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Theorem rexzrexnn0 30328
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 10868 . . . . . . 7  |-  ( x  e.  ZZ  <->  ( x  e.  RR  /\  ( x  e.  NN0  \/  -u x  e.  NN0 ) ) )
21simprbi 464 . . . . . 6  |-  ( x  e.  ZZ  ->  (
x  e.  NN0  \/  -u x  e.  NN0 )
)
32adantr 465 . . . . 5  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( x  e.  NN0  \/  -u x  e.  NN0 ) )
4 simpr 461 . . . . . . . 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 1739 . . . . . . . . . 10  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
87bicomd 201 . . . . . . . . 9  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
98rspcev 3207 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  ph )  ->  E. y  e.  NN0  ps )
104, 5, 9syl2anc 661 . . . . . . 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 461 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  -> 
-u x  e.  NN0 )
13 zcn 10858 . . . . . . . . . . . . . . 15  |-  ( x  e.  ZZ  ->  x  e.  CC )
1413negnegd 9910 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  ->  -u -u x  =  x )
1514eqcomd 2468 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  x  =  -u -u x )
16 negeq 9801 . . . . . . . . . . . . . 14  |-  ( y  =  -u x  ->  -u y  =  -u -u x )
1716eqeq2d 2474 . . . . . . . . . . . . 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 714 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  /\  y  =  -u x )  ->  ( ch 
<-> 
ph ) )
2412, 23rspcedv 3211 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  -u x  e.  NN0 )  ->  ( ph  ->  E. y  e.  NN0  ch ) )
2524impancom 440 . . . . . 6  |-  ( ( x  e.  ZZ  /\  ph )  ->  ( -u x  e.  NN0  ->  E. y  e.  NN0  ch ) )
2611, 25orim12d 835 . . . . 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 3010 . . . 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 2944 . 2  |-  ( E. x  e.  ZZ  ph  ->  E. y  e.  NN0  ( ps  \/  ch ) )
31 nn0z 10876 . . . . 5  |-  ( y  e.  NN0  ->  y  e.  ZZ )
326rspcev 3207 . . . . 5  |-  ( ( y  e.  ZZ  /\  ps )  ->  E. x  e.  ZZ  ph )
3331, 32sylan 471 . . . 4  |-  ( ( y  e.  NN0  /\  ps )  ->  E. x  e.  ZZ  ph )
34 nn0negz 10890 . . . . 5  |-  ( y  e.  NN0  ->  -u y  e.  ZZ )
3520rspcev 3207 . . . . 5  |-  ( (
-u y  e.  ZZ  /\ 
ch )  ->  E. x  e.  ZZ  ph )
3634, 35sylan 471 . . . 4  |-  ( ( y  e.  NN0  /\  ch )  ->  E. x  e.  ZZ  ph )
3733, 36jaodan 783 . . 3  |-  ( ( y  e.  NN0  /\  ( ps  \/  ch ) )  ->  E. x  e.  ZZ  ph )
3837rexlimiva 2944 . 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 1374    e. wcel 1762   E.wrex 2808   RRcr 9480   -ucneg 9795   NN0cn0 10784   ZZcz 10853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854
This theorem is referenced by:  dvdsrabdioph  30334
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