Theorem dflog2

Description: The natural logarithm function in terms of the exponential function restricted to its principal domain. (Contributed by Paul Chapman, 21-Apr-2008)

	Ref	Expression
Assertion	dflog2	$$⊢log={\left({\exp \upharpoonright }_{\mathrm{ran}log}\right)}^{-1}$$

Proof

Step	Hyp	Ref	Expression
1		df-log	$$⊢log={\left({\exp \upharpoonright }_{{\Im }^{-1}\left[\left(-\mathrm{\pi },\mathrm{\pi }\right]\right]}\right)}^{-1}$$
2		logrn	$$⊢\mathrm{ran}log={\Im }^{-1}\left[\left(-\mathrm{\pi },\mathrm{\pi }\right]\right]$$
3	2	reseq2i	$$⊢{\exp \upharpoonright }_{\mathrm{ran}log}={\exp \upharpoonright }_{{\Im }^{-1}\left[\left(-\mathrm{\pi },\mathrm{\pi }\right]\right]}$$
4	3	cnveqi	$$⊢{\left({\exp \upharpoonright }_{\mathrm{ran}log}\right)}^{-1}={\left({\exp \upharpoonright }_{{\Im }^{-1}\left[\left(-\mathrm{\pi },\mathrm{\pi }\right]\right]}\right)}^{-1}$$
5	1 4	eqtr4i	$$⊢log={\left({\exp \upharpoonright }_{\mathrm{ran}log}\right)}^{-1}$$