Theorem ulmss 26757

Description: A uniform limit of functions is still a uniform limit if restricted to a subset. (Contributed by Mario Carneiro, 3-Mar-2015.)

Hypotheses

Ref	Expression
ulmss.z	⊢ 𝑍 = (ℤ≥‘𝑀)
ulmss.t	⊢ (𝜑 → 𝑇 ⊆ 𝑆)
ulmss.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑊)
ulmss.u	⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴)(⇝𝑢‘𝑆)𝐺)

Assertion

Ref	Expression
ulmss	⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))(⇝𝑢‘𝑇)(𝐺 ↾ 𝑇))

Distinct variable groups:   𝑥,𝑇   𝜑,𝑥   𝑥,𝑆   𝑥,𝑍

Allowed substitution hints:   𝐴(𝑥)   𝐺(𝑥)   𝑀(𝑥)   𝑊(𝑥)

Proof of Theorem ulmss

Dummy variables 𝑗 𝑘 𝑚 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ulmss.u	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴)(⇝𝑢‘𝑆)𝐺)
2		ulmss.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀)
3	2	uztrn2 13014	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
4		ulmss.t	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ⊆ 𝑆)
5	4	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑇 ⊆ 𝑆)
6		ssralv 3982	. . . . . . . . . 10 ⊢ (𝑇 ⊆ 𝑆 → (∀𝑧 ∈ 𝑆 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 → ∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
7	5, 6	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (∀𝑧 ∈ 𝑆 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 → ∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
8		fvres 6930	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑇 → ((𝐴 ↾ 𝑇)‘𝑧) = (𝐴‘𝑧))
9	8	ad2antll 742	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → ((𝐴 ↾ 𝑇)‘𝑧) = (𝐴‘𝑧))
10		simprl 783	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → 𝑥 ∈ 𝑍)
11		ulmss.a	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑊)
12	11	adantrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → 𝐴 ∈ 𝑊)
13	12	resexd 6030	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (𝐴 ↾ 𝑇) ∈ V)
14		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇)) = (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))
15	14	fvmpt2 7031	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑍 ∧ (𝐴 ↾ 𝑇) ∈ V) → ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥) = (𝐴 ↾ 𝑇))
16	10, 13, 15	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥) = (𝐴 ↾ 𝑇))
17	16	fveq1d 6913	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = ((𝐴 ↾ 𝑇)‘𝑧))
18		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐴) = (𝑥 ∈ 𝑍 ↦ 𝐴)
19	18	fvmpt2 7031	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑍 ∧ 𝐴 ∈ 𝑊) → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = 𝐴)
20	10, 12, 19	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = 𝐴)
21	20	fveq1d 6913	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧) = (𝐴‘𝑧))
22	9, 17, 21	3eqtr4d 2808	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧))
23	22	ralrimivva 3209	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 ∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧))
24		nfv 1949	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧)
25		nfcv 2926	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝑇
26		nffvmpt1 6922	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)
27		nfcv 2926	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝑧
28	26, 27	nffv 6921	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧)
29		nffvmpt1 6922	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)
30	29, 27	nffv 6921	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧)
31	28, 30	nfeq 2939	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧)
32	25, 31	nfralw 3313	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧)
33		fveq2 6911	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥) = ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘))
34	33	fveq1d 6913	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧))
35		fveq2 6911	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘))
36	35	fveq1d 6913	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧))
37	34, 36	eqeq12d 2779	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → ((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧) ↔ (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧)))
