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Theorem ulmss 24089
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 = (ℤ𝑀)
ulmss.t (𝜑𝑇𝑆)
ulmss.a ((𝜑𝑥𝑍) → 𝐴𝑊)
ulmss.u (𝜑 → (𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺)
Assertion
Ref Expression
ulmss (𝜑 → (𝑥𝑍 ↦ (𝐴𝑇))(⇝𝑢𝑇)(𝐺𝑇))
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.
StepHypRef Expression
1 ulmss.u . 2 (𝜑 → (𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺)
2 ulmss.z . . . . . . . . 9 𝑍 = (ℤ𝑀)
32uztrn2 11665 . . . . . . . 8 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
4 ulmss.t . . . . . . . . . . 11 (𝜑𝑇𝑆)
54adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝑍) → 𝑇𝑆)
6 ssralv 3651 . . . . . . . . . 10 (𝑇𝑆 → (∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∀𝑧𝑇 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
75, 6syl 17 . . . . . . . . 9 ((𝜑𝑘𝑍) → (∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∀𝑧𝑇 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
8 fvres 6174 . . . . . . . . . . . . . . 15 (𝑧𝑇 → ((𝐴𝑇)‘𝑧) = (𝐴𝑧))
98ad2antll 764 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → ((𝐴𝑇)‘𝑧) = (𝐴𝑧))
10 simprl 793 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → 𝑥𝑍)
11 ulmss.a . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝑍) → 𝐴𝑊)
1211adantrr 752 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → 𝐴𝑊)
13 resexg 5411 . . . . . . . . . . . . . . . . 17 (𝐴𝑊 → (𝐴𝑇) ∈ V)
1412, 13syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → (𝐴𝑇) ∈ V)
15 eqid 2621 . . . . . . . . . . . . . . . . 17 (𝑥𝑍 ↦ (𝐴𝑇)) = (𝑥𝑍 ↦ (𝐴𝑇))
1615fvmpt2 6258 . . . . . . . . . . . . . . . 16 ((𝑥𝑍 ∧ (𝐴𝑇) ∈ V) → ((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥) = (𝐴𝑇))
1710, 14, 16syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → ((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥) = (𝐴𝑇))
1817fveq1d 6160 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = ((𝐴𝑇)‘𝑧))
19 eqid 2621 . . . . . . . . . . . . . . . . 17 (𝑥𝑍𝐴) = (𝑥𝑍𝐴)
2019fvmpt2 6258 . . . . . . . . . . . . . . . 16 ((𝑥𝑍𝐴𝑊) → ((𝑥𝑍𝐴)‘𝑥) = 𝐴)
2110, 12, 20syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → ((𝑥𝑍𝐴)‘𝑥) = 𝐴)
2221fveq1d 6160 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → (((𝑥𝑍𝐴)‘𝑥)‘𝑧) = (𝐴𝑧))
239, 18, 223eqtr4d 2665 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑥)‘𝑧))
2423ralrimivva 2967 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝑍𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑥)‘𝑧))
25 nfv 1840 . . . . . . . . . . . . 13 𝑘𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑥)‘𝑧)
26 nfcv 2761 . . . . . . . . . . . . . 14 𝑥𝑇
27 nffvmpt1 6166 . . . . . . . . . . . . . . . 16 𝑥((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)
28 nfcv 2761 . . . . . . . . . . . . . . . 16 𝑥𝑧
2927, 28nffv 6165 . . . . . . . . . . . . . . 15 𝑥(((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧)
30 nffvmpt1 6166 . . . . . . . . . . . . . . . 16 𝑥((𝑥𝑍𝐴)‘𝑘)
3130, 28nffv 6165 . . . . . . . . . . . . . . 15 𝑥(((𝑥𝑍𝐴)‘𝑘)‘𝑧)
3229, 31nfeq 2772 . . . . . . . . . . . . . 14 𝑥(((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧)
3326, 32nfral 2941 . . . . . . . . . . . . 13 𝑥𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧)
34 fveq2 6158 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑘 → ((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥) = ((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘))
3534fveq1d 6160 . . . . . . . . . . . . . . 15 (𝑥 = 𝑘 → (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧))
36 fveq2 6158 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑘 → ((𝑥𝑍𝐴)‘𝑥) = ((𝑥𝑍𝐴)‘𝑘))
