Theorem rngmulr

Description: The multiplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013) (Revised by Mario Carneiro, 30-Apr-2015)

	Ref	Expression
Hypothesis	rngfn.r	$$⊢R=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}$$
Assertion	rngmulr	$$⊢\underset{\dot{}}{·}\in V\to \underset{\dot{}}{·}={\cdot }_R$$

Proof

Step	Hyp	Ref	Expression
1		rngfn.r	$$⊢R=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}$$
2	1	rngstr	$$⊢R\mathrm{Struct}⟨1,3⟩$$
3		mulridx	$$⊢{\cdot }_{𝑟}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		snsstp3	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}$$
5	4 1	sseqtrri	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\subseteq R$$
6	2 3 5	strfv	$$⊢\underset{\dot{}}{·}\in V\to \underset{\dot{}}{·}={\cdot }_R$$