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| Description: Every function converges uniformly on the empty set. (Contributed by Mario Carneiro, 3-Mar-2015.) | 
| Ref | Expression | 
|---|---|
| ulm0.z | ⊢ 𝑍 = (ℤ≥‘𝑀) | 
| ulm0.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| ulm0.f | ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | 
| ulm0.g | ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) | 
| Ref | Expression | 
|---|---|
| ulm0 | ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹(⇝𝑢‘𝑆)𝐺) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ulm0.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 2 | uzid 12893 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
| 3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) | 
| 4 | ulm0.z | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 5 | 3, 4 | eleqtrrdi 2852 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑍) | 
| 6 | 5 | ne0d 4342 | . . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅) | 
| 7 | ral0 4513 | . . . . . . 7 ⊢ ∀𝑧 ∈ ∅ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 | |
| 8 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑆 = ∅) | |
| 9 | 8 | raleqdv 3326 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 = ∅) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ ∅ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) | 
| 10 | 7, 9 | mpbiri 258 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) | 
| 11 | 10 | ralrimivw 3150 | . . . . 5 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) | 
| 12 | 11 | ralrimivw 3150 | . . . 4 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) | 
| 13 | r19.2z 4495 | . . . 4 ⊢ ((𝑍 ≠ ∅ ∧ ∀𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) | |
| 14 | 6, 12, 13 | syl2an2r 685 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) | 
| 15 | 14 | ralrimivw 3150 | . 2 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) | 
| 16 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑀 ∈ ℤ) | 
| 17 | ulm0.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | |
| 18 | 17 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | 
| 19 | eqidd 2738 | . . 3 ⊢ (((𝜑 ∧ 𝑆 = ∅) ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) | |
| 20 | eqidd 2738 | . . 3 ⊢ (((𝜑 ∧ 𝑆 = ∅) ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
| 21 | ulm0.g | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) | |
| 22 | 21 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐺:𝑆⟶ℂ) | 
| 23 | 0ex 5307 | . . . 4 ⊢ ∅ ∈ V | |
| 24 | 8, 23 | eqeltrdi 2849 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑆 ∈ V) | 
| 25 | 4, 16, 18, 19, 20, 22, 24 | ulm2 26428 | . 2 ⊢ ((𝜑 ∧ 𝑆 = ∅) → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) | 
| 26 | 15, 25 | mpbird 257 | 1 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹(⇝𝑢‘𝑆)𝐺) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∅c0 4333 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 ℂcc 11153 < clt 11295 − cmin 11492 ℤcz 12613 ℤ≥cuz 12878 ℝ+crp 13034 abscabs 15273 ⇝𝑢culm 26419 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 df-ulm 26420 | 
| This theorem is referenced by: pserulm 26465 | 
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