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Theorem rexzrexnn0 26988
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 10054 . . . . . . 7  |-  ( x  e.  ZZ  <->  ( x  e.  RR  /\  ( x  e.  NN0  \/  -u x  e.  NN0 ) ) )
21simprbi 450 . . . . . 6  |-  ( x  e.  ZZ  ->  (
x  e.  NN0  \/  -u x  e.  NN0 )
)
32adantr 451 . . . . 5  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( x  e.  NN0  \/  -u x  e.  NN0 ) )
4 simpr 447 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\ 
ph )  /\  x  e.  NN0 )  ->  x  e.  NN0 )
5 simplr 731 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\ 
ph )  /\  x  e.  NN0 )  ->  ph )
6 rexzrexnn0.1 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
76equcoms 1666 . . . . . . . . . 10  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
87bicomd 192 . . . . . . . . 9  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
98rspcev 2897 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  ph )  ->  E. y  e.  NN0  ps )
104, 5, 9syl2anc 642 . . . . . . 7  |-  ( ( ( x  e.  ZZ  /\ 
ph )  /\  x  e.  NN0 )  ->  E. y  e.  NN0  ps )
1110ex 423 . . . . . 6  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( x  e.  NN0  ->  E. y  e.  NN0  ps ) )
12 simpr 447 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  -> 
-u x  e.  NN0 )
13 zcn 10045 . . . . . . . . . . . . . . 15  |-  ( x  e.  ZZ  ->  x  e.  CC )
1413negnegd 9164 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  ->  -u -u x  =  x )
1514eqcomd 2301 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  x  =  -u -u x )
16 negeq 9060 . . . . . . . . . . . . . 14  |-  ( y  =  -u x  ->  -u y  =  -u -u x )
1716eqeq2d 2307 . . . . . . . . . . . . 13  |-  ( y  =  -u x  ->  (
x  =  -u y  <->  x  =  -u -u x ) )
1815, 17syl5ibrcom 213 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  (
y  =  -u x  ->  x  =  -u y
) )
1918imp 418 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  x  =  -u y )
20 rexzrexnn0.2 . . . . . . . . . . 11  |-  ( x  =  -u y  ->  ( ph 
<->  ch ) )
2119, 20syl 15 . . . . . . . . . 10  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  ( ph  <->  ch )
)
2221bicomd 192 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  ( ch  <->  ph ) )
2322adantlr 695 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  /\  y  =  -u x )  ->  ( ch 
<-> 
ph ) )
2412, 23rspcedv 2901 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  ->  ( ph  ->  E. y  e.  NN0  ch ) )
2524impancom 427 . . . . . 6  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( -u x  e.  NN0  ->  E. y  e.  NN0  ch ) )
2611, 25orim12d 811 . . . . 5  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( (
x  e.  NN0  \/  -u x  e.  NN0 )  ->  ( E. y  e. 
NN0  ps  \/  E. y  e.  NN0  ch ) ) )
273, 26mpd 14 . . . 4  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( E. y  e.  NN0  ps  \/  E. y  e.  NN0  ch ) )
28 r19.43 2708 . . . 4  |-  ( E. y  e.  NN0  ( ps  \/  ch )  <->  ( E. y  e.  NN0  ps  \/  E. y  e.  NN0  ch ) )
2927, 28sylibr 203 . . 3  |-  ( ( x  e.  ZZ  /\  ph )  ->  E. y  e.  NN0  ( ps  \/  ch ) )
3029rexlimiva 2675 . 2  |-  ( E. x  e.  ZZ  ph  ->  E. y  e.  NN0  ( ps  \/  ch ) )
31 nn0z 10062 . . . . 5  |-  ( y  e.  NN0  ->  y  e.  ZZ )
326rspcev 2897 . . . . 5  |-  ( ( y  e.  ZZ  /\  ps )  ->  E. x  e.  ZZ  ph )
3331, 32sylan 457 . . . 4  |-  ( ( y  e.  NN0  /\  ps )  ->  E. x  e.  ZZ  ph )
34 nn0negz 10073 . . . . 5  |-  ( y  e.  NN0  ->  -u y  e.  ZZ )
3520rspcev 2897 . . . . 5  |-  ( (
-u y  e.  ZZ  /\ 
ch )  ->  E. x  e.  ZZ  ph )
3634, 35sylan 457 . . . 4  |-  ( ( y  e.  NN0  /\  ch )  ->  E. x  e.  ZZ  ph )
3733, 36jaodan 760 . . 3  |-  ( ( y  e.  NN0  /\  ( ps  \/  ch ) )  ->  E. x  e.  ZZ  ph )
3837rexlimiva 2675 . 2  |-  ( E. y  e.  NN0  ( ps  \/  ch )  ->  E. x  e.  ZZ  ph )
3930, 38impbii 180 1  |-  ( E. x  e.  ZZ  ph  <->  E. y  e.  NN0  ( ps  \/  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   RRcr 8752   -ucneg 9054   NN0cn0 9981   ZZcz 10040
This theorem is referenced by:  dvdsrabdioph  26994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041
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