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Theorem divalgb 11932

Description: Express the division algorithm as stated in divalg 11931 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 980	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
2	1	rexbii 2484	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
3		r19.42v 2634	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
4	2, 3	bitri 184	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
5		zsubcl 9296	. . . . . . . . . . . 12 $/\$
6		divides 11798	. . . . . . . . . . . 12 $/\$
7	5, 6	sylan2 286	. . . . . . . . . . 11 $/\$ $/\$
8	7	3impb 1199	. . . . . . . . . 10 $/\$ $/\$
9	8	3com12 1207	. . . . . . . . 9 $/\$ $/\$
10		zcn 9260	. . . . . . . . . . . . . . . . . 18
11		zcn 9260	. . . . . . . . . . . . . . . . . 18
12		zmulcl 9308	. . . . . . . . . . . . . . . . . . 19 $/\$
13	12	zcnd 9378	. . . . . . . . . . . . . . . . . 18 $/\$
14		subadd 8162	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
15	10, 11, 13, 14	syl3an 1280	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
16		addcom 8096	. . . . . . . . . . . . . . . . . . . 20 $/\$
17	11, 13, 16	syl2an 289	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
18	17	3adant1 1015	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$
19	18	eqeq1d 2186	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
20	15, 19	bitrd 188	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
21		eqcom 2179	. . . . . . . . . . . . . . . 16
22		eqcom 2179	. . . . . . . . . . . . . . . 16
23	20, 21, 22	3bitr3g 222	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
24	23	3expia 1205	. . . . . . . . . . . . . 14 $/\$ $/\$
25	24	expcomd 1441	. . . . . . . . . . . . 13 $/\$
26	25	3impia 1200	. . . . . . . . . . . 12 $/\$ $/\$
27	26	imp 124	. . . . . . . . . . 11 $/\$ $/\$ $/\$
28	27	rexbidva 2474	. . . . . . . . . 10 $/\$ $/\$
29	28	3com23 1209	. . . . . . . . 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 1203	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	35	reubidva 2660	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
37		elnn0z 9268	. . . . . . 7 $/\$
38	37	anbi1i 458	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
39		anass 401	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	38, 39	bitri 184	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
41	40	eubii 2035	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
42		df-reu 2462	. . . 4 $/\$ $/\$ $/\$
43		df-reu 2462	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
44	41, 42, 43	3bitr4ri 213	. . 3 $/\$ $/\$ $/\$
45	36, 44	bitrdi 196	. 2 $/\$ $/\$ $/\$ $/\$
46	45	3adant3 1017	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 978

wceq 1353

weu 2026

wcel 2148

wne 2347

wrex 2456

wreu 2457 class class class wbr 4005

cfv 5218 (class class class)co 5877

cc 7811

cc0 7813

caddc 7816

cmul 7818

clt 7994

cle 7995

cmin 8130

cn0 9178

cz 9255

cabs 11008

cdvds 11796

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-ltadd 7929

This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-inn 8922 df-n0 9179 df-z 9256 df-dvds 11797

This theorem is referenced by: divalg2 11933

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