Theorem xlimpnfxnegmnf2

Description: A sequence converges to +oo if and only if its negation converges to -oo . (Contributed by Glauco Siliprandi, 23-Apr-2023)

	Ref	Expression
Hypotheses	xlimpnfxnegmnf2.j	$$⊢\underset{\_}{Ⅎ}jF$$
	xlimpnfxnegmnf2.m	$$⊢\phi \to M\in \mathbb{Z}$$
	xlimpnfxnegmnf2.z	$$⊢Z={\mathbb{Z}}_{\ge M}$$
	xlimpnfxnegmnf2.f	$$⊢\phi \to F:Z⟶{ℝ}^{*}$$
Assertion	xlimpnfxnegmnf2	$$⊢\phi \to \left(F\underset{*}{⇝}\mathrm{+\infty }\leftrightarrow \left(j\in Z⟼-F\left(j\right)\right)\underset{*}{⇝}\mathrm{-\infty }\right)$$

Proof

Step	Hyp	Ref	Expression
1		xlimpnfxnegmnf2.j	$$⊢\underset{\_}{Ⅎ}jF$$
2		xlimpnfxnegmnf2.m	$$⊢\phi \to M\in \mathbb{Z}$$
3		xlimpnfxnegmnf2.z	$$⊢Z={\mathbb{Z}}_{\ge M}$$
4		xlimpnfxnegmnf2.f	$$⊢\phi \to F:Z⟶{ℝ}^{*}$$
5	1 3 4	xlimpnfxnegmnf	$$⊢\phi \to \left(\forall x\in ℝ\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}x\le F\left(j\right)\leftrightarrow \forall x\in ℝ\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}-F\left(j\right)\le x\right)$$
6	1 2 3 4	xlimpnf	$$⊢\phi \to \left(F\underset{*}{⇝}\mathrm{+\infty }\leftrightarrow \forall x\in ℝ\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}x\le F\left(j\right)\right)$$
7		nfmpt1	$$⊢\underset{\_}{Ⅎ}j\left(j\in Z⟼-F\left(j\right)\right)$$
8	4	ffvelcdmda	$$⊢\left(\phi \wedge k\in Z\right)\to F\left(k\right)\in {ℝ}^{*}$$
9	8	xnegcld	$$⊢\left(\phi \wedge k\in Z\right)\to -F\left(k\right)\in {ℝ}^{*}$$
10		nfcv	$$⊢\underset{\_}{Ⅎ}k\left(-F\left(j\right)\right)$$
11		nfcv	$$⊢\underset{\_}{Ⅎ}jk$$
12	1 11	nffv	$$⊢\underset{\_}{Ⅎ}jF\left(k\right)$$
13	12	nfxneg	$$⊢\underset{\_}{Ⅎ}j\left(-F\left(k\right)\right)$$
14		fveq2	$$⊢j=k\to F\left(j\right)=F\left(k\right)$$
15	14	xnegeqd	$$⊢j=k\to -F\left(j\right)=-F\left(k\right)$$
16	10 13 15	cbvmpt	$$⊢\left(j\in Z⟼-F\left(j\right)\right)=\left(k\in Z⟼-F\left(k\right)\right)$$
17	9 16	fmptd	$$⊢\phi \to \left(j\in Z⟼-F\left(j\right)\right):Z⟶{ℝ}^{*}$$
18	7 2 3 17	xlimmnf	$$⊢\phi \to \left(\left(j\in Z⟼-F\left(j\right)\right)\underset{*}{⇝}\mathrm{-\infty }\leftrightarrow \forall x\in ℝ\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}\left(j\in Z⟼-F\left(j\right)\right)\left(j\right)\le x\right)$$
19	3	uztrn2	$$⊢\left(k\in Z\wedge j\in {\mathbb{Z}}_{\ge k}\right)\to j\in Z$$
20		xnegex	$$⊢-F\left(j\right)\in \mathrm{V}$$
21		fvmpt4	$$⊢\left(j\in Z\wedge -F\left(j\right)\in \mathrm{V}\right)\to \left(j\in Z⟼-F\left(j\right)\right)\left(j\right)=-F\left(j\right)$$
22	20 21	mpan2	$$⊢j\in Z\to \left(j\in Z⟼-F\left(j\right)\right)\left(j\right)=-F\left(j\right)$$
23	22	breq1d	$$⊢j\in Z\to \left(\left(j\in Z⟼-F\left(j\right)\right)\left(j\right)\le x\leftrightarrow -F\left(j\right)\le x\right)$$
24	19 23	syl	$$⊢\left(k\in Z\wedge j\in {\mathbb{Z}}_{\ge k}\right)\to \left(\left(j\in Z⟼-F\left(j\right)\right)\left(j\right)\le x\leftrightarrow -F\left(j\right)\le x\right)$$
25	24	ralbidva	$$⊢k\in Z\to \left(\forall j\in {\mathbb{Z}}_{\ge k}\left(j\in Z⟼-F\left(j\right)\right)\left(j\right)\le x\leftrightarrow \forall j\in {\mathbb{Z}}_{\ge k}-F\left(j\right)\le x\right)$$
26	25	rexbiia	$$⊢\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}\left(j\in Z⟼-F\left(j\right)\right)\left(j\right)\le x\leftrightarrow \exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}-F\left(j\right)\le x$$
27	26	ralbii	$$⊢\forall x\in ℝ\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}\left(j\in Z⟼-F\left(j\right)\right)\left(j\right)\le x\leftrightarrow \forall x\in ℝ\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}-F\left(j\right)\le x$$
28	18 27	bitrdi	$$⊢\phi \to \left(\left(j\in Z⟼-F\left(j\right)\right)\underset{*}{⇝}\mathrm{-\infty }\leftrightarrow \forall x\in ℝ\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}-F\left(j\right)\le x\right)$$
29	5 6 28	3bitr4d	$$⊢\phi \to \left(F\underset{*}{⇝}\mathrm{+\infty }\leftrightarrow \left(j\in Z⟼-F\left(j\right)\right)\underset{*}{⇝}\mathrm{-\infty }\right)$$