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Theorem ulmss 26363

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 12871	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
4		ulmss.t	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ⊆ 𝑆)
5	4	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑇 ⊆ 𝑆)
6		ssralv 4046	. . . . . . . . . 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 17	. . . . . . . . 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 6913	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑇 → ((𝐴 ↾ 𝑇)‘𝑧) = (𝐴‘𝑧))
9	8	ad2antll 727	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → ((𝐴 ↾ 𝑇)‘𝑧) = (𝐴‘𝑧))
10		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → 𝑥 ∈ 𝑍)
11		ulmss.a	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑊)
12	11	adantrr 715	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → 𝐴 ∈ 𝑊)
13	12	resexd 6032	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (𝐴 ↾ 𝑇) ∈ V)
14		eqid 2725	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇)) = (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))
15	14	fvmpt2 7013	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑍 ∧ (𝐴 ↾ 𝑇) ∈ V) → ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥) = (𝐴 ↾ 𝑇))
16	10, 13, 15	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥) = (𝐴 ↾ 𝑇))
17	16	fveq1d 6896	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = ((𝐴 ↾ 𝑇)‘𝑧))
18		eqid 2725	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐴) = (𝑥 ∈ 𝑍 ↦ 𝐴)
19	18	fvmpt2 7013	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑍 ∧ 𝐴 ∈ 𝑊) → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = 𝐴)
20	10, 12, 19	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = 𝐴)
21	20	fveq1d 6896	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧) = (𝐴‘𝑧))
22	9, 17, 21	3eqtr4d 2775	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧))
23	22	ralrimivva 3191	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 ∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧))
24		nfv 1909	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧)
25		nfcv 2892	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝑇
26		nffvmpt1 6905	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)
27		nfcv 2892	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝑧
28	26, 27	nffv 6904	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧)
29		nffvmpt1 6905	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)
30	29, 27	nffv 6904	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧)
31	28, 30	nfeq 2906	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧)
32	25, 31	nfralw 3299	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧)
33		fveq2 6894	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥) = ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘))
34	33	fveq1d 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧))
35		fveq2 6894	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘))
36	35	fveq1d 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧))
37	34, 36	eqeq12d 2741	. . . . . . . . . . . . . 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 3168	. . . . . . . . . . . . 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 3294	. . . . . . . . . . . 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 217	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 ∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧))
41	40	r19.21bi 3239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧))
42		fvoveq1 7440	. . . . . . . . . . . 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 5158	. . . . . . . . . . 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 3073	. . . . . . . . . 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 3093	. . . . . . . . . 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 18	. . . . . . . . 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 258	. . . . . . . 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 591	. . . . . . 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 466	. . . . . 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 3157	. . . . 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 3158	. . . 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 3159	. . 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 26348	. . . . . 6 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴)(⇝_𝑢‘𝑆)𝐺 → ∃𝑚 ∈ ℤ (𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ_≥‘𝑚)⟶(ℂ ↑_m 𝑆))
54	1, 53	syl 17	. . . . 5 ⊢ (𝜑 → ∃𝑚 ∈ ℤ (𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ_≥‘𝑚)⟶(ℂ ↑_m 𝑆))
55		fdm 6730	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ_≥‘𝑚)⟶(ℂ ↑_m 𝑆) → dom (𝑥 ∈ 𝑍 ↦ 𝐴) = (ℤ_≥‘𝑚))
