Theorem divalglem4 15119

Description: Lemma for divalg 15126. (Contributed by Paul Chapman, 21-Mar-2011.)

Hypotheses

Ref	Expression
divalglem0.1	|- N e. ZZ
divalglem0.2	|- D e. ZZ
divalglem1.3	|- D =/= 0
divalglem2.4	|- S = { r e. NN0 | D || ( N - r ) }

Assertion

Ref	Expression
divalglem4	|- S = { r e. NN0 | E. q e. ZZ N = ( ( q x. D ) + r ) }

Distinct variable groups:   D, r   N, r   D, q, r   N, q

Allowed substitution hints:   S( r, q)

Proof of Theorem divalglem4

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		divalglem0.2	. . . . . 6 |- D e. ZZ
2		divalglem0.1	. . . . . . 7 |- N e. ZZ
3		nn0z 11400	. . . . . . 7 |- ( z e. NN0 -> z e. ZZ )
4		zsubcl 11419	. . . . . . 7 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( N - z ) e. ZZ )
5	2, 3, 4	sylancr 695	. . . . . 6 |- ( z e. NN0 -> ( N - z ) e. ZZ )
6		divides 14985	. . . . . 6 |- ( ( D e. ZZ /\ ( N - z ) e. ZZ ) -> ( D || ( N - z ) <-> E. q e. ZZ ( q x. D ) = ( N - z ) ) )
7	1, 5, 6	sylancr 695	. . . . 5 |- ( z e. NN0 -> ( D || ( N - z ) <-> E. q e. ZZ ( q x. D ) = ( N - z ) ) )
8		nn0cn 11302	. . . . . . . 8 |- ( z e. NN0 -> z e. CC )
9		zmulcl 11426	. . . . . . . . . 10 |- ( ( q e. ZZ /\ D e. ZZ ) -> ( q x. D ) e. ZZ )
10	1, 9	mpan2 707	. . . . . . . . 9 |- ( q e. ZZ -> ( q x. D ) e. ZZ )
11	10	zcnd 11483	. . . . . . . 8 |- ( q e. ZZ -> ( q x. D ) e. CC )
12		zcn 11382	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
13	2, 12	ax-mp 5	. . . . . . . . . 10 |- N e. CC
14		subadd 10284	. . . . . . . . . 10 |- ( ( N e. CC /\ z e. CC /\ ( q x. D ) e. CC ) -> ( ( N - z ) = ( q x. D ) <-> ( z + ( q x. D ) ) = N ) )
15	13, 14	mp3an1 1411	. . . . . . . . 9 |- ( ( z e. CC /\ ( q x. D ) e. CC ) -> ( ( N - z ) = ( q x. D ) <-> ( z + ( q x. D ) ) = N ) )
16		addcom 10222	. . . . . . . . . 10 |- ( ( z e. CC /\ ( q x. D ) e. CC ) -> ( z + ( q x. D ) ) = ( ( q x. D ) + z ) )
17	16	eqeq1d 2624	. . . . . . . . 9 |- ( ( z e. CC /\ ( q x. D ) e. CC ) -> ( ( z + ( q x. D ) ) = N <-> ( ( q x. D ) + z ) = N ) )
18	15, 17	bitrd 268	. . . . . . . 8 |- ( ( z e. CC /\ ( q x. D ) e. CC ) -> ( ( N - z ) = ( q x. D ) <-> ( ( q x. D ) + z ) = N ) )
19	8, 11, 18	syl2an 494	. . . . . . 7 |- ( ( z e. NN0 /\ q e. ZZ ) -> ( ( N - z ) = ( q x. D ) <-> ( ( q x. D ) + z ) = N ) )
20		eqcom 2629	. . . . . . 7 |- ( ( N - z ) = ( q x. D ) <-> ( q x. D ) = ( N - z ) )
21		eqcom 2629	. . . . . . 7 |- ( ( ( q x. D ) + z ) = N <-> N = ( ( q x. D ) + z ) )
22	19, 20, 21	3bitr3g 302	. . . . . 6 |- ( ( z e. NN0 /\ q e. ZZ ) -> ( ( q x. D ) = ( N - z ) <-> N = ( ( q x. D ) + z ) ) )
23	22	rexbidva 3049	. . . . 5 |- ( z e. NN0 -> ( E. q e. ZZ ( q x. D ) = ( N - z ) <-> E. q e. ZZ N = ( ( q x. D ) + z ) ) )
24	7, 23	bitrd 268	. . . 4 |- ( z e. NN0 -> ( D || ( N - z ) <-> E. q e. ZZ N = ( ( q x. D ) + z ) ) )
25	24	pm5.32i 669	. . 3 |- ( ( z e. NN0 /\ D || ( N - z ) ) <-> ( z e. NN0 /\ E. q e. ZZ N = ( ( q x. D ) + z ) ) )
26		oveq2 6658	. . . . 5 |- ( r = z -> ( N - r ) = ( N - z ) )
27	26	breq2d 4665	. . . 4 |- ( r = z -> ( D || ( N - r ) <-> D || ( N - z ) ) )
28		divalglem2.4	. . . 4 |- S = { r e. NN0 | D || ( N - r ) }
29	27, 28	elrab2 3366	. . 3 |- ( z e. S <-> ( z e. NN0 /\ D || ( N - z ) ) )
30		oveq2 6658	. . . . . 6 |- ( r = z -> ( ( q x. D ) + r ) = ( ( q x. D ) + z ) )
31	30	eqeq2d 2632	. . . . 5 |- ( r = z -> ( N = ( ( q x. D ) + r ) <-> N = ( ( q x. D ) + z ) ) )
32	31	rexbidv 3052	. . . 4 |- ( r = z -> ( E. q e. ZZ N = ( ( q x. D ) + r ) <-> E. q e. ZZ N = ( ( q x. D ) + z ) ) )
33	32	elrab 3363	. . 3 |- ( z e. { r e. NN0 | E. q e. ZZ N = ( ( q x. D ) + r ) } <-> ( z e. NN0 /\ E. q e. ZZ N = ( ( q x. D ) + z ) ) )
34	25, 29, 33	3bitr4i 292	. 2 |- ( z e. S <-> z e. { r e. NN0 | E. q e. ZZ N = ( ( q x. D ) + r ) } )
35	34	eqriv 2619	1 |- S = { r e. NN0 | E. q e. ZZ N = ( ( q x. D ) + r ) }

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   class class class wbr 4653  (class class class)co 6650   CCcc 9934   0cc0 9936   + caddc 9939   x. cmul 9941   - cmin 10266   NN0cn0 11292   ZZcz 11377   || cdvds 14983

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  df-dvds 14984

This theorem is referenced by:  divalglem10  15125