Theorem climeldmeqmpt

Description: Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	climeldmeqmpt.k	⊢ Ⅎ 𝑘 𝜑
	climeldmeqmpt.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	climeldmeqmpt.z	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	climeldmeqmpt.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑅 )
	climeldmeqmpt.i	⊢ ( 𝜑 → 𝑍 ⊆ 𝐴 )
	climeldmeqmpt.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
	climeldmeqmpt.t	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
	climeldmeqmpt.l	⊢ ( 𝜑 → 𝑍 ⊆ 𝐶 )
	climeldmeqmpt.c	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) → 𝐷 ∈ 𝑊 )
	climeldmeqmpt.e	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐵 = 𝐷 )
Assertion	climeldmeqmpt	⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∈ dom ⇝ ↔ ( 𝑘 ∈ 𝐶 ↦ 𝐷 ) ∈ dom ⇝ ) )

Proof

Step	Hyp	Ref	Expression
1		climeldmeqmpt.k	⊢ Ⅎ 𝑘 𝜑
2		climeldmeqmpt.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		climeldmeqmpt.z	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
4		climeldmeqmpt.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑅 )
5		climeldmeqmpt.i	⊢ ( 𝜑 → 𝑍 ⊆ 𝐴 )
6		climeldmeqmpt.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
7		climeldmeqmpt.t	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
8		climeldmeqmpt.l	⊢ ( 𝜑 → 𝑍 ⊆ 𝐶 )
9		climeldmeqmpt.c	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) → 𝐷 ∈ 𝑊 )
10		climeldmeqmpt.e	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐵 = 𝐷 )
11	4	mptexd	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∈ V )
12	7	mptexd	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐶 ↦ 𝐷 ) ∈ V )
13		nfv	⊢ Ⅎ 𝑘 𝑗 ∈ 𝑍
14	1 13	nfan	⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑗 ∈ 𝑍 )
15		nfcsb1v	⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐵
16		nfcv	⊢ Ⅎ 𝑘 𝑗
17	16	nfcsb1	⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐷
18	15 17	nfeq	⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐷
19	14 18	nfim	⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐷 )
20		eleq1w	⊢ ( 𝑘 = 𝑗 → ( 𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍 ) )
21	20	anbi2d	⊢ ( 𝑘 = 𝑗 → ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ↔ ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ) )
22		csbeq1a	⊢ ( 𝑘 = 𝑗 → 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 )
23		csbeq1a	⊢ ( 𝑘 = 𝑗 → 𝐷 = ⦋ 𝑗 / 𝑘 ⦌ 𝐷 )
24	22 23	eqeq12d	⊢ ( 𝑘 = 𝑗 → ( 𝐵 = 𝐷 ↔ ⦋ 𝑗 / 𝑘 ⦌ 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐷 ) )
25	21 24	imbi12d	⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐵 = 𝐷 ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐷 ) ) )
26	19 25 10	chvarfv	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐷 )
27	5	sselda	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ 𝐴 )
28		nfv	⊢ Ⅎ 𝑘 𝑗 ∈ 𝐴
29	1 28	nfan	⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑗 ∈ 𝐴 )
30		nfcv	⊢ Ⅎ 𝑘 𝑉
31	15 30	nfel	⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑉
32	29 31	nfim	⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑉 )
33		eleq1w	⊢ ( 𝑘 = 𝑗 → ( 𝑘 ∈ 𝐴 ↔ 𝑗 ∈ 𝐴 ) )
34	33	anbi2d	⊢ ( 𝑘 = 𝑗 → ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ↔ ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) ) )
35	22	eleq1d	⊢ ( 𝑘 = 𝑗 → ( 𝐵 ∈ 𝑉 ↔ ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑉 ) )
36	34 35	imbi12d	⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑉 ) ) )
37	32 36 6	chvarfv	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑉 )
38	27 37	syldan	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑉 )
39	16	nfcsb1	⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐵
40		eqid	⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 )
41	16 39 22 40	fvmptf	⊢ ( ( 𝑗 ∈ 𝐴 ∧ ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑉 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 )
42	27 38 41	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 )
43	8	sselda	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ 𝐶 )
44		nfv	⊢ Ⅎ 𝑘 𝑗 ∈ 𝐶
45	1 44	nfan	⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑗 ∈ 𝐶 )
46		nfcv	⊢ Ⅎ 𝑘 𝑊
47	17 46	nfel	⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐷 ∈ 𝑊
48	45 47	nfim	⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑗 ∈ 𝐶 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐷 ∈ 𝑊 )
49		eleq1w	⊢ ( 𝑘 = 𝑗 → ( 𝑘 ∈ 𝐶 ↔ 𝑗 ∈ 𝐶 ) )
50	49	anbi2d	⊢ ( 𝑘 = 𝑗 → ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) ↔ ( 𝜑 ∧ 𝑗 ∈ 𝐶 ) ) )
51	23	eleq1d	⊢ ( 𝑘 = 𝑗 → ( 𝐷 ∈ 𝑊 ↔ ⦋ 𝑗 / 𝑘 ⦌ 𝐷 ∈ 𝑊 ) )
52	50 51	imbi12d	⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) → 𝐷 ∈ 𝑊 ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝐶 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐷 ∈ 𝑊 ) ) )
53	48 52 9	chvarfv	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐶 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐷 ∈ 𝑊 )
54	43 53	syldan	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐷 ∈ 𝑊 )
55		eqid	⊢ ( 𝑘 ∈ 𝐶 ↦ 𝐷 ) = ( 𝑘 ∈ 𝐶 ↦ 𝐷 )
56	16 17 23 55	fvmptf	⊢ ( ( 𝑗 ∈ 𝐶 ∧ ⦋ 𝑗 / 𝑘 ⦌ 𝐷 ∈ 𝑊 ) → ( ( 𝑘 ∈ 𝐶 ↦ 𝐷 ) ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ 𝐷 )
57	43 54 56	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ( 𝑘 ∈ 𝐶 ↦ 𝐷 ) ‘ 𝑗 ) = ⦋ 𝑗 / 𝑘 ⦌ 𝐷 )
58	26 42 57	3eqtr4d	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 ) = ( ( 𝑘 ∈ 𝐶 ↦ 𝐷 ) ‘ 𝑗 ) )
59	3 11 12 2 58	climeldmeq	⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∈ dom ⇝ ↔ ( 𝑘 ∈ 𝐶 ↦ 𝐷 ) ∈ dom ⇝ ) )