3736fveq1d 6160 . . . . . . . . . . . . . . 15 (𝑥 = 𝑘 → (((𝑥𝑍𝐴)‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧))
3835, 37eqeq12d 2636 . . . . . . . . . . . . . 14 (𝑥 = 𝑘 → ((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑥)‘𝑧) ↔ (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧)))
3938ralbidv 2982 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → (∀𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑥)‘𝑧) ↔ ∀𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧)))
4025, 33, 39cbvral 3159 . . . . . . . . . . . 12 (∀𝑥𝑍𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑥)‘𝑧) ↔ ∀𝑘𝑍𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧))
4124, 40sylib 208 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝑍𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧))
4241r19.21bi 2928 . . . . . . . . . 10 ((𝜑𝑘𝑍) → ∀𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧))
43 oveq1 6622 . . . . . . . . . . . . 13 ((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧) → ((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧)) = ((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧)))
4443fveq2d 6162 . . . . . . . . . . . 12 ((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧) → (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) = (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))))
4544breq1d 4633 . . . . . . . . . . 11 ((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧) → ((abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 ↔ (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
4645ralimi 2948 . . . . . . . . . 10 (∀𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧) → ∀𝑧𝑇 ((abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 ↔ (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
47 ralbi 3063 . . . . . . . . . 10 (∀𝑧𝑇 ((abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 ↔ (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟) → (∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 ↔ ∀𝑧𝑇 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
4842, 46, 473syl 18 . . . . . . . . 9 ((𝜑𝑘𝑍) → (∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 ↔ ∀𝑧𝑇 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
497, 48sylibrd 249 . . . . . . . 8 ((𝜑𝑘𝑍) → (∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
503, 49sylan2 491 . . . . . . 7 ((𝜑 ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
5150anassrs 679 . . . . . 6 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
5251ralimdva 2958 . . . . 5 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
5352reximdva 3013 . . . 4 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
5453ralimdv 2959 . . 3 (𝜑 → (∀𝑟 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∀𝑟 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
55 ulmf 24074 . . . . . 6 ((𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺 → ∃𝑚 ∈ ℤ (𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑𝑚 𝑆))
561, 55syl 17 . . . . 5 (𝜑 → ∃𝑚 ∈ ℤ (𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑𝑚 𝑆))
57 fdm 6018 . . . . . . . 8 ((𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑𝑚 𝑆) → dom (𝑥𝑍𝐴) = (ℤ𝑚))
5819dmmptss 5600 . . . . . . . 8 dom (𝑥𝑍𝐴) ⊆ 𝑍
5957, 58syl6eqssr 3641 . . . . . . 7 ((𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑𝑚 𝑆) → (ℤ𝑚) ⊆ 𝑍)
60 uzid 11662 . . . . . . . . 9 (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ𝑚))
6160adantl 482 . . . . . . . 8 ((𝜑𝑚 ∈ ℤ) → 𝑚 ∈ (ℤ𝑚))
62 ssel 3582 . . . . . . . . 9 ((ℤ𝑚) ⊆ 𝑍 → (𝑚 ∈ (ℤ𝑚) → 𝑚𝑍))
63 eluzel2 11652 . . . . . . . . . 10 (𝑚 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
