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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climlimsupcex | Structured version Visualization version GIF version |
Description: Counterexample for climlimsup 45716, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 11171 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 6790 | . . . 4 ⊢ ∅:∅⟶ℝ | |
2 | climlimsupcex.3 | . . . . 5 ⊢ 𝐹 = ∅ | |
3 | climlimsupcex.2 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | climlimsupcex.1 | . . . . . . 7 ⊢ ¬ 𝑀 ∈ ℤ | |
5 | uz0 45362 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (ℤ≥‘𝑀) = ∅ |
7 | 3, 6 | eqtri 2763 | . . . . 5 ⊢ 𝑍 = ∅ |
8 | 2, 7 | feq12i 6730 | . . . 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 15525 | . . . . 5 ⊢ Rel ⇝ | |
12 | 11 | a1i 11 | . . . 4 ⊢ (∅ ∈ ℂ → Rel ⇝ ) |
13 | 0cnv 45698 | . . . . 5 ⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) | |
14 | 2, 13 | eqbrtrid 5183 | . . . 4 ⊢ (∅ ∈ ℂ → 𝐹 ⇝ ∅) |
15 | releldm 5958 | . . . 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 6910 | . . . . . . . . . 10 ⊢ (lim sup‘𝐹) = (lim sup‘∅) |
22 | limsup0 45650 | . . . . . . . . . 10 ⊢ (lim sup‘∅) = -∞ | |
23 | 21, 22 | eqtri 2763 | . . . . . . . . 9 ⊢ (lim sup‘𝐹) = -∞ |
24 | 2, 23 | breq12i 5157 | . . . . . . . 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 15585 | . . . . . 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 2863 | . . . . 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 1127 | 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 1086 = wceq 1537 ∈ wcel 2106 ∅c0 4339 class class class wbr 5148 dom cdm 5689 Rel wrel 5694 ⟶wf 6559 ‘cfv 6563 ℂcc 11151 ℝcr 11152 -∞cmnf 11291 ℤcz 12611 ℤ≥cuz 12876 lim supclsp 15503 ⇝ cli 15517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 |
This theorem is referenced by: (None) |
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