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Theorem divalgb 12107

Description: Express the division algorithm as stated in divalg 12106 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 982	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
2	1	rexbii 2504	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
3		r19.42v 2654	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
4	2, 3	bitri 184	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
5		zsubcl 9384	. . . . . . . . . . . 12 $/\$
6		divides 11971	. . . . . . . . . . . 12 $/\$
7	5, 6	sylan2 286	. . . . . . . . . . 11 $/\$ $/\$
8	7	3impb 1201	. . . . . . . . . 10 $/\$ $/\$
9	8	3com12 1209	. . . . . . . . 9 $/\$ $/\$
10		zcn 9348	. . . . . . . . . . . . . . . . . 18
11		zcn 9348	. . . . . . . . . . . . . . . . . 18
12		zmulcl 9396	. . . . . . . . . . . . . . . . . . 19 $/\$
13	12	zcnd 9466	. . . . . . . . . . . . . . . . . 18 $/\$
14		subadd 8246	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
15	10, 11, 13, 14	syl3an 1291	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
16		addcom 8180	. . . . . . . . . . . . . . . . . . . 20 $/\$
17	11, 13, 16	syl2an 289	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
18	17	3adant1 1017	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$
19	18	eqeq1d 2205	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
20	15, 19	bitrd 188	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
21		eqcom 2198	. . . . . . . . . . . . . . . 16
22		eqcom 2198	. . . . . . . . . . . . . . . 16
23	20, 21, 22	3bitr3g 222	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
24	23	3expia 1207	. . . . . . . . . . . . . 14 $/\$ $/\$
25	24	expcomd 1452	. . . . . . . . . . . . 13 $/\$
26	25	3impia 1202	. . . . . . . . . . . 12 $/\$ $/\$
27	26	imp 124	. . . . . . . . . . 11 $/\$ $/\$ $/\$
28	27	rexbidva 2494	. . . . . . . . . 10 $/\$ $/\$
29	28	3com23 1211	. . . . . . . . 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 1205	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	35	reubidva 2680	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
37		elnn0z 9356	. . . . . . 7 $/\$
38	37	anbi1i 458	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
39		anass 401	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	38, 39	bitri 184	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
41	40	eubii 2054	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
42		df-reu 2482	. . . 4 $/\$ $/\$ $/\$
43		df-reu 2482	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
44	41, 42, 43	3bitr4ri 213	. . 3 $/\$ $/\$ $/\$
45	36, 44	bitrdi 196	. 2 $/\$ $/\$ $/\$ $/\$
46	45	3adant3 1019	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 980

wceq 1364

weu 2045

wcel 2167

wne 2367

wrex 2476

wreu 2477 class class class wbr 4034

cfv 5259 (class class class)co 5925

cc 7894

cc0 7896

caddc 7899

cmul 7901

clt 8078

cle 8079

cmin 8214

cn0 9266

cz 9343

cabs 11179

cdvds 11969

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-ltadd 8012

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-br 4035 df-opab 4096 df-id 4329 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-iota 5220 df-fun 5261 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-inn 9008 df-n0 9267 df-z 9344 df-dvds 11970

This theorem is referenced by: divalg2 12108

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