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Metamath Proof Explorer

Theorem rngolz

Description: The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009) (New usage is discouraged.)

Ref Expression
Hypotheses ringlz.1 ⊢ 𝑍 = ( GId ‘ 𝐺 )
ringlz.2 ⊢ 𝑋 = ran 𝐺
ringlz.3 ⊢ 𝐺 = ( 1st ‘ 𝑅 )
ringlz.4 ⊢ 𝐻 = ( 2nd ‘ 𝑅 )
Assertion rngolz ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝐻 𝐴 ) = 𝑍 )

Proof

Step Hyp Ref Expression
1 ringlz.1 ⊢ 𝑍 = ( GId ‘ 𝐺 )
2 ringlz.2 ⊢ 𝑋 = ran 𝐺
3 ringlz.3 ⊢ 𝐺 = ( 1st ‘ 𝑅 )
4 ringlz.4 ⊢ 𝐻 = ( 2nd ‘ 𝑅 )
5 3 rngogrpo ⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp )
6 2 1 grpoidcl ⊢ ( 𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋 )
7 2 1 grpolid ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝑍 ∈ 𝑋 ) → ( 𝑍 𝐺 𝑍 ) = 𝑍 )
8 5 6 7 syl2anc2 ⊢ ( 𝑅 ∈ RingOps → ( 𝑍 𝐺 𝑍 ) = 𝑍 )
9 8 adantr ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝐺 𝑍 ) = 𝑍 )
10 9 oveq1d ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑍 𝐺 𝑍 ) 𝐻 𝐴 ) = ( 𝑍 𝐻 𝐴 ) )
11 3 2 1 rngo0cl ⊢ ( 𝑅 ∈ RingOps → 𝑍 ∈ 𝑋 )
12 11 adantr ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → 𝑍 ∈ 𝑋 )
13 simpr ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ∈ 𝑋 )
14 12 12 13 3jca ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) )
15 3 4 2 rngodir ⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ( 𝑍 𝐺 𝑍 ) 𝐻 𝐴 ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 ( 𝑍 𝐻 𝐴 ) ) )
16 14 15 syldan ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑍 𝐺 𝑍 ) 𝐻 𝐴 ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 ( 𝑍 𝐻 𝐴 ) ) )
17 5 adantr ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → 𝐺 ∈ GrpOp )
18 simpl ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → 𝑅 ∈ RingOps )
19 3 4 2 rngocl ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝐻 𝐴 ) ∈ 𝑋 )
20 18 12 13 19 syl3anc ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝐻 𝐴 ) ∈ 𝑋 )
21 2 1 grporid ⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝑍 𝐻 𝐴 ) ∈ 𝑋 ) → ( ( 𝑍 𝐻 𝐴 ) 𝐺 𝑍 ) = ( 𝑍 𝐻 𝐴 ) )
22 21 eqcomd ⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝑍 𝐻 𝐴 ) ∈ 𝑋 ) → ( 𝑍 𝐻 𝐴 ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 𝑍 ) )
23 17 20 22 syl2anc ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝐻 𝐴 ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 𝑍 ) )
24 10 16 23 3eqtr3d ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑍 𝐻 𝐴 ) 𝐺 ( 𝑍 𝐻 𝐴 ) ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 𝑍 ) )
25 2 grpolcan ⊢ ( ( 𝐺 ∈ GrpOp ∧ ( ( 𝑍 𝐻 𝐴 ) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ ( 𝑍 𝐻 𝐴 ) ∈ 𝑋 ) ) → ( ( ( 𝑍 𝐻 𝐴 ) 𝐺 ( 𝑍 𝐻 𝐴 ) ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 𝑍 ) ↔ ( 𝑍 𝐻 𝐴 ) = 𝑍 ) )
26 17 20 12 20 25 syl13anc ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( ( 𝑍 𝐻 𝐴 ) 𝐺 ( 𝑍 𝐻 𝐴 ) ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 𝑍 ) ↔ ( 𝑍 𝐻 𝐴 ) = 𝑍 ) )
27 24 26 mpbid ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝐻 𝐴 ) = 𝑍 )