Theorem rngodm1dm2

Description: In a unital ring the domain of the first variable of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010) (New usage is discouraged.)

	Ref	Expression
Hypotheses	rnplrnml0.1	⊢ 𝐻 = ( 2nd ‘ 𝑅 )
	rnplrnml0.2	⊢ 𝐺 = ( 1st ‘ 𝑅 )
Assertion	rngodm1dm2	⊢ ( 𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻 )

Proof

Step	Hyp	Ref	Expression
1		rnplrnml0.1	⊢ 𝐻 = ( 2nd ‘ 𝑅 )
2		rnplrnml0.2	⊢ 𝐺 = ( 1st ‘ 𝑅 )
3	2	rngogrpo	⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp )
4		eqid	⊢ ran 𝐺 = ran 𝐺
5	4	grpofo	⊢ ( 𝐺 ∈ GrpOp → 𝐺 : ( ran 𝐺 × ran 𝐺 ) –onto→ ran 𝐺 )
6	3 5	syl	⊢ ( 𝑅 ∈ RingOps → 𝐺 : ( ran 𝐺 × ran 𝐺 ) –onto→ ran 𝐺 )
7	2 1 4	rngosm	⊢ ( 𝑅 ∈ RingOps → 𝐻 : ( ran 𝐺 × ran 𝐺 ) ⟶ ran 𝐺 )
8		fof	⊢ ( 𝐺 : ( ran 𝐺 × ran 𝐺 ) –onto→ ran 𝐺 → 𝐺 : ( ran 𝐺 × ran 𝐺 ) ⟶ ran 𝐺 )
9	8	fdmd	⊢ ( 𝐺 : ( ran 𝐺 × ran 𝐺 ) –onto→ ran 𝐺 → dom 𝐺 = ( ran 𝐺 × ran 𝐺 ) )
10		fdm	⊢ ( 𝐻 : ( ran 𝐺 × ran 𝐺 ) ⟶ ran 𝐺 → dom 𝐻 = ( ran 𝐺 × ran 𝐺 ) )
11		eqtr	⊢ ( ( dom 𝐺 = ( ran 𝐺 × ran 𝐺 ) ∧ ( ran 𝐺 × ran 𝐺 ) = dom 𝐻 ) → dom 𝐺 = dom 𝐻 )
12	11	dmeqd	⊢ ( ( dom 𝐺 = ( ran 𝐺 × ran 𝐺 ) ∧ ( ran 𝐺 × ran 𝐺 ) = dom 𝐻 ) → dom dom 𝐺 = dom dom 𝐻 )
13	12	expcom	⊢ ( ( ran 𝐺 × ran 𝐺 ) = dom 𝐻 → ( dom 𝐺 = ( ran 𝐺 × ran 𝐺 ) → dom dom 𝐺 = dom dom 𝐻 ) )
14	13	eqcoms	⊢ ( dom 𝐻 = ( ran 𝐺 × ran 𝐺 ) → ( dom 𝐺 = ( ran 𝐺 × ran 𝐺 ) → dom dom 𝐺 = dom dom 𝐻 ) )
15	10 14	syl	⊢ ( 𝐻 : ( ran 𝐺 × ran 𝐺 ) ⟶ ran 𝐺 → ( dom 𝐺 = ( ran 𝐺 × ran 𝐺 ) → dom dom 𝐺 = dom dom 𝐻 ) )
16	9 15	syl5com	⊢ ( 𝐺 : ( ran 𝐺 × ran 𝐺 ) –onto→ ran 𝐺 → ( 𝐻 : ( ran 𝐺 × ran 𝐺 ) ⟶ ran 𝐺 → dom dom 𝐺 = dom dom 𝐻 ) )
17	6 7 16	sylc	⊢ ( 𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻 )