Theorem rngoueqz 33739

Description: Obsolete as of 23-Jan-2020. Use 0ring01eqbi 19273 instead. In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem rngoueqz

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uznzr.1	. . . 4 |- G = ( 1st ` R )
2		uznzr.5	. . . 4 |- X = ran G
3		uznzr.3	. . . 4 |- Z = (GId ` G )
4	1, 2, 3	rngo0cl 33718	. . 3 |- ( R e. RingOps -> Z e. X )
5		en1eqsn 8190	. . . . . 6 |- ( ( Z e. X /\ X ~~ 1o ) -> X = { Z } )
6	1	rneqi 5352	. . . . . . . 8 |- ran G = ran ( 1st ` R )
7		uznzr.2	. . . . . . . 8 |- H = ( 2nd ` R )
8		uznzr.4	. . . . . . . 8 |- U = (GId ` H )
9	6, 7, 8	rngo1cl 33738	. . . . . . 7 |- ( R e. RingOps -> U e. ran G )
10		eleq2 2690	. . . . . . . . . 10 |- ( X = { Z } -> ( U e. X <-> U e. { Z } ) )
11	10	biimpd 219	. . . . . . . . 9 |- ( X = { Z } -> ( U e. X -> U e. { Z } ) )
12		elsni 4194	. . . . . . . . 9 |- ( U e. { Z } -> U = Z )
13	11, 12	syl6com 37	. . . . . . . 8 |- ( U e. X -> ( X = { Z } -> U = Z ) )
14	2	eqcomi 2631	. . . . . . . 8 |- ran G = X
15	13, 14	eleq2s 2719	. . . . . . 7 |- ( U e. ran G -> ( X = { Z } -> U = Z ) )
16	9, 15	syl 17	. . . . . 6 |- ( R e. RingOps -> ( X = { Z } -> U = Z ) )
17	5, 16	syl5com 31	. . . . 5 |- ( ( Z e. X /\ X ~~ 1o ) -> ( R e. RingOps -> U = Z ) )
18	17	ex 450	. . . 4 |- ( Z e. X -> ( X ~~ 1o -> ( R e. RingOps -> U = Z ) ) )
19	18	com23 86	. . 3 |- ( Z e. X -> ( R e. RingOps -> ( X ~~ 1o -> U = Z ) ) )
20	4, 19	mpcom 38	. 2 |- ( R e. RingOps -> ( X ~~ 1o -> U = Z ) )
21	1, 2	rngone0 33710	. . 3 |- ( R e. RingOps -> X =/= (/) )
22		oveq2 6658	. . . . . 6 |- ( U = Z -> ( x H U ) = ( x H Z ) )
23	22	ralrimivw 2967	. . . . 5 |- ( U = Z -> A. x e. X ( x H U ) = ( x H Z ) )
24	3, 2, 1, 7	rngorz 33722	. . . . . . 7 |- ( ( R e. RingOps /\ x e. X ) -> ( x H Z ) = Z )
25	24	ralrimiva 2966	. . . . . 6 |- ( R e. RingOps -> A. x e. X ( x H Z ) = Z )
26	2, 6	eqtri 2644	. . . . . . . . 9 |- X = ran ( 1st ` R )
27	7, 26, 8	rngoridm 33737	. . . . . . . 8 |- ( ( R e. RingOps /\ x e. X ) -> ( x H U ) = x )
28	27	ralrimiva 2966	. . . . . . 7 |- ( R e. RingOps -> A. x e. X ( x H U ) = x )
29		r19.26 3064	. . . . . . . . . 10 |- ( A. x e. X ( ( x H U ) = x /\ ( x H U ) = ( x H Z ) ) <-> ( A. x e. X ( x H U ) = x /\ A. x e. X ( x H U ) = ( x H Z ) ) )
30		r19.26 3064	. . . . . . . . . . . 12 |- ( A. x e. X ( ( ( x H U ) = x /\ ( x H U ) = ( x H Z ) ) /\ ( x H Z ) = Z ) <-> ( A. x e. X ( ( x H U ) = x /\ ( x H U ) = ( x H Z ) ) /\ A. x e. X ( x H Z ) = Z ) )
31		eqtr 2641	. . . . . . . . . . . . . . . . . 18 |- ( ( x = ( x H U ) /\ ( x H U ) = ( x H Z ) ) -> x = ( x H Z ) )
32		eqtr 2641	. . . . . . . . . . . . . . . . . . 19 |- ( ( x = ( x H Z ) /\ ( x H Z ) = Z ) -> x = Z )
33	32	ex 450	. . . . . . . . . . . . . . . . . 18 |- ( x = ( x H Z ) -> ( ( x H Z ) = Z -> x = Z ) )
34	31, 33	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( x = ( x H U ) /\ ( x H U ) = ( x H Z ) ) -> ( ( x H Z ) = Z -> x = Z ) )
35	34	ex 450	. . . . . . . . . . . . . . . 16 |- ( x = ( x H U ) -> ( ( x H U ) = ( x H Z ) -> ( ( x H Z ) = Z -> x = Z ) ) )
36	35	eqcoms 2630	. . . . . . . . . . . . . . 15 |- ( ( x H U ) = x -> ( ( x H U ) = ( x H Z ) -> ( ( x H Z ) = Z -> x = Z ) ) )
