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Mirrors > Home > MPE Home > Th. List > Mathboxes > climlimsupcex | Structured version Visualization version GIF version |
Description: Counterexample for climlimsup 42048, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 10557 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 6562 | . . . 4 ⊢ ∅:∅⟶ℝ | |
2 | climlimsupcex.3 | . . . . 5 ⊢ 𝐹 = ∅ | |
3 | climlimsupcex.2 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | climlimsupcex.1 | . . . . . . 7 ⊢ ¬ 𝑀 ∈ ℤ | |
5 | uz0 41693 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (ℤ≥‘𝑀) = ∅ |
7 | 3, 6 | eqtri 2846 | . . . . 5 ⊢ 𝑍 = ∅ |
8 | 2, 7 | feq12i 6509 | . . . 4 ⊢ (𝐹:𝑍⟶ℝ ↔ ∅:∅⟶ℝ) |
9 | 1, 8 | mpbir 233 | . . 3 ⊢ 𝐹:𝑍⟶ℝ |
10 | 9 | a1i 11 | . 2 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ) |
11 | climrel 14851 | . . . . 5 ⊢ Rel ⇝ | |
12 | 11 | a1i 11 | . . . 4 ⊢ (∅ ∈ ℂ → Rel ⇝ ) |
13 | 0cnv 42030 | . . . . 5 ⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) | |
14 | 2, 13 | eqbrtrid 5103 | . . . 4 ⊢ (∅ ∈ ℂ → 𝐹 ⇝ ∅) |
15 | releldm 5816 | . . . 4 ⊢ ((Rel ⇝ ∧ 𝐹 ⇝ ∅) → 𝐹 ∈ dom ⇝ ) | |
16 | 12, 14, 15 | syl2anc 586 | . . 3 ⊢ (∅ ∈ ℂ → 𝐹 ∈ dom ⇝ ) |
17 | 16 | adantr 483 | . 2 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → 𝐹 ∈ dom ⇝ ) |
18 | 13 | adantr 483 | . . . 4 ⊢ ((∅ ∈ ℂ ∧ 𝐹 ⇝ (lim sup‘𝐹)) → ∅ ⇝ ∅) |
19 | 18 | adantlr 713 | . . 3 ⊢ (((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) ∧ 𝐹 ⇝ (lim sup‘𝐹)) → ∅ ⇝ ∅) |
20 | simpr 487 | . . . . . 6 ⊢ ((𝐹 ⇝ (lim sup‘𝐹) ∧ ∅ ⇝ ∅) → ∅ ⇝ ∅) | |
21 | 2 | fveq2i 6675 | . . . . . . . . . 10 ⊢ (lim sup‘𝐹) = (lim sup‘∅) |
22 | limsup0 41982 | . . . . . . . . . 10 ⊢ (lim sup‘∅) = -∞ | |
23 | 21, 22 | eqtri 2846 | . . . . . . . . 9 ⊢ (lim sup‘𝐹) = -∞ |
24 | 2, 23 | breq12i 5077 | . . . . . . . 8 ⊢ (𝐹 ⇝ (lim sup‘𝐹) ↔ ∅ ⇝ -∞) |
25 | 24 | biimpi 218 | . . . . . . 7 ⊢ (𝐹 ⇝ (lim sup‘𝐹) → ∅ ⇝ -∞) |
26 | 25 | adantr 483 | . . . . . 6 ⊢ ((𝐹 ⇝ (lim sup‘𝐹) ∧ ∅ ⇝ ∅) → ∅ ⇝ -∞) |
27 | climuni 14911 | . . . . . 6 ⊢ ((∅ ⇝ ∅ ∧ ∅ ⇝ -∞) → ∅ = -∞) | |
28 | 20, 26, 27 | syl2anc 586 | . . . . 5 ⊢ ((𝐹 ⇝ (lim sup‘𝐹) ∧ ∅ ⇝ ∅) → ∅ = -∞) |
29 | 28 | adantll 712 | . . . 4 ⊢ ((((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) ∧ 𝐹 ⇝ (lim sup‘𝐹)) ∧ ∅ ⇝ ∅) → ∅ = -∞) |
30 | nelneq 2939 | . . . . 5 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → ¬ ∅ = -∞) | |
31 | 30 | ad2antrr 724 | . . . 4 ⊢ ((((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) ∧ 𝐹 ⇝ (lim sup‘𝐹)) ∧ ∅ ⇝ ∅) → ¬ ∅ = -∞) |
32 | 29, 31 | pm2.65da 815 | . . 3 ⊢ (((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) ∧ 𝐹 ⇝ (lim sup‘𝐹)) → ¬ ∅ ⇝ ∅) |
33 | 19, 32 | pm2.65da 815 | . 2 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → ¬ 𝐹 ⇝ (lim sup‘𝐹)) |
34 | 10, 17, 33 | 3jca 1124 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∅c0 4293 class class class wbr 5068 dom cdm 5557 Rel wrel 5562 ⟶wf 6353 ‘cfv 6357 ℂcc 10537 ℝcr 10538 -∞cmnf 10675 ℤcz 11984 ℤ≥cuz 12246 lim supclsp 14829 ⇝ cli 14843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 |
This theorem is referenced by: (None) |
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