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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limcdm0 | Structured version Visualization version GIF version |
Description: If a function has empty domain, every complex number is a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
limcdm0.f | ⊢ (𝜑 → 𝐹:∅⟶ℂ) |
limcdm0.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
limcdm0 | ⊢ (𝜑 → (𝐹 limℂ 𝐵) = ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limccl 25361 | . . . . 5 ⊢ (𝐹 limℂ 𝐵) ⊆ ℂ | |
2 | 1 | sseli 3976 | . . . 4 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → 𝑥 ∈ ℂ) |
3 | 2 | adantl 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → 𝑥 ∈ ℂ) |
4 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
5 | 1rp 12965 | . . . . . . 7 ⊢ 1 ∈ ℝ+ | |
6 | ral0 4508 | . . . . . . 7 ⊢ ∀𝑧 ∈ ∅ ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 1) → (abs‘((𝐹‘𝑧) − 𝑥)) < 𝑦) | |
7 | brimralrspcev 5205 | . . . . . . 7 ⊢ ((1 ∈ ℝ+ ∧ ∀𝑧 ∈ ∅ ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 1) → (abs‘((𝐹‘𝑧) − 𝑥)) < 𝑦)) → ∃𝑤 ∈ ℝ+ ∀𝑧 ∈ ∅ ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑤) → (abs‘((𝐹‘𝑧) − 𝑥)) < 𝑦)) | |
8 | 5, 6, 7 | mp2an 691 | . . . . . 6 ⊢ ∃𝑤 ∈ ℝ+ ∀𝑧 ∈ ∅ ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑤) → (abs‘((𝐹‘𝑧) − 𝑥)) < 𝑦) |
9 | 8 | rgenw 3066 | . . . . 5 ⊢ ∀𝑦 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑧 ∈ ∅ ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑤) → (abs‘((𝐹‘𝑧) − 𝑥)) < 𝑦) |
10 | 9 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∀𝑦 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑧 ∈ ∅ ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑤) → (abs‘((𝐹‘𝑧) − 𝑥)) < 𝑦)) |
11 | limcdm0.f | . . . . . 6 ⊢ (𝜑 → 𝐹:∅⟶ℂ) | |
12 | 11 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐹:∅⟶ℂ) |
13 | 0ss 4394 | . . . . . 6 ⊢ ∅ ⊆ ℂ | |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∅ ⊆ ℂ) |
15 | limcdm0.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
16 | 15 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ) |
17 | 12, 14, 16 | ellimc3 25365 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ (𝑥 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑧 ∈ ∅ ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑤) → (abs‘((𝐹‘𝑧) − 𝑥)) < 𝑦)))) |
18 | 4, 10, 17 | mpbir2and 712 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ (𝐹 limℂ 𝐵)) |
19 | 3, 18 | impbida 800 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ 𝑥 ∈ ℂ)) |
20 | 19 | eqrdv 2731 | 1 ⊢ (𝜑 → (𝐹 limℂ 𝐵) = ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ⊆ wss 3946 ∅c0 4320 class class class wbr 5144 ⟶wf 6531 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 1c1 11098 < clt 11235 − cmin 11431 ℝ+crp 12961 abscabs 15168 limℂ climc 25348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-pm 8811 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fi 9393 df-sup 9424 df-inf 9425 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-q 12920 df-rp 12962 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-fz 13472 df-seq 13954 df-exp 14015 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 df-struct 17067 df-slot 17102 df-ndx 17114 df-base 17132 df-plusg 17197 df-mulr 17198 df-starv 17199 df-tset 17203 df-ple 17204 df-ds 17206 df-unif 17207 df-rest 17355 df-topn 17356 df-topgen 17376 df-psmet 20910 df-xmet 20911 df-met 20912 df-bl 20913 df-mopn 20914 df-cnfld 20919 df-top 22365 df-topon 22382 df-topsp 22404 df-bases 22418 df-cnp 22701 df-xms 23795 df-ms 23796 df-limc 25352 |
This theorem is referenced by: ioodvbdlimc1 44522 ioodvbdlimc2 44524 |
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