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Theorem divalgb 12566

Description: Express the division algorithm as stated in divalg 12565 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 2540	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
3		r19.42v 2691	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
4	2, 3	bitri 184	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
5		zsubcl 9581	. . . . . . . . . . . 12 $/\$
6		divides 12430	. . . . . . . . . . . 12 $/\$
7	5, 6	sylan2 286	. . . . . . . . . . 11 $/\$ $/\$
8	7	3impb 1226	. . . . . . . . . 10 $/\$ $/\$
9	8	3com12 1234	. . . . . . . . 9 $/\$ $/\$
10		zcn 9545	. . . . . . . . . . . . . . . . . 18
11		zcn 9545	. . . . . . . . . . . . . . . . . 18
12		zmulcl 9594	. . . . . . . . . . . . . . . . . . 19 $/\$
13	12	zcnd 9664	. . . . . . . . . . . . . . . . . 18 $/\$
14		subadd 8441	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
15	10, 11, 13, 14	syl3an 1316	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
16		addcom 8375	. . . . . . . . . . . . . . . . . . . 20 $/\$
17	11, 13, 16	syl2an 289	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
18	17	3adant1 1042	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$
19	18	eqeq1d 2240	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
20	15, 19	bitrd 188	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
21		eqcom 2233	. . . . . . . . . . . . . . . 16
22		eqcom 2233	. . . . . . . . . . . . . . . 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 2530	. . . . . . . . . 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 2718	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
37		elnn0z 9553	. . . . . . 7 $/\$
38	37	anbi1i 458	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
39		anass 401	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	38, 39	bitri 184	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
41	40	eubii 2088	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
42		df-reu 2518	. . . 4 $/\$ $/\$ $/\$
43		df-reu 2518	. . . 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 2079

wcel 2202

wne 2403

wrex 2512

wreu 2513 class class class wbr 4093

cfv 5333 (class class class)co 6028

cc 8090

cc0 8092

caddc 8095

cmul 8097

clt 8273

cle 8274

cmin 8409

cn0 9461

cz 9540

cabs 11637

cdvds 12428

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 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-ltadd 8208

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-br 4094 df-opab 4156 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-iota 5293 df-fun 5335 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-sub 8411 df-neg 8412 df-inn 9203 df-n0 9462 df-z 9541 df-dvds 12429

This theorem is referenced by: divalg2 12567

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