38	37	ralbidv 3189	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧) ↔ ∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧)))
39	24, 32, 38	cbvralw 3308	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑍 ∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧) ↔ ∀𝑘 ∈ 𝑍 ∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧))
40	23, 39	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 ∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧))
41	40	r19.21bi 3258	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧))
42		fvoveq1 7469	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) → (abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) = (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))))
43	42	breq1d 5116	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) → ((abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 ↔ (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
44	43	ralimi 3103	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) → ∀𝑧 ∈ 𝑇 ((abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 ↔ (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
45		ralbi 3121	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑇 ((abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 ↔ (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟) → (∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 ↔ ∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
46	41, 44, 45	3syl 19	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 ↔ ∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
47	7, 46	sylibrd 262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (∀𝑧 ∈ 𝑆 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 → ∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
48	3, 47	sylan2 605	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗))) → (∀𝑧 ∈ 𝑆 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 → ∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
49	48	anassrs 473	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (∀𝑧 ∈ 𝑆 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 → ∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
50	49	ralimdva 3178	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 → ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
51	50	reximdva 3179	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
52	51	ralimdv 3180	. . 3 ⊢ (𝜑 → (∀𝑟 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟 → ∀𝑟 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
53		ulmf 26742	. . . . . 6 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴)(⇝𝑢‘𝑆)𝐺 → ∃𝑚 ∈ ℤ (𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ≥‘𝑚)⟶(ℂ ↑m 𝑆))
54	1, 53	syl 18	. . . . 5 ⊢ (𝜑 → ∃𝑚 ∈ ℤ (𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ≥‘𝑚)⟶(ℂ ↑m 𝑆))
55		fdm 6744	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ≥‘𝑚)⟶(ℂ ↑m 𝑆) → dom (𝑥 ∈ 𝑍 ↦ 𝐴) = (ℤ≥‘𝑚))
56	18	dmmptss 6256	. . . . . . . 8 ⊢ dom (𝑥 ∈ 𝑍 ↦ 𝐴) ⊆ 𝑍
57	55, 56	eqsstrrdi 3958	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ≥‘𝑚)⟶(ℂ ↑m 𝑆) → (ℤ≥‘𝑚) ⊆ 𝑍)
58		uzid 13010	. . . . . . . 8 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ≥‘𝑚))
59	58	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ (ℤ≥‘𝑚))
60		ssel 3907	. . . . . . . 8 ⊢ ((ℤ≥‘𝑚) ⊆ 𝑍 → (𝑚 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ 𝑍))
61		eluzel2 13000	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
62	61, 2	eleq2s 2882	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 → 𝑀 ∈ ℤ)
63	60, 62	syl6 36	. . . . . . 7 ⊢ ((ℤ≥‘𝑚) ⊆ 𝑍 → (𝑚 ∈ (ℤ≥‘𝑚) → 𝑀 ∈ ℤ))
64	57, 59, 63	syl2imc 42	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → ((𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ≥‘𝑚)⟶(ℂ ↑m 𝑆) → 𝑀 ∈ ℤ))
65	64	rexlimdva 3167	. . . . 5 ⊢ (𝜑 → (∃𝑚 ∈ ℤ (𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ≥‘𝑚)⟶(ℂ ↑m 𝑆) → 𝑀 ∈ ℤ))
66	54, 65	mpd 16	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
67	11	ralrimiva 3158	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 𝐴 ∈ 𝑊)
