Theorem cmpidelt 33658

Description: A magma right and left identity element keeps the other elements unchanged. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cmpidelt.1	|- X = ran G
cmpidelt.2	|- U = (GId ` G )

Assertion

Ref	Expression
cmpidelt	|- ( ( G e. ( Magma i^i ExId ) /\ A e. X ) -> ( ( U G A ) = A /\ ( A G U ) = A ) )

Proof of Theorem cmpidelt

Dummy variables x u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmpidelt.1	. . . . 5 |- X = ran G
2		cmpidelt.2	. . . . 5 |- U = (GId ` G )
3	1, 2	idrval 33656	. . . 4 |- ( G e. ( Magma i^i ExId ) -> U = ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) )
4	3	eqcomd 2628	. . 3 |- ( G e. ( Magma i^i ExId ) -> ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) = U )
5	1, 2	iorlid 33657	. . . 4 |- ( G e. ( Magma i^i ExId ) -> U e. X )
6	1	exidu1 33655	. . . 4 |- ( G e. ( Magma i^i ExId ) -> E! u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) )
7		oveq1 6657	. . . . . . . 8 |- ( u = U -> ( u G x ) = ( U G x ) )
8	7	eqeq1d 2624	. . . . . . 7 |- ( u = U -> ( ( u G x ) = x <-> ( U G x ) = x ) )
9		oveq2 6658	. . . . . . . 8 |- ( u = U -> ( x G u ) = ( x G U ) )
10	9	eqeq1d 2624	. . . . . . 7 |- ( u = U -> ( ( x G u ) = x <-> ( x G U ) = x ) )
11	8, 10	anbi12d 747	. . . . . 6 |- ( u = U -> ( ( ( u G x ) = x /\ ( x G u ) = x ) <-> ( ( U G x ) = x /\ ( x G U ) = x ) ) )
12	11	ralbidv 2986	. . . . 5 |- ( u = U -> ( A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) <-> A. x e. X ( ( U G x ) = x /\ ( x G U ) = x ) ) )
13	12	riota2 6633	. . . 4 |- ( ( U e. X /\ E! u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) -> ( A. x e. X ( ( U G x ) = x /\ ( x G U ) = x ) <-> ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) = U ) )
14	5, 6, 13	syl2anc 693	. . 3 |- ( G e. ( Magma i^i ExId ) -> ( A. x e. X ( ( U G x ) = x /\ ( x G U ) = x ) <-> ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) = U ) )
15	4, 14	mpbird 247	. 2 |- ( G e. ( Magma i^i ExId ) -> A. x e. X ( ( U G x ) = x /\ ( x G U ) = x ) )
16		oveq2 6658	. . . . 5 |- ( x = A -> ( U G x ) = ( U G A ) )
17		id 22	. . . . 5 |- ( x = A -> x = A )
18	16, 17	eqeq12d 2637	. . . 4 |- ( x = A -> ( ( U G x ) = x <-> ( U G A ) = A ) )
19		oveq1 6657	. . . . 5 |- ( x = A -> ( x G U ) = ( A G U ) )
20	19, 17	eqeq12d 2637	. . . 4 |- ( x = A -> ( ( x G U ) = x <-> ( A G U ) = A ) )
21	18, 20	anbi12d 747	. . 3 |- ( x = A -> ( ( ( U G x ) = x /\ ( x G U ) = x ) <-> ( ( U G A ) = A /\ ( A G U ) = A ) ) )
22	21	rspccva 3308	. 2 |- ( ( A. x e. X ( ( U G x ) = x /\ ( x G U ) = x ) /\ A e. X ) -> ( ( U G A ) = A /\ ( A G U ) = A ) )
23	15, 22	sylan 488	1 |- ( ( G e. ( Magma i^i ExId ) /\ A e. X ) -> ( ( U G A ) = A /\ ( A G U ) = A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   i^i cin 3573   ran crn 5115   ` cfv 5888   iota_crio 6610  (class class class)co 6650  GIdcgi 27344   ExId cexid 33643   Magmacmagm 33647

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-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-gid 27348  df-exid 33644  df-mgmOLD 33648

This theorem is referenced by:  exidreslem  33676  rngoidmlem  33735