->  (
x  =  -u y  <->  x  =  -u -u x ) )
1815, 17syl5ibrcom 214 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  (
y  =  -u x  ->  x  =  -u y
) )
1918imp 419 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  x  =  -u y )
20 rexzrexnn0.2 . . . . . . . . . . 11  |-  ( x  =  -u y  ->  ( ph 
<->  ch ) )
2119, 20syl 16 . . . . . . . . . 10  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  ( ph  <->  ch )
)
2221bicomd 193 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  ( ch  <->  ph ) )
2322adantlr 696 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  /\  y  =  -u x )  ->  ( ch 
<-> 
ph ) )
2412, 23rspcedv 3016 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  ->  ( ph  ->  E. y  e.  NN0  ch ) )
2524impancom 428 . . . . . 6  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( -u x  e.  NN0  ->  E. y  e.  NN0  ch ) )
2611, 25orim12d 812 . . . . 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 2823 . . . 4  |-  ( E. y  e.  NN0  ( ps  \/  ch )  <->  ( E. y  e.  NN0  ps  \/  E. y  e.  NN0  ch ) )
2927, 28sylibr 204 . . 3  |-  ( ( x  e.  ZZ  /\  ph )  ->  E. y  e.  NN0  ( ps  \/  ch ) )
3029rexlimiva 2785 . 2  |-  ( E. x  e.  ZZ  ph  ->  E. y  e.  NN0  ( ps  \/  ch ) )
31 nn0z 10260 . . . . 5  |-  ( y  e.  NN0  ->  y  e.  ZZ )
326rspcev 3012 . . . . 5  |-  ( ( y  e.  ZZ  /\  ps )  ->  E. x  e.  ZZ  ph )
3331, 32sylan 458 . . . 4  |-  ( ( y  e.  NN0  /\  ps )  ->  E. x  e.  ZZ  ph )
34 nn0negz 10271 . . . . 5  |-  ( y  e.  NN0  ->  -u y  e.  ZZ )
3520rspcev 3012 . . . . 5  |-  ( (
-u y  e.  ZZ  /\ 
ch )  ->  E. x  e.  ZZ  ph )
3634, 35sylan 458 . . . 4  |-  ( ( y  e.  NN0  /\  ch )  ->  E. x  e.  ZZ  ph )
3733, 36jaodan 761 . . 3  |-  ( ( y  e.  NN0  /\  ( ps  \/  ch ) )  ->  E. x  e.  ZZ  ph )
3837rexlimiva 2785 . 2  |-  ( E. y  e.  NN0  ( ps  \/  ch )  ->  E. x  e.  ZZ  ph )
3930, 38impbii 181 1  |-  ( E. x  e.  ZZ  ph  <->  E. y  e.  NN0  ( ps  \/  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   RRcr 8945   -ucneg 9248   NN0cn0 10177   ZZcz 10238
This theorem is referenced by:  dvdsrabdioph  26760
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-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
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-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-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-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239
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