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Theorem ulmss 24971
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 12240 . . . . . . . 8 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
4 ulmss.t . . . . . . . . . . 11 (𝜑𝑇𝑆)
54adantr 484 . . . . . . . . . 10 ((𝜑𝑘𝑍) → 𝑇𝑆)
6 ssralv 4009 . . . . . . . . . 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 6662 . . . . . . . . . . . . . . 15 (𝑧𝑇 → ((𝐴𝑇)‘𝑧) = (𝐴𝑧))
98ad2antll 728 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → ((𝐴𝑇)‘𝑧) = (𝐴𝑧))
10 simprl 770 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → 𝑥𝑍)
11 ulmss.a . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝑍) → 𝐴𝑊)
1211adantrr 716 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → 𝐴𝑊)
13 resexg 5871 . . . . . . . . . . . . . . . . 17 (𝐴𝑊 → (𝐴𝑇) ∈ V)
1412, 13syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → (𝐴𝑇) ∈ V)
15 eqid 2821 . . . . . . . . . . . . . . . . 17 (𝑥𝑍 ↦ (𝐴𝑇)) = (𝑥𝑍 ↦ (𝐴𝑇))
1615fvmpt2 6752 . . . . . . . . . . . . . . . 16 ((𝑥𝑍 ∧ (𝐴𝑇) ∈ V) → ((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥) = (𝐴𝑇))
1710, 14, 16syl2anc 587 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → ((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥) = (𝐴𝑇))
1817fveq1d 6645 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = ((𝐴𝑇)‘𝑧))
19 eqid 2821 . . . . . . . . . . . . . . . . 17 (𝑥𝑍𝐴) = (𝑥𝑍𝐴)
2019fvmpt2 6752 . . . . . . . . . . . . . . . 16 ((𝑥𝑍𝐴𝑊) → ((𝑥𝑍𝐴)‘𝑥) = 𝐴)
2110, 12, 20syl2anc 587 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → ((𝑥𝑍𝐴)‘𝑥) = 𝐴)
2221fveq1d 6645 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → (((𝑥𝑍𝐴)‘𝑥)‘𝑧) = (𝐴𝑧))
239, 18, 223eqtr4d 2866 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑍𝑧𝑇)) → (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑥)‘𝑧))
2423ralrimivva 3179 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝑍𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑥)‘𝑧))
25 nfv 1916 . . . . . . . . . . . . 13 𝑘𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑥)‘𝑧)
26 nfcv 2974 . . . . . . . . . . . . . 14 𝑥𝑇
27 nffvmpt1 6654 . . . . . . . . . . . . . . . 16 𝑥((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)
28 nfcv 2974 . . . . . . . . . . . . . . . 16 𝑥𝑧
2927, 28nffv 6653 . . . . . . . . . . . . . . 15 𝑥(((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧)
30 nffvmpt1 6654 . . . . . . . . . . . . . . . 16 𝑥((𝑥𝑍𝐴)‘𝑘)
3130, 28nffv 6653 . . . . . . . . . . . . . . 15 𝑥(((𝑥𝑍𝐴)‘𝑘)‘𝑧)
3229, 31nfeq 2987 . . . . . . . . . . . . . 14 𝑥(((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧)
3326, 32nfralw 3213 . . . . . . . . . . . . 13 𝑥𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧)
34 fveq2 6643 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑘 → ((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥) = ((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘))
3534fveq1d 6645 . . . . . . . . . . . . . . 15 (𝑥 = 𝑘 → (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧))
36 fveq2 6643 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑘 → ((𝑥𝑍𝐴)‘𝑥) = ((𝑥𝑍𝐴)‘𝑘))
3736fveq1d 6645 . . . . . . . . . . . . . . 15 (𝑥 = 𝑘 → (((𝑥𝑍𝐴)‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧))
