Theorem rexzrexnn0 37368

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

Step	Hyp	Ref	Expression
1		elznn0 11392	. . . . . . 7 |- ( x e. ZZ <-> ( x e. RR /\ ( x e. NN0 \/ -u x e. NN0 ) ) )
2	1	simprbi 480	. . . . . 6 |- ( x e. ZZ -> ( x e. NN0 \/ -u x e. NN0 ) )
3	2	adantr 481	. . . . 5 |- ( ( x e. ZZ /\ ph ) -> ( x e. NN0 \/ -u x e. NN0 ) )
4		simpr 477	. . . . . . . 8 |- ( ( ( x e. ZZ /\ ph ) /\ x e. NN0 ) -> x e. NN0 )
5		simplr 792	. . . . . . . 8 |- ( ( ( x e. ZZ /\ ph ) /\ x e. NN0 ) -> ph )
6		rexzrexnn0.1	. . . . . . . . . . 11 |- ( x = y -> ( ph <-> ps ) )
7	6	equcoms 1947	. . . . . . . . . 10 |- ( y = x -> ( ph <-> ps ) )
8	7	bicomd 213	. . . . . . . . 9 |- ( y = x -> ( ps <-> ph ) )
9	8	rspcev 3309	. . . . . . . 8 |- ( ( x e. NN0 /\ ph ) -> E. y e. NN0 ps )
10	4, 5, 9	syl2anc 693	. . . . . . 7 |- ( ( ( x e. ZZ /\ ph ) /\ x e. NN0 ) -> E. y e. NN0 ps )
11	10	ex 450	. . . . . 6 |- ( ( x e. ZZ /\ ph ) -> ( x e. NN0 -> E. y e. NN0 ps ) )
12		simpr 477	. . . . . . . 8 |- ( ( x e. ZZ /\ -u x e. NN0 ) -> -u x e. NN0 )
13		zcn 11382	. . . . . . . . . . . . . . 15 |- ( x e. ZZ -> x e. CC )
14	13	negnegd 10383	. . . . . . . . . . . . . 14 |- ( x e. ZZ -> -u -u x = x )
15	14	eqcomd 2628	. . . . . . . . . . . . 13 |- ( x e. ZZ -> x = -u -u x )
16		negeq 10273	. . . . . . . . . . . . . 14 |- ( y = -u x -> -u y = -u -u x )
17	16	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( y = -u x -> ( x = -u y <-> x = -u -u x ) )
18	15, 17	syl5ibrcom 237	. . . . . . . . . . . 12 |- ( x e. ZZ -> ( y = -u x -> x = -u y ) )
19	18	imp 445	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ y = -u x ) -> x = -u y )
20		rexzrexnn0.2	. . . . . . . . . . 11 |- ( x = -u y -> ( ph <-> ch ) )
21	19, 20	syl 17	. . . . . . . . . 10 |- ( ( x e. ZZ /\ y = -u x ) -> ( ph <-> ch ) )
22	21	bicomd 213	. . . . . . . . 9 |- ( ( x e. ZZ /\ y = -u x ) -> ( ch <-> ph ) )
23	22	adantlr 751	. . . . . . . 8 |- ( ( ( x e. ZZ /\ -u x e. NN0 ) /\ y = -u x ) -> ( ch <-> ph ) )
24	12, 23	rspcedv 3313	. . . . . . 7 |- ( ( x e. ZZ /\ -u x e. NN0 ) -> ( ph -> E. y e. NN0 ch ) )
25	24	impancom 456	. . . . . 6 |- ( ( x e. ZZ /\ ph ) -> ( -u x e. NN0 -> E. y e. NN0 ch ) )
26	11, 25	orim12d 883	. . . . 5 |- ( ( x e. ZZ /\ ph ) -> ( ( x e. NN0 \/ -u x e. NN0 ) -> ( E. y e. NN0 ps \/ E. y e. NN0 ch ) ) )
27	3, 26	mpd 15	. . . 4 |- ( ( x e. ZZ /\ ph ) -> ( E. y e. NN0 ps \/ E. y e. NN0 ch ) )
28		r19.43 3093	. . . 4 |- ( E. y e. NN0 ( ps \/ ch ) <-> ( E. y e. NN0 ps \/ E. y e. NN0 ch ) )
29	27, 28	sylibr 224	. . 3 |- ( ( x e. ZZ /\ ph ) -> E. y e. NN0 ( ps \/ ch ) )
30	29	rexlimiva 3028	. 2 |- ( E. x e. ZZ ph -> E. y e. NN0 ( ps \/ ch ) )
31		nn0z 11400	. . . . 5 |- ( y e. NN0 -> y e. ZZ )
32	6	rspcev 3309	. . . . 5 |- ( ( y e. ZZ /\ ps ) -> E. x e. ZZ ph )
33	31, 32	sylan 488	. . . 4 |- ( ( y e. NN0 /\ ps ) -> E. x e. ZZ ph )
34		nn0negz 11415	. . . . 5 |- ( y e. NN0 -> -u y e. ZZ )
35	20	rspcev 3309	. . . . 5 |- ( ( -u y e. ZZ /\ ch ) -> E. x e. ZZ ph )
36	34, 35	sylan 488	. . . 4 |- ( ( y e. NN0 /\ ch ) -> E. x e. ZZ ph )
37	33, 36	jaodan 826	. . 3 |- ( ( y e. NN0 /\ ( ps \/ ch ) ) -> E. x e. ZZ ph )
38	37	rexlimiva 3028	. 2 |- ( E. y e. NN0 ( ps \/ ch ) -> E. x e. ZZ ph )
39	30, 38	impbii 199	1 |- ( E. x e. ZZ ph <-> E. y e. NN0 ( ps \/ ch ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   RRcr 9935   -ucneg 10267   NN0cn0 11292   ZZcz 11377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378

This theorem is referenced by:  dvdsrabdioph  37374