37	36	imp31 448	. . . . . . . . . . . . . 14 |- ( ( ( ( x H U ) = x /\ ( x H U ) = ( x H Z ) ) /\ ( x H Z ) = Z ) -> x = Z )
38	37	ralimi 2952	. . . . . . . . . . . . 13 |- ( A. x e. X ( ( ( x H U ) = x /\ ( x H U ) = ( x H Z ) ) /\ ( x H Z ) = Z ) -> A. x e. X x = Z )
39		eqsn 4361	. . . . . . . . . . . . . . 15 |- ( X =/= (/) -> ( X = { Z } <-> A. x e. X x = Z ) )
40		ensn1g 8021	. . . . . . . . . . . . . . . . 17 |- ( Z e. X -> { Z } ~~ 1o )
41	4, 40	syl 17	. . . . . . . . . . . . . . . 16 |- ( R e. RingOps -> { Z } ~~ 1o )
42		breq1 4656	. . . . . . . . . . . . . . . 16 |- ( X = { Z } -> ( X ~~ 1o <-> { Z } ~~ 1o ) )
43	41, 42	syl5ibr 236	. . . . . . . . . . . . . . 15 |- ( X = { Z } -> ( R e. RingOps -> X ~~ 1o ) )
44	39, 43	syl6bir 244	. . . . . . . . . . . . . 14 |- ( X =/= (/) -> ( A. x e. X x = Z -> ( R e. RingOps -> X ~~ 1o ) ) )
45	44	com3l 89	. . . . . . . . . . . . 13 |- ( A. x e. X x = Z -> ( R e. RingOps -> ( X =/= (/) -> X ~~ 1o ) ) )
46	38, 45	syl 17	. . . . . . . . . . . 12 |- ( A. x e. X ( ( ( x H U ) = x /\ ( x H U ) = ( x H Z ) ) /\ ( x H Z ) = Z ) -> ( R e. RingOps -> ( X =/= (/) -> X ~~ 1o ) ) )
47	30, 46	sylbir 225	. . . . . . . . . . 11 |- ( ( A. x e. X ( ( x H U ) = x /\ ( x H U ) = ( x H Z ) ) /\ A. x e. X ( x H Z ) = Z ) -> ( R e. RingOps -> ( X =/= (/) -> X ~~ 1o ) ) )
48	47	ex 450	. . . . . . . . . 10 |- ( A. x e. X ( ( x H U ) = x /\ ( x H U ) = ( x H Z ) ) -> ( A. x e. X ( x H Z ) = Z -> ( R e. RingOps -> ( X =/= (/) -> X ~~ 1o ) ) ) )
49	29, 48	sylbir 225	. . . . . . . . 9 |- ( ( A. x e. X ( x H U ) = x /\ A. x e. X ( x H U ) = ( x H Z ) ) -> ( A. x e. X ( x H Z ) = Z -> ( R e. RingOps -> ( X =/= (/) -> X ~~ 1o ) ) ) )
50	49	ex 450	. . . . . . . 8 |- ( A. x e. X ( x H U ) = x -> ( A. x e. X ( x H U ) = ( x H Z ) -> ( A. x e. X ( x H Z ) = Z -> ( R e. RingOps -> ( X =/= (/) -> X ~~ 1o ) ) ) ) )
51	50	com24 95	. . . . . . 7 |- ( A. x e. X ( x H U ) = x -> ( R e. RingOps -> ( A. x e. X ( x H Z ) = Z -> ( A. x e. X ( x H U ) = ( x H Z ) -> ( X =/= (/) -> X ~~ 1o ) ) ) ) )
52	28, 51	mpcom 38	. . . . . 6 |- ( R e. RingOps -> ( A. x e. X ( x H Z ) = Z -> ( A. x e. X ( x H U ) = ( x H Z ) -> ( X =/= (/) -> X ~~ 1o ) ) ) )
53	25, 52	mpd 15	. . . . 5 |- ( R e. RingOps -> ( A. x e. X ( x H U ) = ( x H Z ) -> ( X =/= (/) -> X ~~ 1o ) ) )
54	23, 53	syl5com 31	. . . 4 |- ( U = Z -> ( R e. RingOps -> ( X =/= (/) -> X ~~ 1o ) ) )
55	54	com13 88	. . 3 |- ( X =/= (/) -> ( R e. RingOps -> ( U = Z -> X ~~ 1o ) ) )
56	21, 55	mpcom 38	. 2 |- ( R e. RingOps -> ( U = Z -> X ~~ 1o ) )
57	20, 56	impbid 202	1 |- ( R e. RingOps -> ( X ~~ 1o <-> U = Z ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)c0 3915   {csn 4177   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   1oc1o 7553   ~~ cen 7952  GIdcgi 27344   RingOpscrngo 33693

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-3or 1038  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-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-grpo 27347  df-gid 27348  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694

This theorem is referenced by:  dvrunz  33753  isdmn3  33873