3835, 37eqeq12d 2837 . . . . . . . . . . . . . 14 (𝑥 = 𝑘 → ((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑥)‘𝑧) ↔ (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧)))
3938ralbidv 3185 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → (∀𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑥)‘𝑧) ↔ ∀𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧)))
4025, 33, 39cbvralw 3418 . . . . . . . . . . . 12 (∀𝑥𝑍𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑥)‘𝑧) = (((𝑥𝑍𝐴)‘𝑥)‘𝑧) ↔ ∀𝑘𝑍𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧))
4124, 40sylib 221 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝑍𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧))
4241r19.21bi 3196 . . . . . . . . . 10 ((𝜑𝑘𝑍) → ∀𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧))
43 fvoveq1 7153 . . . . . . . . . . . 12 ((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧) → (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) = (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))))
4443breq1d 5049 . . . . . . . . . . 11 ((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧) → ((abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 ↔ (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
4544ralimi 3148 . . . . . . . . . 10 (∀𝑧𝑇 (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧) → ∀𝑧𝑇 ((abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 ↔ (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
46 ralbi 3155 . . . . . . . . . 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))
4742, 45, 463syl 18 . . . . . . . . 9 ((𝜑𝑘𝑍) → (∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 ↔ ∀𝑧𝑇 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
487, 47sylibrd 262 . . . . . . . 8 ((𝜑𝑘𝑍) → (∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
493, 48sylan2 595 . . . . . . 7 ((𝜑 ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
5049anassrs 471 . . . . . 6 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
5150ralimdva 3165 . . . . 5 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
5251reximdva 3260 . . . 4 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
5352ralimdv 3166 . . 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))
54 ulmf 24956 . . . . . 6 ((𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺 → ∃𝑚 ∈ ℤ (𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑m 𝑆))
551, 54syl 17 . . . . 5 (𝜑 → ∃𝑚 ∈ ℤ (𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑m 𝑆))
56 fdm 6495 . . . . . . . 8 ((𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑m 𝑆) → dom (𝑥𝑍𝐴) = (ℤ𝑚))
5719dmmptss 6068 . . . . . . . 8 dom (𝑥𝑍𝐴) ⊆ 𝑍
5856, 57eqsstrrdi 3998 . . . . . . 7 ((𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑m 𝑆) → (ℤ𝑚) ⊆ 𝑍)
59 uzid 12236 . . . . . . . 8 (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ𝑚))
6059adantl 485 . . . . . . 7 ((𝜑𝑚 ∈ ℤ) → 𝑚 ∈ (ℤ𝑚))
61 ssel 3937 . . . . . . . 8 ((ℤ𝑚) ⊆ 𝑍 → (𝑚 ∈ (ℤ𝑚) → 𝑚𝑍))
62 eluzel2 12226 . . . . . . . . 9 (𝑚 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
6362, 2eleq2s 2930 . . . . . . . 8 (𝑚𝑍𝑀 ∈ ℤ)
6461, 63syl6 35 . . . . . . 7 ((ℤ𝑚) ⊆ 𝑍 → (𝑚 ∈ (ℤ𝑚) → 𝑀 ∈ ℤ))
6558, 60, 64syl2imc 41 . . . . . 6 ((𝜑𝑚 ∈ ℤ) → ((𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑m 𝑆) → 𝑀 ∈ ℤ))
