Theorem uznzr 10416

Description: In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.)

Hypotheses

Ref	Expression
uznzr.1	|- G = (1st` R)
uznzr.2	|- H = (2nd` R)
uznzr.3	|- Z = (Id` G)
uznzr.4	|- U = (Id` H)
uznzr.5	|- X = ran G

Assertion

Ref	Expression
uznzr	|- (R e. Ring -> (X ~~ 1o <-> U = Z))

Proof of Theorem uznzr

Step	Hyp	Ref	Expression
1		uznzr.1	. . . 4 |- G = (1st` R)
2		uznzr.5	. . . 4 |- X = ran G
3		uznzr.3	. . . 4 |- Z = (Id` G)
4	1, 2, 3	ring0cl 9484	. . 3 |- (R e. Ring -> Z e. X)
5	1	rneqi 4187	. . . . . . . 8 |- ran G = ran (1st` R)
6		uznzr.2	. . . . . . . 8 |- H = (2nd` R)
7		uznzr.4	. . . . . . . 8 |- U = (Id` H)
8	5, 6, 7	ring1cl 10415	. . . . . . 7 |- (R e. Ring -> U e. ran G)
9	2	eqcomi 1888	. . . . . . . . 9 |- ran G = X
10	9	eleq2i 1961	. . . . . . . 8 |- (U e. ran G <-> U e. X)
11		eleq2 1958	. . . . . . . . . 10 |- (X = {Z} -> (U e. X <-> U e. {Z}))
12	11	biimpd 170	. . . . . . . . 9 |- (X = {Z} -> (U e. X -> U e. {Z}))
13		elsni 3066	. . . . . . . . 9 |- (U e. {Z} -> U = Z)
14	12, 13	syl6com 64	. . . . . . . 8 |- (U e. X -> (X = {Z} -> U = Z))
15	10, 14	sylbi 216	. . . . . . 7 |- (U e. ran G -> (X = {Z} -> U = Z))
16	8, 15	syl 12	. . . . . 6 |- (R e. Ring -> (X = {Z} -> U = Z))
17		setwoe 10170	. . . . . 6 |- ((Z e. X /\ X ~~ 1o) -> X = {Z})
18	16, 17	syl5com 63	. . . . 5 |- ((Z e. X /\ X ~~ 1o) -> (R e. Ring -> U = Z))
19	18	ex 402	. . . 4 |- (Z e. X -> (X ~~ 1o -> (R e. Ring -> U = Z)))
20	19	com23 36	. . 3 |- (Z e. X -> (R e. Ring -> (X ~~ 1o -> U = Z)))
21	4, 20	mpcom 60	. 2 |- (R e. Ring -> (X ~~ 1o -> U = Z))
22	1, 2	rngn0 10400	. . 3 |- (R e. Ring -> X =/= (/))
23	3, 2, 1, 6	ringrz 9488	. . . . . . 7 |- ((R e. Ring /\ x e. X) -> (xHZ) = Z)
24	23	r19.21aiva 2176	. . . . . 6 |- (R e. Ring -> A.x e. X (xHZ) = Z)
25	2, 5	eqtri 1908	. . . . . . . . . 10 |- X = ran (1st` R)
26	6, 25, 7	ringidmlem 10409	. . . . . . . . 9 |- ((R e. Ring /\ x e. X) -> ((UHx) = x /\ (xHU) = x))
27	26	simprd 352	. . . . . . . 8 |- ((R e. Ring /\ x e. X) -> (xHU) = x)
28	27	r19.21aiva 2176	. . . . . . 7 |- (R e. Ring -> A.x e. X (xHU) = x)
29		r19.26 2219	. . . . . . . . . 10 |- (A.x e. X ((xHU) = x /\ (xHU) = (xHZ)) <-> (A.x e. X (xHU) = x /\ A.x e. X (xHU) = (xHZ)))
30		r19.26 2219	. . . . . . . . . . . 12 |- (A.x e. X (((xHU) = x /\ (xHU) = (xHZ)) /\ (xHZ) = Z) <-> (A.x e. X ((xHU) = x /\ (xHU) = (xHZ)) /\ A.x e. X (xHZ) = Z))
31		eqtr 1904	. . . . . . . . . . . . . . . . . 18 |- ((x = (xHU) /\ (xHU) = (xHZ)) -> x = (xHZ))
32		eqtr 1904	. . . . . . . . . . . . . . . . . . 19 |- ((x = (xHZ) /\ (xHZ) = Z) -> x = Z)
33	32	ex 402	. . . . . . . . . . . . . . . . . 18 |- (x = (xHZ) -> ((xHZ) = Z -> x = Z))
34	31, 33	syl 12	. . . . . . . . . . . . . . . . 17 |- ((x = (xHU) /\ (xHU) = (xHZ)) -> ((xHZ) = Z -> x = Z))
35	34	ex 402	. . . . . . . . . . . . . . . 16 |- (x = (xHU) -> ((xHU) = (xHZ) -> ((xHZ) = Z -> x = Z)))
36	35	eqcoms 1887	. . . . . . . . . . . . . . 15 |- ((xHU) = x -> ((xHU) = (xHZ) -> ((xHZ) = Z -> x = Z)))
37	36	imp31 389	. . . . . . . . . . . . . 14 |- ((((xHU) = x /\ (xHU) = (xHZ)) /\ (xHZ) = Z) -> x = Z)
