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Theorem rexzrexnn0 28987
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 10649 . . . . . . 7  |-  ( x  e.  ZZ  <->  ( x  e.  RR  /\  ( x  e.  NN0  \/  -u x  e.  NN0 ) ) )
21simprbi 461 . . . . . 6  |-  ( x  e.  ZZ  ->  (
x  e.  NN0  \/  -u x  e.  NN0 )
)
32adantr 462 . . . . 5  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( x  e.  NN0  \/  -u x  e.  NN0 ) )
4 simpr 458 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\ 
ph )  /\  x  e.  NN0 )  ->  x  e.  NN0 )
5 simplr 747 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\ 
ph )  /\  x  e.  NN0 )  ->  ph )
6 rexzrexnn0.1 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
76equcoms 1732 . . . . . . . . . 10  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
87bicomd 201 . . . . . . . . 9  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
98rspcev 3062 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  ph )  ->  E. y  e.  NN0  ps )
104, 5, 9syl2anc 654 . . . . . . 7  |-  ( ( ( x  e.  ZZ  /\ 
ph )  /\  x  e.  NN0 )  ->  E. y  e.  NN0  ps )
1110ex 434 . . . . . 6  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( x  e.  NN0  ->  E. y  e.  NN0  ps ) )
12 simpr 458 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  -> 
-u x  e.  NN0 )
13 zcn 10639 . . . . . . . . . . . . . . 15  |-  ( x  e.  ZZ  ->  x  e.  CC )
1413negnegd 9698 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  ->  -u -u x  =  x )
1514eqcomd 2438 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  x  =  -u -u x )
16 negeq 9590 . . . . . . . . . . . . . 14  |-  ( y  =  -u x  ->  -u y  =  -u -u x )
1716eqeq2d 2444 . . . . . . . . . . . . 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 429 . . . . . . . . . . 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 201 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  y  =  -u x )  ->  ( ch  <->  ph ) )
2322adantlr 707 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  /\  y  =  -u x )  ->  ( ch 
<-> 
ph ) )
2412, 23rspcedv 3066 . . . . . . 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 827 . . . . 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 2866 . . . 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 2826 . 2  |-  ( E. x  e.  ZZ  ph  ->  E. y  e.  NN0  ( ps  \/  ch ) )
31 nn0z 10657 . . . . 5  |-  ( y  e.  NN0  ->  y  e.  ZZ )
326rspcev 3062 . . . . 5  |-  ( ( y  e.  ZZ  /\  ps )  ->  E. x  e.  ZZ  ph )
3331, 32sylan 468 . . . 4  |-  ( ( y  e.  NN0  /\  ps )  ->  E. x  e.  ZZ  ph )
34 nn0negz 10671 . . . . 5  |-  ( y  e.  NN0  ->  -u y  e.  ZZ )
3520rspcev 3062 . . . . 5  |-  ( (
-u y  e.  ZZ  /\ 
ch )  ->  E. x  e.  ZZ  ph )
3634, 35sylan 468 . . . 4  |-  ( ( y  e.  NN0  /\  ch )  ->  E. x  e.  ZZ  ph )
3733, 36jaodan 776 . . 3  |-  ( ( y  e.  NN0  /\  ( ps  \/  ch ) )  ->  E. x  e.  ZZ  ph )
3837rexlimiva 2826 . 2  |-  ( E. y  e.  NN0  ( ps  \/  ch )  ->  E. x  e.  ZZ  ph )
3930, 38impbii 188 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 368    /\ wa 369    = wceq 1362    e. wcel 1755   E.wrex 2706   RRcr 9269   -ucneg 9584   NN0cn0 10567   ZZcz 10634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-n0 10568  df-z 10635
This theorem is referenced by:  dvdsrabdioph  28993
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