Theorem k0004val

Description: The topological simplex of dimension N is the set of real vectors where the components are nonnegative and sum to 1. (Contributed by RP, 29-Mar-2021)

	Ref	Expression
Hypothesis	k0004.a	$$⊢A=\left(n\in {\mathbb{N}}_0⟼\left\{t\in \left({\left[0,1\right]}^{\left(1\dots n+1\right)}\right)|\sum _{k=1}^{n+1}t\left(k\right)=1\right\}\right)$$
Assertion	k0004val	$$⊢N\in {\mathbb{N}}_0\to A\left(N\right)=\left\{t\in \left({\left[0,1\right]}^{\left(1\dots N+1\right)}\right)|\sum _{k=1}^{N+1}t\left(k\right)=1\right\}$$

Proof

Step	Hyp	Ref	Expression
1		k0004.a	$$⊢A=\left(n\in {\mathbb{N}}_0⟼\left\{t\in \left({\left[0,1\right]}^{\left(1\dots n+1\right)}\right)|\sum _{k=1}^{n+1}t\left(k\right)=1\right\}\right)$$
2		oveq1	$$⊢n=N\to n+1=N+1$$
3	2	oveq2d	$$⊢n=N\to \left(1\dots n+1\right)=\left(1\dots N+1\right)$$
4	3	oveq2d	$$⊢n=N\to {\left[0,1\right]}^{\left(1\dots n+1\right)}={\left[0,1\right]}^{\left(1\dots N+1\right)}$$
5	3	sumeq1d	$$⊢n=N\to \sum _{k=1}^{n+1}t\left(k\right)=\sum _{k=1}^{N+1}t\left(k\right)$$
6	5	eqeq1d	$$⊢n=N\to \left(\sum _{k=1}^{n+1}t\left(k\right)=1\leftrightarrow \sum _{k=1}^{N+1}t\left(k\right)=1\right)$$
7	4 6	rabeqbidv	$$⊢n=N\to \left\{t\in \left({\left[0,1\right]}^{\left(1\dots n+1\right)}\right)|\sum _{k=1}^{n+1}t\left(k\right)=1\right\}=\left\{t\in \left({\left[0,1\right]}^{\left(1\dots N+1\right)}\right)|\sum _{k=1}^{N+1}t\left(k\right)=1\right\}$$
8		ovex	$$⊢{\left[0,1\right]}^{\left(1\dots N+1\right)}\in \mathrm{V}$$
9	8	rabex	$$⊢\left\{t\in \left({\left[0,1\right]}^{\left(1\dots N+1\right)}\right)|\sum _{k=1}^{N+1}t\left(k\right)=1\right\}\in \mathrm{V}$$
10	7 1 9	fvmpt	$$⊢N\in {\mathbb{N}}_0\to A\left(N\right)=\left\{t\in \left({\left[0,1\right]}^{\left(1\dots N+1\right)}\right)|\sum _{k=1}^{N+1}t\left(k\right)=1\right\}$$