Theorem climsubc2mpt

Description: Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021)

	Ref	Expression
Hypotheses	climsubc2mpt.k	$$⊢Ⅎk\phi$$
	climsubc2mpt.z	$$⊢Z={\mathbb{Z}}_{\ge M}$$
	climsubc2mpt.m	$$⊢\phi \to M\in \mathbb{Z}$$
	climsubc2mpt.a	$$⊢\left(\phi \wedge k\in Z\right)\to A\in ℂ$$
	climsubc2mpt.c	$$⊢\phi \to \left(k\in Z⟼A\right)⇝C$$
	climsubc2mpt.b	$$⊢\phi \to B\in ℂ$$
Assertion	climsubc2mpt	$$⊢\phi \to \left(k\in Z⟼A-B\right)⇝\left(C-B\right)$$

Proof

Step	Hyp	Ref	Expression
1		climsubc2mpt.k	$$⊢Ⅎk\phi$$
2		climsubc2mpt.z	$$⊢Z={\mathbb{Z}}_{\ge M}$$
3		climsubc2mpt.m	$$⊢\phi \to M\in \mathbb{Z}$$
4		climsubc2mpt.a	$$⊢\left(\phi \wedge k\in Z\right)\to A\in ℂ$$
5		climsubc2mpt.c	$$⊢\phi \to \left(k\in Z⟼A\right)⇝C$$
6		climsubc2mpt.b	$$⊢\phi \to B\in ℂ$$
7	6	adantr	$$⊢\left(\phi \wedge k\in Z\right)\to B\in ℂ$$
8	3 2 6	climconstmpt	$$⊢\phi \to \left(k\in Z⟼B\right)⇝B$$
9	1 2 3 4 7 5 8	climsubmpt	$$⊢\phi \to \left(k\in Z⟼A-B\right)⇝\left(C-B\right)$$