Theorem wrecseq123

Description: General equality theorem for the well-ordered recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018) (Proof shortened by Scott Fenton, 17-Nov-2024)

	Ref	Expression
Assertion	wrecseq123	$$⊢\left(R=S\wedge A=B\wedge F=G\right)\to \mathrm{wrecs}\left(R,A,F\right)=\mathrm{wrecs}\left(S,B,G\right)$$

Proof

Step	Hyp	Ref	Expression
1		coeq1	$$⊢F=G\to F\circ 2^{nd}=G\circ 2^{nd}$$
2		frecseq123	$$⊢\left(R=S\wedge A=B\wedge F\circ 2^{nd}=G\circ 2^{nd}\right)\to \mathrm{frecs}\left(R,A,F\circ 2^{nd}\right)=\mathrm{frecs}\left(S,B,G\circ 2^{nd}\right)$$
3	1 2	syl3an3	$$⊢\left(R=S\wedge A=B\wedge F=G\right)\to \mathrm{frecs}\left(R,A,F\circ 2^{nd}\right)=\mathrm{frecs}\left(S,B,G\circ 2^{nd}\right)$$
4		df-wrecs	$$⊢\mathrm{wrecs}\left(R,A,F\right)=\mathrm{frecs}\left(R,A,F\circ 2^{nd}\right)$$
5		df-wrecs	$$⊢\mathrm{wrecs}\left(S,B,G\right)=\mathrm{frecs}\left(S,B,G\circ 2^{nd}\right)$$
6	3 4 5	3eqtr4g	$$⊢\left(R=S\wedge A=B\wedge F=G\right)\to \mathrm{wrecs}\left(R,A,F\right)=\mathrm{wrecs}\left(S,B,G\right)$$