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Theorem divalgb 12436

Description: Express the division algorithm as stated in divalg 12435 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 1004	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
2	1	rexbii 2537	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
3		r19.42v 2688	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
4	2, 3	bitri 184	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
5		zsubcl 9487	. . . . . . . . . . . 12 $/\$
6		divides 12300	. . . . . . . . . . . 12 $/\$
7	5, 6	sylan2 286	. . . . . . . . . . 11 $/\$ $/\$
8	7	3impb 1223	. . . . . . . . . 10 $/\$ $/\$
9	8	3com12 1231	. . . . . . . . 9 $/\$ $/\$
10		zcn 9451	. . . . . . . . . . . . . . . . . 18
11		zcn 9451	. . . . . . . . . . . . . . . . . 18
12		zmulcl 9500	. . . . . . . . . . . . . . . . . . 19 $/\$
13	12	zcnd 9570	. . . . . . . . . . . . . . . . . 18 $/\$
14		subadd 8349	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
15	10, 11, 13, 14	syl3an 1313	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
16		addcom 8283	. . . . . . . . . . . . . . . . . . . 20 $/\$
17	11, 13, 16	syl2an 289	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
18	17	3adant1 1039	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$
19	18	eqeq1d 2238	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
20	15, 19	bitrd 188	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
21		eqcom 2231	. . . . . . . . . . . . . . . 16
22		eqcom 2231	. . . . . . . . . . . . . . . 16
23	20, 21, 22	3bitr3g 222	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
24	23	3expia 1229	. . . . . . . . . . . . . 14 $/\$ $/\$
25	24	expcomd 1484	. . . . . . . . . . . . 13 $/\$
26	25	3impia 1224	. . . . . . . . . . . 12 $/\$ $/\$
27	26	imp 124	. . . . . . . . . . 11 $/\$ $/\$ $/\$
28	27	rexbidva 2527	. . . . . . . . . 10 $/\$ $/\$
29	28	3com23 1233	. . . . . . . . 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 1227	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	35	reubidva 2715	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
37		elnn0z 9459	. . . . . . 7 $/\$
38	37	anbi1i 458	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
39		anass 401	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	38, 39	bitri 184	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
41	40	eubii 2086	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
42		df-reu 2515	. . . 4 $/\$ $/\$ $/\$
43		df-reu 2515	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
44	41, 42, 43	3bitr4ri 213	. . 3 $/\$ $/\$ $/\$
45	36, 44	bitrdi 196	. 2 $/\$ $/\$ $/\$ $/\$
46	45	3adant3 1041	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1002

wceq 1395

weu 2077

wcel 2200

wne 2400

wrex 2509

wreu 2510 class class class wbr 4083

cfv 5318 (class class class)co 6001

cc 7997

cc0 7999

caddc 8002

cmul 8004

clt 8181

cle 8182

cmin 8317

cn0 9369

cz 9446

cabs 11508

cdvds 12298

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-ltadd 8115

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-br 4084 df-opab 4146 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-inn 9111 df-n0 9370 df-z 9447 df-dvds 12299

This theorem is referenced by: divalg2 12437

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