Theorem divrngcl 16110

Description: The product of two nonzero elements of a division ring is nonzero.

Hypotheses

Ref	Expression
isdivrng1.1	|- G = (1st` R)
isdivrng1.2	|- H = (2nd` R)
isdivrng1.3	|- Z = (Id` G)
isdivrng1.4	|- X = ran G

Assertion

Ref	Expression
divrngcl	|- ((R e. DivRing /\ A e. (X \ {Z}) /\ B e. (X \ {Z})) -> (AHB) e. (X \ {Z}))

Proof of Theorem divrngcl

Step	Hyp	Ref	Expression
1		oprvres 4963	. . . . 5 |- ((A e. (X \ {Z}) /\ B e. (X \ {Z})) -> (A(H |` ((X \ {Z}) X. (X \ {Z})))B) = (AHB))
2	1	adantl 424	. . . 4 |- (((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) /\ (A e. (X \ {Z}) /\ B e. (X \ {Z}))) -> (A(H |` ((X \ {Z}) X. (X \ {Z})))B) = (AHB))
3		eqid 1884	. . . . . . . . 9 |- ran ( H |` ((X \ {Z}) X. (X \ {Z}))) = ran ( H |` ((X \ {Z}) X. (X \ {Z})))
4	3	grpcl 9324	. . . . . . . 8 |- (((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp /\ A e. ran ( H |` ((X \ {Z}) X. (X \ {Z}))) /\ B e. ran ( H |` ((X \ {Z}) X. (X \ {Z})))) -> (A(H |` ((X \ {Z}) X. (X \ {Z})))B) e. ran ( H |` ((X \ {Z}) X. (X \ {Z}))))
5	4	3expib 1070	. . . . . . 7 |- ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp -> ((A e. ran ( H |` ((X \ {Z}) X. (X \ {Z}))) /\ B e. ran ( H |` ((X \ {Z}) X. (X \ {Z})))) -> (A(H |` ((X \ {Z}) X. (X \ {Z})))B) e. ran ( H |` ((X \ {Z}) X. (X \ {Z})))))
6	5	adantl 424	. . . . . 6 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> ((A e. ran ( H |` ((X \ {Z}) X. (X \ {Z}))) /\ B e. ran ( H |` ((X \ {Z}) X. (X \ {Z})))) -> (A(H |` ((X \ {Z}) X. (X \ {Z})))B) e. ran ( H |` ((X \ {Z}) X. (X \ {Z})))))
7		grprndm 9334	. . . . . . . . . 10 |- ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp -> ran ( H |` ((X \ {Z}) X. (X \ {Z}))) = dom dom ( H |` ((X \ {Z}) X. (X \ {Z}))))
8	7	adantl 424	. . . . . . . . 9 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> ran ( H |` ((X \ {Z}) X. (X \ {Z}))) = dom dom ( H |` ((X \ {Z}) X. (X \ {Z}))))
9		difss 2735	. . . . . . . . . . . . . . 15 |- (X \ {Z}) C_ X
10		xpss12 4089	. . . . . . . . . . . . . . 15 |- (((X \ {Z}) C_ X /\ (X \ {Z}) C_ X) -> ((X \ {Z}) X. (X \ {Z})) C_ (X X. X))
11	9, 9, 10	mp2an 761	. . . . . . . . . . . . . 14 |- ((X \ {Z}) X. (X \ {Z})) C_ (X X. X)
12		isdivrng1.1	. . . . . . . . . . . . . . . . 17 |- G = (1st` R)
13		isdivrng1.2	. . . . . . . . . . . . . . . . 17 |- H = (2nd` R)
14		isdivrng1.4	. . . . . . . . . . . . . . . . 17 |- X = ran G
15	12, 13, 14	ringsm 9467	. . . . . . . . . . . . . . . 16 |- (R e. Ring -> H:(X X. X)-->X)
16		fdm 4567	. . . . . . . . . . . . . . . 16 |- (H:(X X. X)-->X -> dom H = (X X. X))
17	15, 16	syl 12	. . . . . . . . . . . . . . 15 |- (R e. Ring -> dom H = (X X. X))
18	17	sseq2d 2645	. . . . . . . . . . . . . 14 |- (R e. Ring -> (((X \ {Z}) X. (X \ {Z})) C_ dom H <-> ((X \ {Z}) X. (X \ {Z})) C_ (X X. X)))
19	11, 18	mpbiri 211	. . . . . . . . . . . . 13 |- (R e. Ring -> ((X \ {Z}) X. (X \ {Z})) C_ dom H)
