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| Mirrors > Home > MPE Home > Th. List > Mathboxes > climlimsupcex | Structured version Visualization version GIF version | ||
| Description: Counterexample for climlimsup 45775, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 11173 and its comment). (Contributed by Glauco Siliprandi, 2-Jan-2022.) | 
| Ref | Expression | 
|---|---|
| climlimsupcex.1 | ⊢ ¬ 𝑀 ∈ ℤ | 
| climlimsupcex.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) | 
| climlimsupcex.3 | ⊢ 𝐹 = ∅ | 
| Ref | Expression | 
|---|---|
| climlimsupcex | ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹:𝑍⟶ℝ ∧ 𝐹 ∈ dom ⇝ ∧ ¬ 𝐹 ⇝ (lim sup‘𝐹))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | f0 6789 | . . . 4 ⊢ ∅:∅⟶ℝ | |
| 2 | climlimsupcex.3 | . . . . 5 ⊢ 𝐹 = ∅ | |
| 3 | climlimsupcex.2 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 4 | climlimsupcex.1 | . . . . . . 7 ⊢ ¬ 𝑀 ∈ ℤ | |
| 5 | uz0 45423 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) | |
| 6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (ℤ≥‘𝑀) = ∅ | 
| 7 | 3, 6 | eqtri 2765 | . . . . 5 ⊢ 𝑍 = ∅ | 
| 8 | 2, 7 | feq12i 6729 | . . . 4 ⊢ (𝐹:𝑍⟶ℝ ↔ ∅:∅⟶ℝ) | 
| 9 | 1, 8 | mpbir 231 | . . 3 ⊢ 𝐹:𝑍⟶ℝ | 
| 10 | 9 | a1i 11 | . 2 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ) | 
| 11 | climrel 15528 | . . . . 5 ⊢ Rel ⇝ | |
| 12 | 11 | a1i 11 | . . . 4 ⊢ (∅ ∈ ℂ → Rel ⇝ ) | 
| 13 | 0cnv 45757 | . . . . 5 ⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) | |
| 14 | 2, 13 | eqbrtrid 5178 | . . . 4 ⊢ (∅ ∈ ℂ → 𝐹 ⇝ ∅) | 
| 15 | releldm 5955 | . . . 4 ⊢ ((Rel ⇝ ∧ 𝐹 ⇝ ∅) → 𝐹 ∈ dom ⇝ ) | |
| 16 | 12, 14, 15 | syl2anc 584 | . . 3 ⊢ (∅ ∈ ℂ → 𝐹 ∈ dom ⇝ ) | 
| 17 | 16 | adantr 480 | . 2 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → 𝐹 ∈ dom ⇝ ) | 
| 18 | 13 | adantr 480 | . . . 4 ⊢ ((∅ ∈ ℂ ∧ 𝐹 ⇝ (lim sup‘𝐹)) → ∅ ⇝ ∅) | 
| 19 | 18 | adantlr 715 | . . 3 ⊢ (((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) ∧ 𝐹 ⇝ (lim sup‘𝐹)) → ∅ ⇝ ∅) | 
| 20 | simpr 484 | . . . . . 6 ⊢ ((𝐹 ⇝ (lim sup‘𝐹) ∧ ∅ ⇝ ∅) → ∅ ⇝ ∅) | |
| 21 | 2 | fveq2i 6909 | . . . . . . . . . 10 ⊢ (lim sup‘𝐹) = (lim sup‘∅) | 
| 22 | limsup0 45709 | . . . . . . . . . 10 ⊢ (lim sup‘∅) = -∞ | |
| 23 | 21, 22 | eqtri 2765 | . . . . . . . . 9 ⊢ (lim sup‘𝐹) = -∞ | 
| 24 | 2, 23 | breq12i 5152 | . . . . . . . 8 ⊢ (𝐹 ⇝ (lim sup‘𝐹) ↔ ∅ ⇝ -∞) | 
| 25 | 24 | biimpi 216 | . . . . . . 7 ⊢ (𝐹 ⇝ (lim sup‘𝐹) → ∅ ⇝ -∞) | 
| 26 | 25 | adantr 480 | . . . . . 6 ⊢ ((𝐹 ⇝ (lim sup‘𝐹) ∧ ∅ ⇝ ∅) → ∅ ⇝ -∞) | 
| 27 | climuni 15588 | . . . . . 6 ⊢ ((∅ ⇝ ∅ ∧ ∅ ⇝ -∞) → ∅ = -∞) | |
| 28 | 20, 26, 27 | syl2anc 584 | . . . . 5 ⊢ ((𝐹 ⇝ (lim sup‘𝐹) ∧ ∅ ⇝ ∅) → ∅ = -∞) | 
| 29 | 28 | adantll 714 | . . . 4 ⊢ ((((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) ∧ 𝐹 ⇝ (lim sup‘𝐹)) ∧ ∅ ⇝ ∅) → ∅ = -∞) | 
| 30 | nelneq 2865 | . . . . 5 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → ¬ ∅ = -∞) | |
| 31 | 30 | ad2antrr 726 | . . . 4 ⊢ ((((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) ∧ 𝐹 ⇝ (lim sup‘𝐹)) ∧ ∅ ⇝ ∅) → ¬ ∅ = -∞) | 
| 32 | 29, 31 | pm2.65da 817 | . . 3 ⊢ (((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) ∧ 𝐹 ⇝ (lim sup‘𝐹)) → ¬ ∅ ⇝ ∅) | 
| 33 | 19, 32 | pm2.65da 817 | . 2 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → ¬ 𝐹 ⇝ (lim sup‘𝐹)) | 
| 34 | 10, 17, 33 | 3jca 1129 | 1 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹:𝑍⟶ℝ ∧ 𝐹 ∈ dom ⇝ ∧ ¬ 𝐹 ⇝ (lim sup‘𝐹))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∅c0 4333 class class class wbr 5143 dom cdm 5685 Rel wrel 5690 ⟶wf 6557 ‘cfv 6561 ℂcc 11153 ℝcr 11154 -∞cmnf 11293 ℤcz 12613 ℤ≥cuz 12878 lim supclsp 15506 ⇝ cli 15520 | 
| 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-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| 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-rmo 3380 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-pss 3971 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-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 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-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 | 
| This theorem is referenced by: (None) | 
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