6665rexlimdva 3270 . . . . 5 (𝜑 → (∃𝑚 ∈ ℤ (𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑m 𝑆) → 𝑀 ∈ ℤ))
6755, 66mpd 15 . . . 4 (𝜑𝑀 ∈ ℤ)
6811ralrimiva 3170 . . . . . 6 (𝜑 → ∀𝑥𝑍 𝐴𝑊)
6919fnmpt 6461 . . . . . 6 (∀𝑥𝑍 𝐴𝑊 → (𝑥𝑍𝐴) Fn 𝑍)
7068, 69syl 17 . . . . 5 (𝜑 → (𝑥𝑍𝐴) Fn 𝑍)
71 frn 6493 . . . . . . 7 ((𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑m 𝑆) → ran (𝑥𝑍𝐴) ⊆ (ℂ ↑m 𝑆))
7271rexlimivw 3268 . . . . . 6 (∃𝑚 ∈ ℤ (𝑥𝑍𝐴):(ℤ𝑚)⟶(ℂ ↑m 𝑆) → ran (𝑥𝑍𝐴) ⊆ (ℂ ↑m 𝑆))
7355, 72syl 17 . . . . 5 (𝜑 → ran (𝑥𝑍𝐴) ⊆ (ℂ ↑m 𝑆))
74 df-f 6332 . . . . 5 ((𝑥𝑍𝐴):𝑍⟶(ℂ ↑m 𝑆) ↔ ((𝑥𝑍𝐴) Fn 𝑍 ∧ ran (𝑥𝑍𝐴) ⊆ (ℂ ↑m 𝑆)))
7570, 73, 74sylanbrc 586 . . . 4 (𝜑 → (𝑥𝑍𝐴):𝑍⟶(ℂ ↑m 𝑆))
76 eqidd 2822 . . . 4 ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → (((𝑥𝑍𝐴)‘𝑘)‘𝑧) = (((𝑥𝑍𝐴)‘𝑘)‘𝑧))
77 eqidd 2822 . . . 4 ((𝜑𝑧𝑆) → (𝐺𝑧) = (𝐺𝑧))
78 ulmcl 24955 . . . . 5 ((𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺𝐺:𝑆⟶ℂ)
791, 78syl 17 . . . 4 (𝜑𝐺:𝑆⟶ℂ)
80 ulmscl 24953 . . . . 5 ((𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺𝑆 ∈ V)
811, 80syl 17 . . . 4 (𝜑𝑆 ∈ V)
822, 67, 75, 76, 77, 79, 81ulm2 24959 . . 3 (𝜑 → ((𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺 ↔ ∀𝑟 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘((((𝑥𝑍𝐴)‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
8375fvmptelrn 6850 . . . . . . . 8 ((𝜑𝑥𝑍) → 𝐴 ∈ (ℂ ↑m 𝑆))
84 elmapi 8403 . . . . . . . 8 (𝐴 ∈ (ℂ ↑m 𝑆) → 𝐴:𝑆⟶ℂ)
8583, 84syl 17 . . . . . . 7 ((𝜑𝑥𝑍) → 𝐴:𝑆⟶ℂ)
864adantr 484 . . . . . . 7 ((𝜑𝑥𝑍) → 𝑇𝑆)
8785, 86fssresd 6518 . . . . . 6 ((𝜑𝑥𝑍) → (𝐴𝑇):𝑇⟶ℂ)
88 cnex 10595 . . . . . . 7 ℂ ∈ V
8981, 4ssexd 5201 . . . . . . . 8 (𝜑𝑇 ∈ V)
9089adantr 484 . . . . . . 7 ((𝜑𝑥𝑍) → 𝑇 ∈ V)
91 elmapg 8394 . . . . . . 7 ((ℂ ∈ V ∧ 𝑇 ∈ V) → ((𝐴𝑇) ∈ (ℂ ↑m 𝑇) ↔ (𝐴𝑇):𝑇⟶ℂ))
9288, 90, 91sylancr 590 . . . . . 6 ((𝜑𝑥𝑍) → ((𝐴𝑇) ∈ (ℂ ↑m 𝑇) ↔ (𝐴𝑇):𝑇⟶ℂ))
9387, 92mpbird 260 . . . . 5 ((𝜑𝑥𝑍) → (𝐴𝑇) ∈ (ℂ ↑m 𝑇))
9493fmpttd 6852 . . . 4 (𝜑 → (𝑥𝑍 ↦ (𝐴𝑇)):𝑍⟶(ℂ ↑m 𝑇))
95 eqidd 2822 . . . 4 ((𝜑 ∧ (𝑘𝑍𝑧𝑇)) → (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) = (((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧))
96 fvres 6662 . . . . 5 (𝑧𝑇 → ((𝐺𝑇)‘𝑧) = (𝐺𝑧))
9796adantl 485 . . . 4 ((𝜑𝑧𝑇) → ((𝐺𝑇)‘𝑧) = (𝐺𝑧))
9879, 4fssresd 6518 . . . 4 (𝜑 → (𝐺𝑇):𝑇⟶ℂ)
992, 67, 94, 95, 97, 98, 89ulm2 24959 . . 3 (𝜑 → ((𝑥𝑍 ↦ (𝐴𝑇))(⇝𝑢𝑇)(𝐺𝑇) ↔ ∀𝑟 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑇 (abs‘((((𝑥𝑍 ↦ (𝐴𝑇))‘𝑘)‘𝑧) − (𝐺𝑧))) < 𝑟))
10053, 82, 993imtr4d 297 . 2 (𝜑 → ((𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺 → (𝑥𝑍 ↦ (𝐴𝑇))(⇝𝑢𝑇)(𝐺𝑇)))
1011, 100mpd 15 1 (𝜑 → (𝑥𝑍 ↦ (𝐴𝑇))(⇝𝑢𝑇)(𝐺𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3126  wrex 3127  Vcvv 3471  wss 3910   class class class wbr 5039  cmpt 5119  dom cdm 5528  ran crn 5529  cres 5530   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7130  m cmap 8381  cc 10512   < clt 10652  cmin 10847  cz 11959  cuz 12221  +crp 12367  abscabs 14572  𝑢culm 24950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-pre-lttri 10588  ax-pre-lttrn 10589
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-er 8264  df-map 8383  df-pm 8384  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-neg 10850  df-z 11960  df-uz 12222  df-ulm 24951
This theorem is referenced by: (None)
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