Theorem rngosn3 33723

Description: Obsolete as of 25-Jan-2020. Use ring1zr 19275 or srg1zr 18529 instead. The only unital ring with a base set consisting in one element is the zero ring. (Contributed by FL, 13-Feb-2010.) (Proof shortened by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
on1el3.1	|- G = ( 1st ` R )
on1el3.2	|- X = ran G

Assertion

Ref	Expression
rngosn3	|- ( ( R e. RingOps /\ A e. B ) -> ( X = { A } <-> R = <. { <. <. A , A >. , A >. } , { <. <. A , A >. , A >. } >. ) )

Proof of Theorem rngosn3

Step	Hyp	Ref	Expression
1		on1el3.1	. . . . . . . . . 10 |- G = ( 1st ` R )
2	1	rngogrpo 33709	. . . . . . . . 9 |- ( R e. RingOps -> G e. GrpOp )
3		on1el3.2	. . . . . . . . . 10 |- X = ran G
4	3	grpofo 27353	. . . . . . . . 9 |- ( G e. GrpOp -> G : ( X X. X ) -onto-> X )
5		fof 6115	. . . . . . . . 9 |- ( G : ( X X. X ) -onto-> X -> G : ( X X. X ) --> X )
6	2, 4, 5	3syl 18	. . . . . . . 8 |- ( R e. RingOps -> G : ( X X. X ) --> X )
7	6	adantr 481	. . . . . . 7 |- ( ( R e. RingOps /\ A e. B ) -> G : ( X X. X ) --> X )
8		id 22	. . . . . . . . 9 |- ( X = { A } -> X = { A } )
9	8	sqxpeqd 5141	. . . . . . . 8 |- ( X = { A } -> ( X X. X ) = ( { A } X. { A } ) )
10	9, 8	feq23d 6040	. . . . . . 7 |- ( X = { A } -> ( G : ( X X. X ) --> X <-> G : ( { A } X. { A } ) --> { A } ) )
11	7, 10	syl5ibcom 235	. . . . . 6 |- ( ( R e. RingOps /\ A e. B ) -> ( X = { A } -> G : ( { A } X. { A } ) --> { A } ) )
12		fdm 6051	. . . . . . . . . 10 |- ( G : ( X X. X ) --> X -> dom G = ( X X. X ) )
13	7, 12	syl 17	. . . . . . . . 9 |- ( ( R e. RingOps /\ A e. B ) -> dom G = ( X X. X ) )
14	13	eqcomd 2628	. . . . . . . 8 |- ( ( R e. RingOps /\ A e. B ) -> ( X X. X ) = dom G )
15		fdm 6051	. . . . . . . . 9 |- ( G : ( { A } X. { A } ) --> { A } -> dom G = ( { A } X. { A } ) )
16	15	eqeq2d 2632	. . . . . . . 8 |- ( G : ( { A } X. { A } ) --> { A } -> ( ( X X. X ) = dom G <-> ( X X. X ) = ( { A } X. { A } ) ) )
17	14, 16	syl5ibcom 235	. . . . . . 7 |- ( ( R e. RingOps /\ A e. B ) -> ( G : ( { A } X. { A } ) --> { A } -> ( X X. X ) = ( { A } X. { A } ) ) )
18		xpid11 5347	. . . . . . 7 |- ( ( X X. X ) = ( { A } X. { A } ) <-> X = { A } )
19	17, 18	syl6ib 241	. . . . . 6 |- ( ( R e. RingOps /\ A e. B ) -> ( G : ( { A } X. { A } ) --> { A } -> X = { A } ) )
20	11, 19	impbid 202	. . . . 5 |- ( ( R e. RingOps /\ A e. B ) -> ( X = { A } <-> G : ( { A } X. { A } ) --> { A } ) )
21		simpr 477	. . . . . . 7 |- ( ( R e. RingOps /\ A e. B ) -> A e. B )
22		xpsng 6406	. . . . . . 7 |- ( ( A e. B /\ A e. B ) -> ( { A } X. { A } ) = { <. A , A >. } )
23	21, 22	sylancom 701	. . . . . 6 |- ( ( R e. RingOps /\ A e. B ) -> ( { A } X. { A } ) = { <. A , A >. } )
24	23	feq2d 6031	. . . . 5 |- ( ( R e. RingOps /\ A e. B ) -> ( G : ( { A } X. { A } ) --> { A } <-> G : { <. A , A >. } --> { A } ) )
25		opex 4932	. . . . . 6 |- <. A , A >. e. _V
26		fsng 6404	. . . . . 6 |- ( ( <. A , A >. e. _V /\ A e. B ) -> ( G : { <. A , A >. } --> { A } <-> G = { <. <. A , A >. , A >. } ) )
27	25, 21, 26	sylancr 695	. . . . 5 |- ( ( R e. RingOps /\ A e. B ) -> ( G : { <. A , A >. } --> { A } <-> G = { <. <. A , A >. , A >. } ) )
28	20, 24, 27	3bitrd 294	. . . 4 |- ( ( R e. RingOps /\ A e. B ) -> ( X = { A } <-> G = { <. <. A , A >. , A >. } ) )
