Theorem opsrbas

Description: The base set of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015) (Revised by Mario Carneiro, 30-Aug-2015)

	Ref	Expression
Hypotheses	opsrbas.s	$$⊢S=I\mathrm{mPwSer}R$$
	opsrbas.o	$$⊢O=\left(I\mathrm{ordPwSer}R\right)\left(T\right)$$
	opsrbas.t	$$⊢\phi \to T\subseteq I\times I$$
Assertion	opsrbas	$$⊢\phi \to {\mathrm{Base}}_S={\mathrm{Base}}_O$$

Proof

Step	Hyp	Ref	Expression
1		opsrbas.s	$$⊢S=I\mathrm{mPwSer}R$$
2		opsrbas.o	$$⊢O=\left(I\mathrm{ordPwSer}R\right)\left(T\right)$$
3		opsrbas.t	$$⊢\phi \to T\subseteq I\times I$$
4		df-base	$$⊢\mathrm{Base}=\mathrm{Slot}1$$
5		1nn	$$⊢1\in \mathbb{N}$$
6		1lt10	$$⊢1<10$$
7	1 2 3 4 5 6	opsrbaslem	$$⊢\phi \to {\mathrm{Base}}_S={\mathrm{Base}}_O$$