Theorem ulmss 25554

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 12600	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
4		ulmss.t	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ⊆ 𝑆)
5	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑇 ⊆ 𝑆)
6		ssralv 3992	. . . . . . . . . 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 6790	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑇 → ((𝐴 ↾ 𝑇)‘𝑧) = (𝐴‘𝑧))
9	8	ad2antll 726	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → ((𝐴 ↾ 𝑇)‘𝑧) = (𝐴‘𝑧))
10		simprl 768	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → 𝑥 ∈ 𝑍)
11		ulmss.a	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑊)
12	11	adantrr 714	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → 𝐴 ∈ 𝑊)
13	12	resexd 5937	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (𝐴 ↾ 𝑇) ∈ V)
14		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇)) = (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))
15	14	fvmpt2 6883	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑍 ∧ (𝐴 ↾ 𝑇) ∈ V) → ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥) = (𝐴 ↾ 𝑇))
16	10, 13, 15	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥) = (𝐴 ↾ 𝑇))
17	16	fveq1d 6773	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = ((𝐴 ↾ 𝑇)‘𝑧))
18		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐴) = (𝑥 ∈ 𝑍 ↦ 𝐴)
19	18	fvmpt2 6883	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑍 ∧ 𝐴 ∈ 𝑊) → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = 𝐴)
20	10, 12, 19	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = 𝐴)
21	20	fveq1d 6773	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧) = (𝐴‘𝑧))
22	9, 17, 21	3eqtr4d 2790	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧))
23	22	ralrimivva 3117	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 ∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧))
24		nfv 1921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧)
25		nfcv 2909	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝑇
26		nffvmpt1 6782	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)
27		nfcv 2909	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝑧
28	26, 27	nffv 6781	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧)
29		nffvmpt1 6782	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)
30	29, 27	nffv 6781	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧)
31	28, 30	nfeq 2922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧)
32	25, 31	nfralw 3152	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧)
33		fveq2 6771	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥) = ((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘))
34	33	fveq1d 6773	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧))
35		fveq2 6771	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘))
36	35	fveq1d 6773	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧))
37	34, 36	eqeq12d 2756	. . . . . . . . . . . . . 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 3123	. . . . . . . . . . . . 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 3372	. . . . . . . . . . . 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 3135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∀𝑧 ∈ 𝑇 (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧))
42		fvoveq1 7294	. . . . . . . . . . . 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 5089	. . . . . . . . . . 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 3089	. . . . . . . . . 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 3095	. . . . . . . . . 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 593	. . . . . . 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 468	. . . . . 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 3105	. . . . 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 3205	. . . 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 3106	. . 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 25539	. . . . . 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 6607	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ≥‘𝑚)⟶(ℂ ↑m 𝑆) → dom (𝑥 ∈ 𝑍 ↦ 𝐴) = (ℤ≥‘𝑚))
56	18	dmmptss 6143	. . . . . . . 8 ⊢ dom (𝑥 ∈ 𝑍 ↦ 𝐴) ⊆ 𝑍
57	55, 56	eqsstrrdi 3981	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ≥‘𝑚)⟶(ℂ ↑m 𝑆) → (ℤ≥‘𝑚) ⊆ 𝑍)
58		uzid 12596	. . . . . . . 8 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ≥‘𝑚))
59	58	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ (ℤ≥‘𝑚))
60		ssel 3919	. . . . . . . 8 ⊢ ((ℤ≥‘𝑚) ⊆ 𝑍 → (𝑚 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ 𝑍))
61		eluzel2 12586	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
62	61, 2	eleq2s 2859	. . . . . . . 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 3215	. . . . 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 3110	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 𝐴 ∈ 𝑊)
68	18	fnmpt 6571	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑍 𝐴 ∈ 𝑊 → (𝑥 ∈ 𝑍 ↦ 𝐴) Fn 𝑍)
69	67, 68	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴) Fn 𝑍)
70		frn 6605	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴):(ℤ≥‘𝑚)⟶(ℂ ↑m 𝑆) → ran (𝑥 ∈ 𝑍 ↦ 𝐴) ⊆ (ℂ ↑m 𝑆))
71	70	rexlimivw 3213	. . . . . 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 6436	. . . . 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 583	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴):𝑍⟶(ℂ ↑m 𝑆))
75		eqidd 2741	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑘)‘𝑧))
76		eqidd 2741	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧))
77		ulmcl 25538	. . . . 5 ⊢ ((𝑥 ∈ 𝑍 ↦ 𝐴)(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ)
78	1, 77	syl 17	. . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ)
79		ulmscl 25536	. . . . 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 25542	. . 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	fvmptelrn 6984	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ (ℂ ↑m 𝑆))
83		elmapi 8620	. . . . . . . 8 ⊢ (𝐴 ∈ (ℂ ↑m 𝑆) → 𝐴:𝑆⟶ℂ)
84	82, 83	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴:𝑆⟶ℂ)
85	4	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑇 ⊆ 𝑆)
86	84, 85	fssresd 6639	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐴 ↾ 𝑇):𝑇⟶ℂ)
87		cnex 10953	. . . . . . 7 ⊢ ℂ ∈ V
88	80, 4	ssexd 5252	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ V)
89	88	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑇 ∈ V)
90		elmapg 8611	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑇 ∈ V) → ((𝐴 ↾ 𝑇) ∈ (ℂ ↑m 𝑇) ↔ (𝐴 ↾ 𝑇):𝑇⟶ℂ))
91	87, 89, 90	sylancr 587	. . . . . 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 6986	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇)):𝑍⟶(ℂ ↑m 𝑇))
94		eqidd 2741	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑇)) → (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧) = (((𝑥 ∈ 𝑍 ↦ (𝐴 ↾ 𝑇))‘𝑘)‘𝑧))
95		fvres 6790	. . . . 5 ⊢ (𝑧 ∈ 𝑇 → ((𝐺 ↾ 𝑇)‘𝑧) = (𝐺‘𝑧))
96	95	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑇) → ((𝐺 ↾ 𝑇)‘𝑧) = (𝐺‘𝑧))
97	78, 4	fssresd 6639	. . . 4 ⊢ (𝜑 → (𝐺 ↾ 𝑇):𝑇⟶ℂ)
98	2, 66, 93, 94, 96, 97, 88	ulm2 25542	. . 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 294	. 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))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110  ∀wral 3066  ∃wrex 3067  Vcvv 3431   ⊆ wss 3892   class class class wbr 5079   ↦ cmpt 5162  dom cdm 5590  ran crn 5591   ↾ cres 5592   Fn wfn 6427  ⟶wf 6428  ‘cfv 6432  (class class class)co 7271   ↑_m cmap 8598  ℂcc 10870   < clt 11010   − cmin 11205  ℤcz 12319  ℤ_≥cuz 12581  ℝ⁺crp 12729  abscabs 14943  ⇝_𝑢culm 25533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-pre-lttri 10946  ax-pre-lttrn 10947

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-po 5504  df-so 5505  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-er 8481  df-map 8600  df-pm 8601  df-en 8717  df-dom 8718  df-sdom 8719  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-neg 11208  df-z 12320  df-uz 12582  df-ulm 25534

This theorem is referenced by: (None)