6463, 2eleq2s 2716 . . . . . . . . 9 (𝑚𝑍𝑀 ∈ ℤ)
6562, 64syl6 35 . . . . . . . 8 ((ℤ𝑚) ⊆ 𝑍 → (𝑚 ∈ (ℤ𝑚) → 𝑀 ∈ ℤ))
6661, 65syl5com 31 . . . . . . 7 ((𝜑𝑚 ∈ ℤ) → ((ℤ𝑚) ⊆ 𝑍𝑀 ∈ ℤ))
6759, 66syl5 34 . . . . . 6 ((𝜑𝑚 ∈ ℤ) → ((𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑𝑚 𝑆) → 𝑀 ∈ ℤ))
6867rexlimdva 3026 . . . . 5 (𝜑 → (∃𝑚 ∈ ℤ (𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑𝑚 𝑆) → 𝑀 ∈ ℤ))
6956, 68mpd 15 . . . 4 (𝜑𝑀 ∈ ℤ)
7011ralrimiva 2962 . . . . . 6 (𝜑 → ∀𝑥𝑍 𝐴𝑊)
7119fnmpt 5987 . . . . . 6 (∀𝑥𝑍 𝐴𝑊 → (𝑥𝑍𝐴) Fn 𝑍)
7270, 71syl 17 . . . . 5 (𝜑 → (𝑥𝑍𝐴) Fn 𝑍)
73 frn 6020 . . . . . . 7 ((𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑𝑚 𝑆) → ran (𝑥𝑍𝐴) ⊆ (ℂ ↑𝑚 𝑆))
7473rexlimivw 3024 . . . . . 6 (∃𝑚 ∈ ℤ (𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑𝑚 𝑆) → ran (𝑥𝑍𝐴) ⊆ (ℂ ↑𝑚 𝑆))
7556, 74syl 17 . . . . 5 (𝜑 → ran (𝑥𝑍𝐴) ⊆ (ℂ ↑𝑚 𝑆))
76 df-f 5861 . . . . 5 ((𝑥𝑍𝐴):𝑍⟶(ℂ ↑𝑚 𝑆) ↔ ((𝑥𝑍𝐴) Fn 𝑍 ∧ ran (𝑥𝑍𝐴) ⊆ (ℂ ↑𝑚 𝑆)))
7772, 75, 76sylanbrc 697 . . . 4 (𝜑 → (𝑥𝑍𝐴):𝑍⟶(ℂ ↑𝑚 𝑆))
78 eqidd 2622 . . . 4 ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → (((𝑥𝑍𝐴)‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧))
79 eqidd 2622 . . . 4 ((𝜑𝑧𝑆) → (𝐺𝑧) = (𝐺𝑧))
80 ulmcl 24073 . . . . 5 ((𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺𝐺:𝑆⟶ℂ)
811, 80syl 17 . . . 4 (𝜑𝐺:𝑆⟶ℂ)
82 ulmscl 24071 . . . . 5 ((𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺𝑆 ∈ V)
831, 82syl 17 . . . 4 (𝜑𝑆 ∈ V)
842, 69, 77, 78, 79, 81, 83ulm2 24077 . . 3 (𝜑 → ((𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺 ↔ ∀𝑟 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
8519fmpt 6347 . . . . . . . . . 10 (∀𝑥𝑍 𝐴 ∈ (ℂ ↑𝑚 𝑆) ↔ (𝑥𝑍𝐴):𝑍⟶(ℂ ↑𝑚 𝑆))
8677, 85sylibr 224 . . . . . . . . 9 (𝜑 → ∀𝑥𝑍 𝐴 ∈ (ℂ ↑𝑚 𝑆))
8786r19.21bi 2928 . . . . . . . 8 ((𝜑𝑥𝑍) → 𝐴 ∈ (ℂ ↑𝑚 𝑆))
88 elmapi 7839 . . . . . . . 8 (𝐴 ∈ (ℂ ↑𝑚 𝑆) → 𝐴:𝑆⟶ℂ)
8987, 88syl 17 . . . . . . 7 ((𝜑𝑥𝑍) → 𝐴:𝑆⟶ℂ)
904adantr 481 . . . . . . 7 ((𝜑𝑥𝑍) → 𝑇𝑆)
9189, 90fssresd 6038 . . . . . 6 ((𝜑𝑥𝑍) → (𝐴𝑇):𝑇⟶ℂ)
92 cnex 9977 . . . . . . 7 ℂ ∈ V
9383, 4ssexd 4775 . . . . . . . 8 (𝜑𝑇 ∈ V)
9493adantr 481 . . . . . . 7 ((𝜑𝑥𝑍) → 𝑇 ∈ V)
95 elmapg 7830 . . . . . . 7 ((ℂ ∈ V ∧ 𝑇 ∈ V) → ((𝐴𝑇) ∈ (ℂ ↑𝑚 𝑇) ↔ (𝐴𝑇):𝑇⟶ℂ))
9692, 94, 95sylancr 694 . . . . . 6 ((𝜑𝑥𝑍) → ((𝐴𝑇) ∈ (ℂ ↑𝑚 𝑇) ↔ (𝐴𝑇):𝑇⟶ℂ))
9791, 96mpbird 247 . . . . 5 ((𝜑𝑥𝑍) → (𝐴𝑇) ∈ (ℂ ↑𝑚 𝑇))
9897, 15fmptd 6351 . . . 4 (𝜑 → (𝑥𝑍 ↦ (𝐴𝑇)):𝑍⟶(ℂ ↑𝑚 𝑇))
99 eqidd 2622 . . . 4 ((𝜑 ∧ (𝑘𝑍𝑧𝑇)) → (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧))
100 fvres 6174 . . . . 5 (𝑧𝑇 → ((𝐺𝑇)‘𝑧) = (𝐺𝑧))
101100adantl 482 . . . 4 ((𝜑𝑧𝑇) → ((𝐺𝑇)‘𝑧) = (𝐺𝑧))
10281, 4fssresd 6038 . . . 4 (𝜑 → (𝐺𝑇):𝑇⟶ℂ)
1032, 69, 98, 99, 101, 102, 93ulm2 24077 . . 3 (𝜑 → ((𝑥𝑍 ↦ (𝐴𝑇))(⇝𝑢𝑇)(𝐺𝑇) ↔ ∀𝑟 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
10454, 84, 1033imtr4d 283 . 2 (𝜑 → ((𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺 → (𝑥𝑍 ↦ (𝐴𝑇))(⇝𝑢𝑇)(𝐺𝑇)))
1051, 104mpd 15 1 (𝜑 → (𝑥𝑍 ↦ (𝐴𝑇))(⇝𝑢𝑇)(𝐺𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909  Vcvv 3190  wss 3560   class class class wbr 4623  cmpt 4683  dom cdm 5084  ran crn 5085  cres 5086   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  𝑚 cmap 7817  cc 9894   < clt 10034  cmin 10226  cz 11337  cuz 11647  +crp 11792  abscabs 13924  𝑢culm 24068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-pre-lttri 9970  ax-pre-lttrn 9971
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-po 5005  df-so 5006  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-neg 10229  df-z 11338  df-uz 11648  df-ulm 24069
This theorem is referenced by: (None)
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