Theorem rngoidmlem 33735

Description: The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)

Hypotheses

Ref	Expression
uridm.1	|- H = ( 2nd ` R )
uridm.2	|- X = ran ( 1st ` R )
uridm.3	|- U = (GId ` H )

Assertion

Ref	Expression
rngoidmlem	|- ( ( R e. RingOps /\ A e. X ) -> ( ( U H A ) = A /\ ( A H U ) = A ) )

Proof of Theorem rngoidmlem

Step	Hyp	Ref	Expression
1		uridm.1	. . . . 5 |- H = ( 2nd ` R )
2	1	rngomndo 33734	. . . 4 |- ( R e. RingOps -> H e. MndOp )
3		mndomgmid 33670	. . . 4 |- ( H e. MndOp -> H e. ( Magma i^i ExId ) )
4		eqid 2622	. . . . . 6 |- ran H = ran H
5		uridm.3	. . . . . 6 |- U = (GId ` H )
6	4, 5	cmpidelt 33658	. . . . 5 |- ( ( H e. ( Magma i^i ExId ) /\ A e. ran H ) -> ( ( U H A ) = A /\ ( A H U ) = A ) )
7	6	ex 450	. . . 4 |- ( H e. ( Magma i^i ExId ) -> ( A e. ran H -> ( ( U H A ) = A /\ ( A H U ) = A ) ) )
8	2, 3, 7	3syl 18	. . 3 |- ( R e. RingOps -> ( A e. ran H -> ( ( U H A ) = A /\ ( A H U ) = A ) ) )
9		eqid 2622	. . . . 5 |- ( 1st ` R ) = ( 1st ` R )
10	1, 9	rngorn1eq 33733	. . . 4 |- ( R e. RingOps -> ran ( 1st ` R ) = ran H )
11		uridm.2	. . . . 5 |- X = ran ( 1st ` R )
12		eqtr 2641	. . . . . 6 |- ( ( X = ran ( 1st ` R ) /\ ran ( 1st ` R ) = ran H ) -> X = ran H )
13		simpl 473	. . . . . . . . 9 |- ( ( X = ran H /\ R e. RingOps ) -> X = ran H )
14	13	eleq2d 2687	. . . . . . . 8 |- ( ( X = ran H /\ R e. RingOps ) -> ( A e. X <-> A e. ran H ) )
15	14	imbi1d 331	. . . . . . 7 |- ( ( X = ran H /\ R e. RingOps ) -> ( ( A e. X -> ( ( U H A ) = A /\ ( A H U ) = A ) ) <-> ( A e. ran H -> ( ( U H A ) = A /\ ( A H U ) = A ) ) ) )
16	15	ex 450	. . . . . 6 |- ( X = ran H -> ( R e. RingOps -> ( ( A e. X -> ( ( U H A ) = A /\ ( A H U ) = A ) ) <-> ( A e. ran H -> ( ( U H A ) = A /\ ( A H U ) = A ) ) ) ) )
17	12, 16	syl 17	. . . . 5 |- ( ( X = ran ( 1st ` R ) /\ ran ( 1st ` R ) = ran H ) -> ( R e. RingOps -> ( ( A e. X -> ( ( U H A ) = A /\ ( A H U ) = A ) ) <-> ( A e. ran H -> ( ( U H A ) = A /\ ( A H U ) = A ) ) ) ) )
18	11, 17	mpan 706	. . . 4 |- ( ran ( 1st ` R ) = ran H -> ( R e. RingOps -> ( ( A e. X -> ( ( U H A ) = A /\ ( A H U ) = A ) ) <-> ( A e. ran H -> ( ( U H A ) = A /\ ( A H U ) = A ) ) ) ) )
19	10, 18	mpcom 38	. . 3 |- ( R e. RingOps -> ( ( A e. X -> ( ( U H A ) = A /\ ( A H U ) = A ) ) <-> ( A e. ran H -> ( ( U H A ) = A /\ ( A H U ) = A ) ) ) )
20	8, 19	mpbird 247	. 2 |- ( R e. RingOps -> ( A e. X -> ( ( U H A ) = A /\ ( A H U ) = A ) ) )
21	20	imp 445	1 |- ( ( R e. RingOps /\ A e. X ) -> ( ( U H A ) = A /\ ( A H U ) = A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  GIdcgi 27344   ExId cexid 33643   Magmacmagm 33647  MndOpcmndo 33665   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-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-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-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169  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:  rngolidm  33736  rngoridm  33737