38	37	ralimi 2168	. . . . . . . . . . . . 13 |- (A.x e. X (((xHU) = x /\ (xHU) = (xHZ)) /\ (xHZ) = Z) -> A.x e. X x = Z)
39		eqsn 3143	. . . . . . . . . . . . . . 15 |- (X =/= (/) -> (X = {Z} <-> A.x e. X x = Z))
40		breq1 3341	. . . . . . . . . . . . . . . 16 |- (X = {Z} -> (X ~~ 1o <-> {Z} ~~ 1o))
41		ensn1g 5484	. . . . . . . . . . . . . . . . 17 |- (Z e. X -> {Z} ~~ 1o)
42	4, 41	syl 12	. . . . . . . . . . . . . . . 16 |- (R e. Ring -> {Z} ~~ 1o)
43	40, 42	syl5bir 227	. . . . . . . . . . . . . . 15 |- (X = {Z} -> (R e. Ring -> X ~~ 1o))
44	39, 43	syl6bir 232	. . . . . . . . . . . . . 14 |- (X =/= (/) -> (A.x e. X x = Z -> (R e. Ring -> X ~~ 1o)))
45	44	com3l 38	. . . . . . . . . . . . 13 |- (A.x e. X x = Z -> (R e. Ring -> (X =/= (/) -> X ~~ 1o)))
46	38, 45	syl 12	. . . . . . . . . . . 12 |- (A.x e. X (((xHU) = x /\ (xHU) = (xHZ)) /\ (xHZ) = Z) -> (R e. Ring -> (X =/= (/) -> X ~~ 1o)))
47	30, 46	sylbir 218	. . . . . . . . . . 11 |- ((A.x e. X ((xHU) = x /\ (xHU) = (xHZ)) /\ A.x e. X (xHZ) = Z) -> (R e. Ring -> (X =/= (/) -> X ~~ 1o)))
48	47	ex 402	. . . . . . . . . 10 |- (A.x e. X ((xHU) = x /\ (xHU) = (xHZ)) -> (A.x e. X (xHZ) = Z -> (R e. Ring -> (X =/= (/) -> X ~~ 1o))))
49	29, 48	sylbir 218	. . . . . . . . 9 |- ((A.x e. X (xHU) = x /\ A.x e. X (xHU) = (xHZ)) -> (A.x e. X (xHZ) = Z -> (R e. Ring -> (X =/= (/) -> X ~~ 1o))))
50	49	ex 402	. . . . . . . 8 |- (A.x e. X (xHU) = x -> (A.x e. X (xHU) = (xHZ) -> (A.x e. X (xHZ) = Z -> (R e. Ring -> (X =/= (/) -> X ~~ 1o)))))
51	50	com24 41	. . . . . . 7 |- (A.x e. X (xHU) = x -> (R e. Ring -> (A.x e. X (xHZ) = Z -> (A.x e. X (xHU) = (xHZ) -> (X =/= (/) -> X ~~ 1o)))))
52	28, 51	mpcom 60	. . . . . 6 |- (R e. Ring -> (A.x e. X (xHZ) = Z -> (A.x e. X (xHU) = (xHZ) -> (X =/= (/) -> X ~~ 1o))))
53	24, 52	mpd 29	. . . . 5 |- (R e. Ring -> (A.x e. X (xHU) = (xHZ) -> (X =/= (/) -> X ~~ 1o)))
54		opreq2 4890	. . . . . . 7 |- (U = Z -> (xHU) = (xHZ))
55	54	adantr 425	. . . . . 6 |- ((U = Z /\ x e. X) -> (xHU) = (xHZ))
56	55	r19.21aiva 2176	. . . . 5 |- (U = Z -> A.x e. X (xHU) = (xHZ))
57	53, 56	syl5com 63	. . . 4 |- (U = Z -> (R e. Ring -> (X =/= (/) -> X ~~ 1o)))
58	57	com13 37	. . 3 |- (X =/= (/) -> (R e. Ring -> (U = Z -> X ~~ 1o)))
59	22, 58	mpcom 60	. 2 |- (R e. Ring -> (U = Z -> X ~~ 1o))
60	21, 59	impbid 574	1 |- (R e. Ring -> (X ~~ 1o <-> U = Z))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  (/)c0 2875  {csn 3044   class class class wbr 3338  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  1oc1o 5172   ~~ cen 5423  Idcgi 9312  Ringcring 9463

This theorem is referenced by:  dvrunz 10419  isdmn3 16222

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-rep 3428  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-3or 859  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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-1o 5177  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-grp 9316  df-gid 9317  df-abl 9408  df-ring 9464  df-ass 10360  df-exid 10362  df-mgm 10366  df-sgr 10378  df-mnd 10385

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