29	1	eqeq1i 2627	. . . 4 |- ( G = { <. <. A , A >. , A >. } <-> ( 1st ` R ) = { <. <. A , A >. , A >. } )
30	28, 29	syl6bb 276	. . 3 |- ( ( R e. RingOps /\ A e. B ) -> ( X = { A } <-> ( 1st ` R ) = { <. <. A , A >. , A >. } ) )
31	30	anbi1d 741	. 2 |- ( ( R e. RingOps /\ A e. B ) -> ( ( X = { A } /\ ( 2nd ` R ) = { <. <. A , A >. , A >. } ) <-> ( ( 1st ` R ) = { <. <. A , A >. , A >. } /\ ( 2nd ` R ) = { <. <. A , A >. , A >. } ) ) )
32		eqid 2622	. . . . . . 7 |- ( 2nd ` R ) = ( 2nd ` R )
33	1, 32, 3	rngosm 33699	. . . . . 6 |- ( R e. RingOps -> ( 2nd ` R ) : ( X X. X ) --> X )
34	33	adantr 481	. . . . 5 |- ( ( R e. RingOps /\ A e. B ) -> ( 2nd ` R ) : ( X X. X ) --> X )
35	9, 8	feq23d 6040	. . . . 5 |- ( X = { A } -> ( ( 2nd ` R ) : ( X X. X ) --> X <-> ( 2nd ` R ) : ( { A } X. { A } ) --> { A } ) )
36	34, 35	syl5ibcom 235	. . . 4 |- ( ( R e. RingOps /\ A e. B ) -> ( X = { A } -> ( 2nd ` R ) : ( { A } X. { A } ) --> { A } ) )
37	23	feq2d 6031	. . . . 5 |- ( ( R e. RingOps /\ A e. B ) -> ( ( 2nd ` R ) : ( { A } X. { A } ) --> { A } <-> ( 2nd ` R ) : { <. A , A >. } --> { A } ) )
38		fsng 6404	. . . . . 6 |- ( ( <. A , A >. e. _V /\ A e. B ) -> ( ( 2nd ` R ) : { <. A , A >. } --> { A } <-> ( 2nd ` R ) = { <. <. A , A >. , A >. } ) )
39	25, 21, 38	sylancr 695	. . . . 5 |- ( ( R e. RingOps /\ A e. B ) -> ( ( 2nd ` R ) : { <. A , A >. } --> { A } <-> ( 2nd ` R ) = { <. <. A , A >. , A >. } ) )
40	37, 39	bitrd 268	. . . 4 |- ( ( R e. RingOps /\ A e. B ) -> ( ( 2nd ` R ) : ( { A } X. { A } ) --> { A } <-> ( 2nd ` R ) = { <. <. A , A >. , A >. } ) )
41	36, 40	sylibd 229	. . 3 |- ( ( R e. RingOps /\ A e. B ) -> ( X = { A } -> ( 2nd ` R ) = { <. <. A , A >. , A >. } ) )
42	41	pm4.71d 666	. 2 |- ( ( R e. RingOps /\ A e. B ) -> ( X = { A } <-> ( X = { A } /\ ( 2nd ` R ) = { <. <. A , A >. , A >. } ) ) )
43		relrngo 33695	. . . . . 6 |- Rel RingOps
44		df-rel 5121	. . . . . 6 |- ( Rel RingOps <-> RingOps C_ ( _V X. _V ) )
45	43, 44	mpbi 220	. . . . 5 |- RingOps C_ ( _V X. _V )
46	45	sseli 3599	. . . 4 |- ( R e. RingOps -> R e. ( _V X. _V ) )
47	46	adantr 481	. . 3 |- ( ( R e. RingOps /\ A e. B ) -> R e. ( _V X. _V ) )
48		eqop 7208	. . 3 |- ( R e. ( _V X. _V ) -> ( R = <. { <. <. A , A >. , A >. } , { <. <. A , A >. , A >. } >. <-> ( ( 1st ` R ) = { <. <. A , A >. , A >. } /\ ( 2nd ` R ) = { <. <. A , A >. , A >. } ) ) )
49	47, 48	syl 17	. 2 |- ( ( R e. RingOps /\ A e. B ) -> ( R = <. { <. <. A , A >. , A >. } , { <. <. A , A >. , A >. } >. <-> ( ( 1st ` R ) = { <. <. A , A >. , A >. } /\ ( 2nd ` R ) = { <. <. A , A >. , A >. } ) ) )
50	31, 42, 49	3bitr4d 300	1 |- ( ( R e. RingOps /\ A e. B ) -> ( X = { A } <-> R = <. { <. <. A , A >. , A >. } , { <. <. A , A >. , A >. } >. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   <.cop 4183   X. cxp 5112   dom cdm 5114   ran crn 5115   Rel wrel 5119   -->wf 5884   -onto->wfo 5886   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   GrpOpcgr 27343   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-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-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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-grpo 27347  df-ablo 27399  df-rngo 33694

This theorem is referenced by:  rngosn4  33724