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Metamath Proof Explorer

Theorem lmodbase

Description: The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013) (Revised by Mario Carneiro, 29-Aug-2015)

		Ref	Expression
	Hypothesis	lmodstr.w	$⊢ W = \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩, ⟨Scalar (ndx), F⟩\} \cup \{⟨\cdot_{ndx}, \underset{˙}{\cdot}⟩\}$
	Assertion	lmodbase	$⊢ B \in X \to B = {Base}_{W}$

Proof

Step	Hyp	Ref	Expression
1		lmodstr.w	$⊢ W = \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩, ⟨Scalar (ndx), F⟩\} \cup \{⟨\cdot_{ndx}, \underset{˙}{\cdot}⟩\}$
2	1	lmodstr	$⊢ W Struct ⟨1, 6⟩$
3		baseid	$⊢ Base = Slot {Base}_{ndx}$
4		snsstp1	$⊢ \{⟨{Base}_{ndx}, B⟩\} \subseteq \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩, ⟨Scalar (ndx), F⟩\}$
5		ssun1	$⊢ \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩, ⟨Scalar (ndx), F⟩\} \subseteq \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩, ⟨Scalar (ndx), F⟩\} \cup \{⟨\cdot_{ndx}, \underset{˙}{\cdot}⟩\}$
6	5 1	sseqtrri	$⊢ \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩, ⟨Scalar (ndx), F⟩\} \subseteq W$
7	4 6	sstri	$⊢ \{⟨{Base}_{ndx}, B⟩\} \subseteq W$
8	2 3 7	strfv	$⊢ B \in X \to B = {Base}_{W}$