68	18	fnmpt 6704	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑍 𝐴 ∈ 𝑊 → (𝑥 ∈ 𝑍 ↦ 𝐴) Fn 𝑍)
69	67, 68	syl 18	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴) Fn 𝑍)
70		frn 6742	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ≥‘𝑚)⟶(ℂ ↑m 𝑆) → ran (𝑥 ∈ 𝑍 ↦ 𝐴) ⊆ (ℂ ↑m 𝑆))
71	70	rexlimivw 3163	. . . . . 6 ⊢ (∃𝑚 ∈ ℤ (𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ≥‘𝑚)⟶(ℂ ↑m 𝑆) → ran (𝑥 ∈ 𝑍 ↦ 𝐴) ⊆ (ℂ ↑m 𝑆))
72	54, 71	syl 18	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ 𝑍 ↦ 𝐴) ⊆ (ℂ ↑m 𝑆))
73		df-f 6565	. . . . 5 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴):𝑍⟶(ℂ ↑m 𝑆) ↔ ((𝑥 ∈ 𝑍 ↦ 𝐴) Fn 𝑍 ∧ ran (𝑥 ∈ 𝑍 ↦ 𝐴) ⊆ (ℂ ↑m 𝑆)))
74	69, 72, 73	sylanbrc 595	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴):𝑍⟶(ℂ ↑m 𝑆))
75		eqidd 2764	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧))
76		eqidd 2764	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧))
77		ulmcl 26741	. . . . 5 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴)(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ)
78	1, 77	syl 18	. . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ)
79		ulmscl 26739	. . . . 5 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴)(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V)
80	1, 79	syl 18	. . . 4 ⊢ (𝜑 → 𝑆 ∈ V)
81	2, 66, 74, 75, 76, 78, 80	ulm2 26745	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑍 ↦ 𝐴)(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑟 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘((((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
82	74	fvmptelcdm 7140	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ (ℂ ↑m 𝑆))
83		elmapi 8900	. . . . . . . 8 ⊢ (𝐴 ∈ (ℂ ↑m 𝑆) → 𝐴:𝑆⟶ℂ)
84	82, 83	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴:𝑆⟶ℂ)
85	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑇 ⊆ 𝑆)
86	84, 85	fssresd 6774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐴 ↾ 𝑇):𝑇⟶ℂ)
87		cnex 11311	. . . . . . 7 ⊢ ℂ ∈ V
88	80, 4	ssexd 5291	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ V)
89	88	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑇 ∈ V)
90		elmapg 8890	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑇 ∈ V) → ((𝐴 ↾ 𝑇) ∈ (ℂ ↑m 𝑇) ↔ (𝐴 ↾ 𝑇):𝑇⟶ℂ))
91	87, 89, 90	sylancr 599	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝐴 ↾ 𝑇) ∈ (ℂ ↑m 𝑇) ↔ (𝐴 ↾ 𝑇):𝑇⟶ℂ))
92	86, 91	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐴 ↾ 𝑇) ∈ (ℂ ↑m 𝑇))
93	92	fmpttd 7142	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇)):𝑍⟶(ℂ ↑m 𝑇))
94		eqidd 2764	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧))
95		fvres 6930	. . . . 5 ⊢ (𝑧 ∈ 𝑇 → ((𝐺 ↾ 𝑇)‘𝑧) = (𝐺‘𝑧))
96	95	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑇) → ((𝐺 ↾ 𝑇)‘𝑧) = (𝐺‘𝑧))
97	78, 4	fssresd 6774	. . . 4 ⊢ (𝜑 → (𝐺 ↾ 𝑇):𝑇⟶ℂ)
98	2, 66, 93, 94, 96, 97, 88	ulm2 26745	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))(⇝𝑢‘𝑇)(𝐺 ↾ 𝑇) ↔ ∀𝑟 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑇 (abs‘((((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑟))
99	52, 81, 98	3imtr4d 297	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑍 ↦ 𝐴)(⇝𝑢‘𝑆)𝐺 → (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))(⇝𝑢‘𝑇)(𝐺 ↾ 𝑇)))
100	1, 99	mpd 16	1 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))(⇝𝑢‘𝑇)(𝐺 ↾ 𝑇))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   class class class wbr 5106   ↦ cmpt 5190  dom cdm 5656  ran crn 5657   ↾ cres 5658   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446   ↑_m cmap 8878  ℂcc 11228   < clt 11373   − cmin 11571  ℤcz 12723  ℤ_≥cuz 12995  ℝ⁺crp 13150  abscabs 15433  ⇝_𝑢culm 26736

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-pre-lttri 11304  ax-pre-lttrn 11305

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-po 5564  df-so 5565  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-er 8748  df-map 8880  df-pm 8881  df-en 9005  df-dom 9006  df-sdom 9007  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-neg 11574  df-z 12724  df-uz 12996  df-ulm 26737

This theorem is used by: (None)