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Metamath Proof Explorer

Theorem rngonegmn1r

Description: Negation in a ring is the same as right multiplication by -u 1 . (Contributed by Jeff Madsen, 19-Jun-2010)

Ref Expression
Hypotheses ringneg.1 ⊢ 𝐺 = ( 1st ‘ 𝑅 )
ringneg.2 ⊢ 𝐻 = ( 2nd ‘ 𝑅 )
ringneg.3 ⊢ 𝑋 = ran 𝐺
ringneg.4 ⊢ 𝑁 = ( inv ‘ 𝐺 )
ringneg.5 ⊢ 𝑈 = ( GId ‘ 𝐻 )
Assertion rngonegmn1r ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) )

Proof

Step Hyp Ref Expression
1 ringneg.1 ⊢ 𝐺 = ( 1st ‘ 𝑅 )
2 ringneg.2 ⊢ 𝐻 = ( 2nd ‘ 𝑅 )
3 ringneg.3 ⊢ 𝑋 = ran 𝐺
4 ringneg.4 ⊢ 𝑁 = ( inv ‘ 𝐺 )
5 ringneg.5 ⊢ 𝑈 = ( GId ‘ 𝐻 )
6 1 rneqi ⊢ ran 𝐺 = ran ( 1st ‘ 𝑅 )
7 3 6 eqtri ⊢ 𝑋 = ran ( 1st ‘ 𝑅 )
8 7 2 5 rngo1cl ⊢ ( 𝑅 ∈ RingOps → 𝑈 ∈ 𝑋 )
9 1 3 4 rngonegcl ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋 ) → ( 𝑁 ‘ 𝑈 ) ∈ 𝑋 )
10 8 9 mpdan ⊢ ( 𝑅 ∈ RingOps → ( 𝑁 ‘ 𝑈 ) ∈ 𝑋 )
11 10 adantr ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝑈 ) ∈ 𝑋 )
12 8 adantr ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → 𝑈 ∈ 𝑋 )
13 11 12 jca ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝑈 ) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋 ) )
14 1 2 3 rngodi ⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝐴 ∈ 𝑋 ∧ ( 𝑁 ‘ 𝑈 ) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋 ) ) → ( 𝐴 𝐻 ( ( 𝑁 ‘ 𝑈 ) 𝐺 𝑈 ) ) = ( ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) 𝐺 ( 𝐴 𝐻 𝑈 ) ) )
15 14 3exp2 ⊢ ( 𝑅 ∈ RingOps → ( 𝐴 ∈ 𝑋 → ( ( 𝑁 ‘ 𝑈 ) ∈ 𝑋 → ( 𝑈 ∈ 𝑋 → ( 𝐴 𝐻 ( ( 𝑁 ‘ 𝑈 ) 𝐺 𝑈 ) ) = ( ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) 𝐺 ( 𝐴 𝐻 𝑈 ) ) ) ) ) )
16 15 imp43 ⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) ∧ ( ( 𝑁 ‘ 𝑈 ) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋 ) ) → ( 𝐴 𝐻 ( ( 𝑁 ‘ 𝑈 ) 𝐺 𝑈 ) ) = ( ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) 𝐺 ( 𝐴 𝐻 𝑈 ) ) )
17 13 16 mpdan ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐻 ( ( 𝑁 ‘ 𝑈 ) 𝐺 𝑈 ) ) = ( ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) 𝐺 ( 𝐴 𝐻 𝑈 ) ) )
18 eqid ⊢ ( GId ‘ 𝐺 ) = ( GId ‘ 𝐺 )
19 1 3 4 18 rngoaddneg2 ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝑈 ) 𝐺 𝑈 ) = ( GId ‘ 𝐺 ) )
20 8 19 mpdan ⊢ ( 𝑅 ∈ RingOps → ( ( 𝑁 ‘ 𝑈 ) 𝐺 𝑈 ) = ( GId ‘ 𝐺 ) )
21 20 adantr ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝑈 ) 𝐺 𝑈 ) = ( GId ‘ 𝐺 ) )
22 21 oveq2d ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐻 ( ( 𝑁 ‘ 𝑈 ) 𝐺 𝑈 ) ) = ( 𝐴 𝐻 ( GId ‘ 𝐺 ) ) )
23 18 3 1 2 rngorz ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐻 ( GId ‘ 𝐺 ) ) = ( GId ‘ 𝐺 ) )
24 22 23 eqtrd ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐻 ( ( 𝑁 ‘ 𝑈 ) 𝐺 𝑈 ) ) = ( GId ‘ 𝐺 ) )
25 2 7 5 rngoridm ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐻 𝑈 ) = 𝐴 )
26 25 oveq2d ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) 𝐺 ( 𝐴 𝐻 𝑈 ) ) = ( ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) 𝐺 𝐴 ) )
27 17 24 26 3eqtr3rd ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) 𝐺 𝐴 ) = ( GId ‘ 𝐺 ) )
28 1 2 3 rngocl ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑁 ‘ 𝑈 ) ∈ 𝑋 ) → ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) ∈ 𝑋 )
29 11 28 mpd3an3 ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) ∈ 𝑋 )
30 1 rngogrpo ⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp )
31 3 18 4 grpoinvid2 ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) ↔ ( ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) 𝐺 𝐴 ) = ( GId ‘ 𝐺 ) ) )
32 30 31 syl3an1 ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) ↔ ( ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) 𝐺 𝐴 ) = ( GId ‘ 𝐺 ) ) )
33 29 32 mpd3an3 ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) ↔ ( ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) 𝐺 𝐴 ) = ( GId ‘ 𝐺 ) ) )
34 27 33 mpbird ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐻 ( 𝑁 ‘ 𝑈 ) ) )