Theorem topgrpplusg

Description: The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015)

	Ref	Expression
Hypothesis	topgrpfn.w	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}$$
Assertion	topgrpplusg	$$⊢\underset{\dot{}}{+}\in X\to \underset{\dot{}}{+}={+}_W$$

Proof

Step	Hyp	Ref	Expression
1		topgrpfn.w	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}$$
2	1	topgrpstr	$$⊢W\mathrm{Struct}⟨1,9⟩$$
3		plusgid	$$⊢{+}_{𝑔}=\mathrm{Slot}{+}_{\mathrm{ndx}}$$
4		snsstp2	$$⊢\left\{⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}$$
5	4 1	sseqtrri	$$⊢\left\{⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩\right\}\subseteq W$$
6	2 3 5	strfv	$$⊢\underset{\dot{}}{+}\in X\to \underset{\dot{}}{+}={+}_W$$