I need to prove the finite strong completeness theorem for Basic Logic, where $BL$ is: $\vdash_{BL} =\vDash_{FLew}$ + linearity + division. The strong completeness is:
$$\varphi_1,...,\varphi_n \vdash_{BL} \varphi \Leftrightarrow \varphi_1,...,\varphi_n \vDash_{+} \varphi$$ Where $+$ is $\{[0,1]*cut \ t-norms\}$.
Now, I have a few notions, I know I need to use/check that: Any $BL$ chain is locally embeddable into $\{[0,1] \ continuos \ t-norms\}$
I also now I need to check that every $BL$ chain is embeddable into an ultra product of standard $BL$ chains.
My main problem is figuring out the structure of this proof, if someone could point out wither some set of steps that I need to follow, or even better a book or article where a similar proof is worked out for some other Many valued logic (Lukasiewics, product, etc...) I'd very much appreciate your help.