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Theorem divalgb 12615

Description: Express the division algorithm as stated in divalg 12614 in terms of

. (Contributed by Paul Chapman, 31-Mar-2011.)

**Assertion**
Ref	Expression
divalgb	$/\$ $/\$ $/\$ $/\$ $/\$

Distinct variable groups:

**Proof of Theorem divalgb**
Step	Hyp	Ref	Expression
1		df-3an 1007	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
2	1	rexbii 2551	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
3		r19.42v 2702	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
4	2, 3	bitri 184	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
5		zsubcl 9620	. . . . . . . . . . . 12 $/\$
6		divides 12479	. . . . . . . . . . . 12 $/\$
7	5, 6	sylan2 286	. . . . . . . . . . 11 $/\$ $/\$
8	7	3impb 1226	. . . . . . . . . 10 $/\$ $/\$
9	8	3com12 1234	. . . . . . . . 9 $/\$ $/\$
10		zcn 9584	. . . . . . . . . . . . . . . . . 18
11		zcn 9584	. . . . . . . . . . . . . . . . . 18
12		zmulcl 9633	. . . . . . . . . . . . . . . . . . 19 $/\$
13	12	zcnd 9704	. . . . . . . . . . . . . . . . . 18 $/\$
14		subadd 8478	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
15	10, 11, 13, 14	syl3an 1316	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
16		addcom 8412	. . . . . . . . . . . . . . . . . . . 20 $/\$
17	11, 13, 16	syl2an 289	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
18	17	3adant1 1042	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$
19	18	eqeq1d 2243	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
20	15, 19	bitrd 188	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
21		eqcom 2236	. . . . . . . . . . . . . . . 16
22		eqcom 2236	. . . . . . . . . . . . . . . 16
23	20, 21, 22	3bitr3g 222	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
24	23	3expia 1232	. . . . . . . . . . . . . 14 $/\$ $/\$
25	24	expcomd 1487	. . . . . . . . . . . . 13 $/\$
26	25	3impia 1227	. . . . . . . . . . . 12 $/\$ $/\$
27	26	imp 124	. . . . . . . . . . 11 $/\$ $/\$ $/\$
28	27	rexbidva 2541	. . . . . . . . . 10 $/\$ $/\$
29	28	3com23 1236	. . . . . . . . 9 $/\$ $/\$
30	9, 29	bitrd 188	. . . . . . . 8 $/\$ $/\$
31	30	anbi2d 464	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	4, 31	bitr4id 199	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		anass 401	. . . . . 6 $/\$ $/\$ $/\$ $/\$
34	32, 33	bitrdi 196	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	34	3expa 1230	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	35	reubidva 2730	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
37		elnn0z 9592	. . . . . . 7 $/\$
38	37	anbi1i 458	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
39		anass 401	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	38, 39	bitri 184	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
41	40	eubii 2091	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
42		df-reu 2529	. . . 4 $/\$ $/\$ $/\$
43		df-reu 2529	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
44	41, 42, 43	3bitr4ri 213	. . 3 $/\$ $/\$ $/\$
45	36, 44	bitrdi 196	. 2 $/\$ $/\$ $/\$ $/\$
46	45	3adant3 1044	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1005

wceq 1398

weu 2082

wcel 2205

wne 2414

wrex 2523

wreu 2524 class class class wbr 4111

cfv 5354 (class class class)co 6052

cc 8127

cc0 8129

caddc 8132

cmul 8134

clt 8310

cle 8311

cmin 8446

cn0 9498

cz 9579

cabs 11686

cdvds 12477

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4230 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-cnex 8220 ax-resscn 8221 ax-1cn 8222 ax-1re 8223 ax-icn 8224 ax-addcl 8225 ax-addrcl 8226 ax-mulcl 8227 ax-mulrcl 8228 ax-addcom 8229 ax-mulcom 8230 ax-addass 8231 ax-mulass 8232 ax-distr 8233 ax-i2m1 8234 ax-0lt1 8235 ax-1rid 8236 ax-0id 8237 ax-rnegex 8238 ax-cnre 8240 ax-pre-ltirr 8241 ax-pre-ltwlin 8242 ax-pre-lttrn 8243 ax-pre-ltadd 8245

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3045 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-br 4112 df-opab 4174 df-id 4416 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-iota 5314 df-fun 5356 df-fv 5362 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-pnf 8312 df-mnf 8313 df-xr 8314 df-ltxr 8315 df-le 8316 df-sub 8448 df-neg 8449 df-inn 9240 df-n0 9499 df-z 9580 df-dvds 12478

This theorem is referenced by: divalg2 12616

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