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Mirrors > Home > MPE Home > Th. List > Mathboxes > climlimsupcex | Structured version Visualization version GIF version |
Description: Counterexample for climlimsup 42047, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 10558 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 6563 | . . . 4 ⊢ ∅:∅⟶ℝ | |
2 | climlimsupcex.3 | . . . . 5 ⊢ 𝐹 = ∅ | |
3 | climlimsupcex.2 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | climlimsupcex.1 | . . . . . . 7 ⊢ ¬ 𝑀 ∈ ℤ | |
5 | uz0 41692 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (ℤ≥‘𝑀) = ∅ |
7 | 3, 6 | eqtri 2847 | . . . . 5 ⊢ 𝑍 = ∅ |
8 | 2, 7 | feq12i 6510 | . . . 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 14852 | . . . . 5 ⊢ Rel ⇝ | |
12 | 11 | a1i 11 | . . . 4 ⊢ (∅ ∈ ℂ → Rel ⇝ ) |
13 | 0cnv 42029 | . . . . 5 ⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) | |
14 | 2, 13 | eqbrtrid 5104 | . . . 4 ⊢ (∅ ∈ ℂ → 𝐹 ⇝ ∅) |
15 | releldm 5817 | . . . 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 6676 | . . . . . . . . . 10 ⊢ (lim sup‘𝐹) = (lim sup‘∅) |
22 | limsup0 41981 | . . . . . . . . . 10 ⊢ (lim sup‘∅) = -∞ | |
23 | 21, 22 | eqtri 2847 | . . . . . . . . 9 ⊢ (lim sup‘𝐹) = -∞ |
24 | 2, 23 | breq12i 5078 | . . . . . . . 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 14912 | . . . . . 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 2940 | . . . . 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 1536 ∈ wcel 2113 ∅c0 4294 class class class wbr 5069 dom cdm 5558 Rel wrel 5563 ⟶wf 6354 ‘cfv 6358 ℂcc 10538 ℝcr 10539 -∞cmnf 10676 ℤcz 11984 ℤ≥cuz 12246 lim supclsp 14830 ⇝ cli 14844 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 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 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 |
This theorem is referenced by: (None) |
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