56	18	dmmptss 6245	. . . . . . . 8 ⊢ dom (𝑥 ∈ 𝑍 ↦ 𝐴) ⊆ 𝑍
57	55, 56	eqsstrrdi 4033	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ_≥‘𝑚)⟶(ℂ ↑_m 𝑆) → (ℤ_≥‘𝑚) ⊆ 𝑍)
58		uzid 12867	. . . . . . . 8 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ_≥‘𝑚))
59	58	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ (ℤ_≥‘𝑚))
60		ssel 3971	. . . . . . . 8 ⊢ ((ℤ_≥‘𝑚) ⊆ 𝑍 → (𝑚 ∈ (ℤ_≥‘𝑚) → 𝑚 ∈ 𝑍))
61		eluzel2 12857	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
62	61, 2	eleq2s 2843	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 → 𝑀 ∈ ℤ)
63	60, 62	syl6 35	. . . . . . 7 ⊢ ((ℤ_≥‘𝑚) ⊆ 𝑍 → (𝑚 ∈ (ℤ_≥‘𝑚) → 𝑀 ∈ ℤ))
64	57, 59, 63	syl2imc 41	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → ((𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ_≥‘𝑚)⟶(ℂ ↑_m 𝑆) → 𝑀 ∈ ℤ))
65	64	rexlimdva 3145	. . . . 5 ⊢ (𝜑 → (∃𝑚 ∈ ℤ (𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ_≥‘𝑚)⟶(ℂ ↑_m 𝑆) → 𝑀 ∈ ℤ))
66	54, 65	mpd 15	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
67	11	ralrimiva 3136	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 𝐴 ∈ 𝑊)
68	18	fnmpt 6694	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑍 𝐴 ∈ 𝑊 → (𝑥 ∈ 𝑍 ↦ 𝐴) Fn 𝑍)
69	67, 68	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴) Fn 𝑍)
70		frn 6728	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ_≥‘𝑚)⟶(ℂ ↑_m 𝑆) → ran (𝑥 ∈ 𝑍 ↦ 𝐴) ⊆ (ℂ ↑_m 𝑆))
71	70	rexlimivw 3141	. . . . . 6 ⊢ (∃𝑚 ∈ ℤ (𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ_≥‘𝑚)⟶(ℂ ↑_m 𝑆) → ran (𝑥 ∈ 𝑍 ↦ 𝐴) ⊆ (ℂ ↑_m 𝑆))
72	54, 71	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ 𝑍 ↦ 𝐴) ⊆ (ℂ ↑_m 𝑆))
73		df-f 6551	. . . . 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 581	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴):𝑍⟶(ℂ ↑_m 𝑆))
75		eqidd 2726	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧))
76		eqidd 2726	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧))
77		ulmcl 26347	. . . . 5 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴)(⇝_𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ)
78	1, 77	syl 17	. . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ)
79		ulmscl 26345	. . . . 5 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴)(⇝_𝑢‘𝑆)𝐺 → 𝑆 ∈ V)
80	1, 79	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 ∈ V)
81	2, 66, 74, 75, 76, 78, 80	ulm2 26351	. . 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 7120	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ (ℂ ↑_m 𝑆))
83		elmapi 8866	. . . . . . . 8 ⊢ (𝐴 ∈ (ℂ ↑_m 𝑆) → 𝐴:𝑆⟶ℂ)
84	82, 83	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴:𝑆⟶ℂ)
85	4	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑇 ⊆ 𝑆)
86	84, 85	fssresd 6762	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐴 ↾ 𝑇):𝑇⟶ℂ)
87		cnex 11219	. . . . . . 7 ⊢ ℂ ∈ V
88	80, 4	ssexd 5324	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ V)
89	88	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑇 ∈ V)
90		elmapg 8856	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑇 ∈ V) → ((𝐴 ↾ 𝑇) ∈ (ℂ ↑_m 𝑇) ↔ (𝐴 ↾ 𝑇):𝑇⟶ℂ))
91	87, 89, 90	sylancr 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝐴 ↾ 𝑇) ∈ (ℂ ↑_m 𝑇) ↔ (𝐴 ↾ 𝑇):𝑇⟶ℂ))
92	86, 91	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐴 ↾ 𝑇) ∈ (ℂ ↑_m 𝑇))
93	92	fmpttd 7122	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇)):𝑍⟶(ℂ ↑_m 𝑇))
94		eqidd 2726	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧))
95		fvres 6913	. . . . 5 ⊢ (𝑧 ∈ 𝑇 → ((𝐺 ↾ 𝑇)‘𝑧) = (𝐺‘𝑧))
96	95	adantl 480	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑇) → ((𝐺 ↾ 𝑇)‘𝑧) = (𝐺‘𝑧))
97	78, 4	fssresd 6762	. . . 4 ⊢ (𝜑 → (𝐺 ↾ 𝑇):𝑇⟶ℂ)
98	2, 66, 93, 94, 96, 97, 88	ulm2 26351	. . 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 293	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑍 ↦ 𝐴)(⇝_𝑢‘𝑆)𝐺 → (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))(⇝_𝑢‘𝑇)(𝐺 ↾ 𝑇)))
100	1, 99	mpd 15	1 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))(⇝_𝑢‘𝑇)(𝐺 ↾ 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 Vcvv 3463 ⊆ wss 3945 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5677 ran crn 5678 ↾ cres 5679 Fn wfn 6542 ⟶wf 6543 ‘cfv 6547 (class class class)co 7417 ↑_m cmap 8843 ℂcc 11136 < clt 11278 − cmin 11474 ℤcz 12588 ℤ_≥cuz 12852 ℝ⁺crp 13006 abscabs 15213 ⇝_𝑢culm 26342

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-pre-lttri 11212 ax-pre-lttrn 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-ov 7420 df-oprab 7421 df-mpo 7422 df-1st 7992 df-2nd 7993 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-neg 11477 df-z 12589 df-uz 12853 df-ulm 26343

This theorem is referenced by: (None)

W3C validator