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Mirrors > Home > MPE Home > Th. List > Mathboxes > climlimsupcex | Structured version Visualization version GIF version |
Description: Counterexample for climlimsup 39992, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 9954 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 6086 | . . . 4 ⊢ ∅:∅⟶ℝ | |
2 | climlimsupcex.3 | . . . . 5 ⊢ 𝐹 = ∅ | |
3 | climlimsupcex.2 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | climlimsupcex.1 | . . . . . . 7 ⊢ ¬ 𝑀 ∈ ℤ | |
5 | uz0 39639 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (ℤ≥‘𝑀) = ∅ |
7 | 3, 6 | eqtri 2644 | . . . . 5 ⊢ 𝑍 = ∅ |
8 | 2, 7 | feq12i 6038 | . . . 4 ⊢ (𝐹:𝑍⟶ℝ ↔ ∅:∅⟶ℝ) |
9 | 1, 8 | mpbir 221 | . . 3 ⊢ 𝐹:𝑍⟶ℝ |
10 | 9 | a1i 11 | . 2 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ) |
11 | climrel 14223 | . . . . 5 ⊢ Rel ⇝ | |
12 | 11 | a1i 11 | . . . 4 ⊢ (∅ ∈ ℂ → Rel ⇝ ) |
13 | 0cnv 39974 | . . . . 5 ⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) | |
14 | 2, 13 | syl5eqbr 4688 | . . . 4 ⊢ (∅ ∈ ℂ → 𝐹 ⇝ ∅) |
15 | releldm 5358 | . . . 4 ⊢ ((Rel ⇝ ∧ 𝐹 ⇝ ∅) → 𝐹 ∈ dom ⇝ ) | |
16 | 12, 14, 15 | syl2anc 693 | . . 3 ⊢ (∅ ∈ ℂ → 𝐹 ∈ dom ⇝ ) |
17 | 16 | adantr 481 | . 2 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → 𝐹 ∈ dom ⇝ ) |
18 | 13 | adantr 481 | . . . 4 ⊢ ((∅ ∈ ℂ ∧ 𝐹 ⇝ (lim sup‘𝐹)) → ∅ ⇝ ∅) |
19 | 18 | adantlr 751 | . . 3 ⊢ (((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) ∧ 𝐹 ⇝ (lim sup‘𝐹)) → ∅ ⇝ ∅) |
20 | simpr 477 | . . . . . 6 ⊢ ((𝐹 ⇝ (lim sup‘𝐹) ∧ ∅ ⇝ ∅) → ∅ ⇝ ∅) | |
21 | 2 | fveq2i 6194 | . . . . . . . . . 10 ⊢ (lim sup‘𝐹) = (lim sup‘∅) |
22 | limsup0 39926 | . . . . . . . . . 10 ⊢ (lim sup‘∅) = -∞ | |
23 | 21, 22 | eqtri 2644 | . . . . . . . . 9 ⊢ (lim sup‘𝐹) = -∞ |
24 | 2, 23 | breq12i 4662 | . . . . . . . 8 ⊢ (𝐹 ⇝ (lim sup‘𝐹) ↔ ∅ ⇝ -∞) |
25 | 24 | biimpi 206 | . . . . . . 7 ⊢ (𝐹 ⇝ (lim sup‘𝐹) → ∅ ⇝ -∞) |
26 | 25 | adantr 481 | . . . . . 6 ⊢ ((𝐹 ⇝ (lim sup‘𝐹) ∧ ∅ ⇝ ∅) → ∅ ⇝ -∞) |
27 | climuni 14283 | . . . . . 6 ⊢ ((∅ ⇝ ∅ ∧ ∅ ⇝ -∞) → ∅ = -∞) | |
28 | 20, 26, 27 | syl2anc 693 | . . . . 5 ⊢ ((𝐹 ⇝ (lim sup‘𝐹) ∧ ∅ ⇝ ∅) → ∅ = -∞) |
29 | 28 | adantll 750 | . . . 4 ⊢ ((((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) ∧ 𝐹 ⇝ (lim sup‘𝐹)) ∧ ∅ ⇝ ∅) → ∅ = -∞) |
30 | nelneq 2725 | . . . . 5 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → ¬ ∅ = -∞) | |
31 | 30 | ad2antrr 762 | . . . 4 ⊢ ((((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) ∧ 𝐹 ⇝ (lim sup‘𝐹)) ∧ ∅ ⇝ ∅) → ¬ ∅ = -∞) |
32 | 29, 31 | pm2.65da 600 | . . 3 ⊢ (((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) ∧ 𝐹 ⇝ (lim sup‘𝐹)) → ¬ ∅ ⇝ ∅) |
33 | 19, 32 | pm2.65da 600 | . 2 ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → ¬ 𝐹 ⇝ (lim sup‘𝐹)) |
34 | 10, 17, 33 | 3jca 1242 | 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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∅c0 3915 class class class wbr 4653 dom cdm 5114 Rel wrel 5119 ⟶wf 5884 ‘cfv 5888 ℂcc 9934 ℝcr 9935 -∞cmnf 10072 ℤcz 11377 ℤ≥cuz 11687 lim supclsp 14201 ⇝ cli 14215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 |
This theorem is referenced by: (None) |
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