Theorem rlimcn2

Description: Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014)

	Ref	Expression
Hypotheses	rlimcn2.1a	$$⊢\left(\phi \wedge z\in A\right)\to B\in X$$
	rlimcn2.1b	$$⊢\left(\phi \wedge z\in A\right)\to C\in Y$$
	rlimcn2.2a	$$⊢\phi \to R\in X$$
	rlimcn2.2b	$$⊢\phi \to S\in Y$$
	rlimcn2.3a	$$⊢\phi \to \left(z\in A⟼B\right)\underset{ℝ}{⇝}R$$
	rlimcn2.3b	$$⊢\phi \to \left(z\in A⟼C\right)\underset{ℝ}{⇝}S$$
	rlimcn2.4	$$⊢\phi \to F:X\times Y⟶ℂ$$
	rlimcn2.5	$$⊢\left(\phi \wedge x\in {ℝ}^{+}\right)\to \exists r\in {ℝ}^{+}\exists s\in {ℝ}^{+}\forall u\in X\forall v\in Y\left(\left(\left|u-R\right|<r\wedge \left|v-S\right|<s\right)\to \left|\left(uFv\right)-\left(RFS\right)\right|<x\right)$$
Assertion	rlimcn2	$$⊢\phi \to \left(z\in A⟼BFC\right)\underset{ℝ}{⇝}\left(RFS\right)$$

Proof

Step	Hyp	Ref	Expression
1		rlimcn2.1a	$$⊢\left(\phi \wedge z\in A\right)\to B\in X$$
2		rlimcn2.1b	$$⊢\left(\phi \wedge z\in A\right)\to C\in Y$$
3		rlimcn2.2a	$$⊢\phi \to R\in X$$
4		rlimcn2.2b	$$⊢\phi \to S\in Y$$
5		rlimcn2.3a	$$⊢\phi \to \left(z\in A⟼B\right)\underset{ℝ}{⇝}R$$
6		rlimcn2.3b	$$⊢\phi \to \left(z\in A⟼C\right)\underset{ℝ}{⇝}S$$
7		rlimcn2.4	$$⊢\phi \to F:X\times Y⟶ℂ$$
8		rlimcn2.5	$$⊢\left(\phi \wedge x\in {ℝ}^{+}\right)\to \exists r\in {ℝ}^{+}\exists s\in {ℝ}^{+}\forall u\in X\forall v\in Y\left(\left(\left|u-R\right|<r\wedge \left|v-S\right|<s\right)\to \left|\left(uFv\right)-\left(RFS\right)\right|<x\right)$$
9	7	adantr	$$⊢\left(\phi \wedge z\in A\right)\to F:X\times Y⟶ℂ$$
10	9 1 2	fovcdmd	$$⊢\left(\phi \wedge z\in A\right)\to BFC\in ℂ$$
11	7 3 4	fovcdmd	$$⊢\phi \to RFS\in ℂ$$
12	1 2 10 11 5 6 8	rlimcn3	$$⊢\phi \to \left(z\in A⟼BFC\right)\underset{ℝ}{⇝}\left(RFS\right)$$