Theorem rngoideu 33702

Description: The unit element of a ring is unique. (Contributed by NM, 4-Apr-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ringi.1	|- G = ( 1st ` R )
ringi.2	|- H = ( 2nd ` R )
ringi.3	|- X = ran G

Assertion

Ref	Expression
rngoideu	|- ( R e. RingOps -> E! u e. X A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) )

Distinct variable groups:   x, u, G   u, H, x   u, X, x   u, R, x

Proof of Theorem rngoideu

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringi.1	. . . . 5 |- G = ( 1st ` R )
2		ringi.2	. . . . 5 |- H = ( 2nd ` R )
3		ringi.3	. . . . 5 |- X = ran G
4	1, 2, 3	rngoi 33698	. . . 4 |- ( R e. RingOps -> ( ( G e. AbelOp /\ H : ( X X. X ) --> X ) /\ ( A. u e. X A. x e. X A. y e. X ( ( ( u H x ) H y ) = ( u H ( x H y ) ) /\ ( u H ( x G y ) ) = ( ( u H x ) G ( u H y ) ) /\ ( ( u G x ) H y ) = ( ( u H y ) G ( x H y ) ) ) /\ E. u e. X A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) ) ) )
5	4	simprrd 797	. . 3 |- ( R e. RingOps -> E. u e. X A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) )
6		simpl 473	. . . . . . . 8 |- ( ( ( u H x ) = x /\ ( x H u ) = x ) -> ( u H x ) = x )
7	6	ralimi 2952	. . . . . . 7 |- ( A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) -> A. x e. X ( u H x ) = x )
8		oveq2 6658	. . . . . . . . 9 |- ( x = y -> ( u H x ) = ( u H y ) )
9		id 22	. . . . . . . . 9 |- ( x = y -> x = y )
10	8, 9	eqeq12d 2637	. . . . . . . 8 |- ( x = y -> ( ( u H x ) = x <-> ( u H y ) = y ) )
11	10	rspcv 3305	. . . . . . 7 |- ( y e. X -> ( A. x e. X ( u H x ) = x -> ( u H y ) = y ) )
12	7, 11	syl5 34	. . . . . 6 |- ( y e. X -> ( A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) -> ( u H y ) = y ) )
13		simpr 477	. . . . . . . 8 |- ( ( ( y H x ) = x /\ ( x H y ) = x ) -> ( x H y ) = x )
14	13	ralimi 2952	. . . . . . 7 |- ( A. x e. X ( ( y H x ) = x /\ ( x H y ) = x ) -> A. x e. X ( x H y ) = x )
15		oveq1 6657	. . . . . . . . 9 |- ( x = u -> ( x H y ) = ( u H y ) )
16		id 22	. . . . . . . . 9 |- ( x = u -> x = u )
17	15, 16	eqeq12d 2637	. . . . . . . 8 |- ( x = u -> ( ( x H y ) = x <-> ( u H y ) = u ) )
18	17	rspcv 3305	. . . . . . 7 |- ( u e. X -> ( A. x e. X ( x H y ) = x -> ( u H y ) = u ) )
19	14, 18	syl5 34	. . . . . 6 |- ( u e. X -> ( A. x e. X ( ( y H x ) = x /\ ( x H y ) = x ) -> ( u H y ) = u ) )
20	12, 19	im2anan9r 881	. . . . 5 |- ( ( u e. X /\ y e. X ) -> ( ( A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) /\ A. x e. X ( ( y H x ) = x /\ ( x H y ) = x ) ) -> ( ( u H y ) = y /\ ( u H y ) = u ) ) )
21		eqtr2 2642	. . . . . 6 |- ( ( ( u H y ) = y /\ ( u H y ) = u ) -> y = u )
22	21	eqcomd 2628	. . . . 5 |- ( ( ( u H y ) = y /\ ( u H y ) = u ) -> u = y )
23	20, 22	syl6 35	. . . 4 |- ( ( u e. X /\ y e. X ) -> ( ( A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) /\ A. x e. X ( ( y H x ) = x /\ ( x H y ) = x ) ) -> u = y ) )
24	23	rgen2a 2977	. . 3 |- A. u e. X A. y e. X ( ( A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) /\ A. x e. X ( ( y H x ) = x /\ ( x H y ) = x ) ) -> u = y )
25	5, 24	jctir 561	. 2 |- ( R e. RingOps -> ( E. u e. X A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) /\ A. u e. X A. y e. X ( ( A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) /\ A. x e. X ( ( y H x ) = x /\ ( x H y ) = x ) ) -> u = y ) ) )
26		oveq1 6657	. . . . . 6 |- ( u = y -> ( u H x ) = ( y H x ) )
27	26	eqeq1d 2624	. . . . 5 |- ( u = y -> ( ( u H x ) = x <-> ( y H x ) = x ) )
28		oveq2 6658	. . . . . 6 |- ( u = y -> ( x H u ) = ( x H y ) )
29	28	eqeq1d 2624	. . . . 5 |- ( u = y -> ( ( x H u ) = x <-> ( x H y ) = x ) )
30	27, 29	anbi12d 747	. . . 4 |- ( u = y -> ( ( ( u H x ) = x /\ ( x H u ) = x ) <-> ( ( y H x ) = x /\ ( x H y ) = x ) ) )
31	30	ralbidv 2986	. . 3 |- ( u = y -> ( A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) <-> A. x e. X ( ( y H x ) = x /\ ( x H y ) = x ) ) )
32	31	reu4 3400	. 2 |- ( E! u e. X A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) <-> ( E. u e. X A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) /\ A. u e. X A. y e. X ( ( A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) /\ A. x e. X ( ( y H x ) = x /\ ( x H y ) = x ) ) -> u = y ) ) )
33	25, 32	sylibr 224	1 |- ( R e. RingOps -> E! u e. X A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   AbelOpcablo 27398   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-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  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-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-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-rngo 33694

This theorem is referenced by: (None)