Theorem grpplusg

Description: The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013) (Revised by Mario Carneiro, 30-Apr-2015) (Revised by AV, 27-Oct-2024)

	Ref	Expression
Hypothesis	grpfn.g	$$⊢G=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩\right\}$$
Assertion	grpplusg	$$⊢\underset{\dot{}}{+}\in V\to \underset{\dot{}}{+}={+}_G$$

Proof

Step	Hyp	Ref	Expression
1		grpfn.g	$$⊢G=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩\right\}$$
2		basendxltplusgndx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}<{+}_{\mathrm{ndx}}$$
3		plusgndxnn	$$⊢{+}_{\mathrm{ndx}}\in \mathbb{N}$$
4		plusgid	$$⊢{+}_{𝑔}=\mathrm{Slot}{+}_{\mathrm{ndx}}$$
5	1 2 3 4	2strop1	$$⊢\underset{\dot{}}{+}\in V\to \underset{\dot{}}{+}={+}_G$$