20		ssdmres 4235	. . . . . . . . . . . . 13 |- (((X \ {Z}) X. (X \ {Z})) C_ dom H <-> dom ( H |` ((X \ {Z}) X. (X \ {Z}))) = ((X \ {Z}) X. (X \ {Z})))
21	19, 20	sylib 215	. . . . . . . . . . . 12 |- (R e. Ring -> dom ( H |` ((X \ {Z}) X. (X \ {Z}))) = ((X \ {Z}) X. (X \ {Z})))
22	21	adantr 425	. . . . . . . . . . 11 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> dom ( H |` ((X \ {Z}) X. (X \ {Z}))) = ((X \ {Z}) X. (X \ {Z})))
23	22	dmeqd 4159	. . . . . . . . . 10 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> dom dom ( H |` ((X \ {Z}) X. (X \ {Z}))) = dom ((X \ {Z}) X. (X \ {Z})))
24		dmxpid 4179	. . . . . . . . . 10 |- dom ((X \ {Z}) X. (X \ {Z})) = (X \ {Z})
25	23, 24	syl6eq 1944	. . . . . . . . 9 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> dom dom ( H |` ((X \ {Z}) X. (X \ {Z}))) = (X \ {Z}))
26	8, 25	eqtrd 1925	. . . . . . . 8 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> ran ( H |` ((X \ {Z}) X. (X \ {Z}))) = (X \ {Z}))
27	26	eleq2d 1964	. . . . . . 7 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> (A e. ran ( H |` ((X \ {Z}) X. (X \ {Z}))) <-> A e. (X \ {Z})))
28	26	eleq2d 1964	. . . . . . 7 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> (B e. ran ( H |` ((X \ {Z}) X. (X \ {Z}))) <-> B e. (X \ {Z})))
29	27, 28	anbi12d 690	. . . . . 6 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> ((A e. ran ( H |` ((X \ {Z}) X. (X \ {Z}))) /\ B e. ran ( H |` ((X \ {Z}) X. (X \ {Z})))) <-> (A e. (X \ {Z}) /\ B e. (X \ {Z}))))
30	26	eleq2d 1964	. . . . . 6 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> ((A(H |` ((X \ {Z}) X. (X \ {Z})))B) e. ran ( H |` ((X \ {Z}) X. (X \ {Z}))) <-> (A(H |` ((X \ {Z}) X. (X \ {Z})))B) e. (X \ {Z})))
31	6, 29, 30	3imtr3d 601	. . . . 5 |- ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) -> ((A e. (X \ {Z}) /\ B e. (X \ {Z})) -> (A(H |` ((X \ {Z}) X. (X \ {Z})))B) e. (X \ {Z})))
32	31	imp 377	. . . 4 |- (((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) /\ (A e. (X \ {Z}) /\ B e. (X \ {Z}))) -> (A(H |` ((X \ {Z}) X. (X \ {Z})))B) e. (X \ {Z}))
33	2, 32	eqeltrrd 1972	. . 3 |- (((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) /\ (A e. (X \ {Z}) /\ B e. (X \ {Z}))) -> (AHB) e. (X \ {Z}))
34	33	3impb 1063	. 2 |- (((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) /\ A e. (X \ {Z}) /\ B e. (X \ {Z})) -> (AHB) e. (X \ {Z}))
35		isdivrng1.3	. . 3 |- Z = (Id` G)
36	12, 13, 35, 14	isdivrng1 16109	. 2 |- (R e. DivRing <-> (R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp))
37	34, 36	syl3an1b 1133	1 |- ((R e. DivRing /\ A e. (X \ {Z}) /\ B e. (X \ {Z})) -> (AHB) e. (X \ {Z}))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   \ cdif 2590   C_ wss 2593  {csn 3044   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Grpcgr 9311  Idcgi 9312  Ringcring 9463  DivRingcdrng 9491

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-grp 9316  df-ring 9464  df-drng 9492

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