divrngcl - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > divrngcl		Structured version Visualization version Unicode version

Theorem divrngcl 33756

Description: The product of two nonzero elements of a division ring is nonzero. (Contributed by Jeff Madsen, 9-Jun-2010.)

**Hypotheses**
Ref	Expression
isdivrng1.1
isdivrng1.2
isdivrng1.3	GId
isdivrng1.4

**Assertion**
Ref	Expression
divrngcl	$/\$ $\$ $/\$ $\$ $\$

**Proof of Theorem divrngcl**
Step	Hyp	Ref	Expression
1		isdivrng1.1	. . 3
2		isdivrng1.2	. . 3
3		isdivrng1.3	. . 3 GId
4		isdivrng1.4	. . 3
5	1, 2, 3, 4	isdrngo1 33755	. 2 $/\$ $\$ $\$
6		ovres 6800	. . . . 5 $\$ $/\$ $\$ $\$ $\$
7	6	adantl 482	. . . 4 $/\$ $\$ $\$ $/\$ $\$ $/\$ $\$ $\$ $\$
8		eqid 2622	. . . . . . . . 9 $\$ $\$ $\$ $\$
9	8	grpocl 27354	. . . . . . . 8 $\$ $\$ $/\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
10	9	3expib 1268	. . . . . . 7 $\$ $\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
11	10	adantl 482	. . . . . 6 $/\$ $\$ $\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
12		grporndm 27364	. . . . . . . . . 10 $\$ $\$ $\$ $\$ $\$ $\$
13	12	adantl 482	. . . . . . . . 9 $/\$ $\$ $\$ $\$ $\$ $\$ $\$
14		difss 3737	. . . . . . . . . . . . . . 15 $\$
15		xpss12 5225	. . . . . . . . . . . . . . 15 $\$ $/\$ $\$ $\$ $\$
16	14, 14, 15	mp2an 708	. . . . . . . . . . . . . 14 $\$ $\$
17	1, 2, 4	rngosm 33699	. . . . . . . . . . . . . . 15
18		fdm 6051	. . . . . . . . . . . . . . 15
19	17, 18	syl 17	. . . . . . . . . . . . . 14
20	16, 19	syl5sseqr 3654	. . . . . . . . . . . . 13 $\$ $\$
21		ssdmres 5420	. . . . . . . . . . . . 13 $\$ $\$ $\$ $\$ $\$ $\$
22	20, 21	sylib 208	. . . . . . . . . . . 12 $\$ $\$ $\$ $\$
23	22	adantr 481	. . . . . . . . . . 11 $/\$ $\$ $\$ $\$ $\$ $\$ $\$
24	23	dmeqd 5326	. . . . . . . . . 10 $/\$ $\$ $\$ $\$ $\$ $\$ $\$
25		dmxpid 5345	. . . . . . . . . 10 $\$ $\$ $\$
26	24, 25	syl6eq 2672	. . . . . . . . 9 $/\$ $\$ $\$ $\$ $\$ $\$
27	13, 26	eqtrd 2656	. . . . . . . 8 $/\$ $\$ $\$ $\$ $\$ $\$
28	27	eleq2d 2687	. . . . . . 7 $/\$ $\$ $\$ $\$ $\$ $\$
29	27	eleq2d 2687	. . . . . . 7 $/\$ $\$ $\$ $\$ $\$ $\$
30	28, 29	anbi12d 747	. . . . . 6 $/\$ $\$ $\$ $\$ $\$ $/\$ $\$ $\$ $\$ $/\$ $\$
31	27	eleq2d 2687	. . . . . 6 $/\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$
32	11, 30, 31	3imtr3d 282	. . . . 5 $/\$ $\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$
33	32	imp 445	. . . 4 $/\$ $\$ $\$ $/\$ $\$ $/\$ $\$ $\$ $\$ $\$
34	7, 33	eqeltrrd 2702	. . 3 $/\$ $\$ $\$ $/\$ $\$ $/\$ $\$ $\$
35	34	3impb 1260	. 2 $/\$ $\$ $\$ $/\$ $\$ $/\$ $\$ $\$
36	5, 35	syl3an1b 1362	1 $/\$ $\$ $/\$ $\$ $\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990 $\$ cdif 3571

wss 3574

csn 4177

cxp 5112

cdm 5114

crn 5115

cres 5116

wf 5884

cfv 5888 (class class class)co 6650

c1st 7166

c2nd 7167

cgr 27343 GIdcgi 27344

crngo 33693

cdrng 33747

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-ov 6653 df-1st 7168 df-2nd 7169 df-grpo 27347 df-rngo 33694 df-drngo 33748

This theorem is referenced by: (None)

W3C validator