# Metamath Proof Explorer

## Theorem grpoidinvlem4

Description: Lemma for grpoidinv . (Contributed by NM, 14-Oct-2006) (New usage is discouraged.)

Ref Expression
Hypothesis grpfo.1
`|- X = ran G`
Assertion grpoidinvlem4
`|- ( ( ( G e. GrpOp /\ A e. X ) /\ E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) ) -> ( A G U ) = ( U G A ) )`

### Proof

Step Hyp Ref Expression
1 grpfo.1
` |-  X = ran G`
2 simpll
` |-  ( ( ( G e. GrpOp /\ A e. X ) /\ y e. X ) -> G e. GrpOp )`
3 simplr
` |-  ( ( ( G e. GrpOp /\ A e. X ) /\ y e. X ) -> A e. X )`
4 simpr
` |-  ( ( ( G e. GrpOp /\ A e. X ) /\ y e. X ) -> y e. X )`
5 1 grpoass
` |-  ( ( G e. GrpOp /\ ( A e. X /\ y e. X /\ A e. X ) ) -> ( ( A G y ) G A ) = ( A G ( y G A ) ) )`
6 2 3 4 3 5 syl13anc
` |-  ( ( ( G e. GrpOp /\ A e. X ) /\ y e. X ) -> ( ( A G y ) G A ) = ( A G ( y G A ) ) )`
7 oveq2
` |-  ( ( y G A ) = U -> ( A G ( y G A ) ) = ( A G U ) )`
8 6 7 sylan9eq
` |-  ( ( ( ( G e. GrpOp /\ A e. X ) /\ y e. X ) /\ ( y G A ) = U ) -> ( ( A G y ) G A ) = ( A G U ) )`
9 oveq1
` |-  ( ( A G y ) = U -> ( ( A G y ) G A ) = ( U G A ) )`
10 8 9 sylan9req
` |-  ( ( ( ( ( G e. GrpOp /\ A e. X ) /\ y e. X ) /\ ( y G A ) = U ) /\ ( A G y ) = U ) -> ( A G U ) = ( U G A ) )`
11 10 anasss
` |-  ( ( ( ( G e. GrpOp /\ A e. X ) /\ y e. X ) /\ ( ( y G A ) = U /\ ( A G y ) = U ) ) -> ( A G U ) = ( U G A ) )`
12 11 r19.29an
` |-  ( ( ( G e. GrpOp /\ A e. X ) /\ E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) ) -> ( A G U ) = ( U G A ) )`