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| Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupequz | Structured version Visualization version GIF version | ||
| Description: Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) | 
| Ref | Expression | 
|---|---|
| limsupequz.1 | ⊢ Ⅎ𝑘𝜑 | 
| limsupequz.2 | ⊢ Ⅎ𝑘𝐹 | 
| limsupequz.3 | ⊢ Ⅎ𝑘𝐺 | 
| limsupequz.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| limsupequz.5 | ⊢ (𝜑 → 𝐹 Fn (ℤ≥‘𝑀)) | 
| limsupequz.6 | ⊢ (𝜑 → 𝑁 ∈ ℤ) | 
| limsupequz.7 | ⊢ (𝜑 → 𝐺 Fn (ℤ≥‘𝑁)) | 
| limsupequz.8 | ⊢ (𝜑 → 𝐾 ∈ ℤ) | 
| limsupequz.9 | ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘)) | 
| Ref | Expression | 
|---|---|
| limsupequz | ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfv 1914 | . 2 ⊢ Ⅎ𝑗𝜑 | |
| 2 | limsupequz.4 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 3 | limsupequz.5 | . 2 ⊢ (𝜑 → 𝐹 Fn (ℤ≥‘𝑀)) | |
| 4 | limsupequz.6 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
| 5 | limsupequz.7 | . 2 ⊢ (𝜑 → 𝐺 Fn (ℤ≥‘𝑁)) | |
| 6 | limsupequz.8 | . 2 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
| 7 | limsupequz.1 | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
| 8 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ (ℤ≥‘𝐾) | |
| 9 | 7, 8 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) | 
| 10 | limsupequz.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
| 11 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
| 12 | 10, 11 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) | 
| 13 | limsupequz.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
| 14 | 13, 11 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) | 
| 15 | 12, 14 | nfeq 2919 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗) | 
| 16 | 9, 15 | nfim 1896 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑗) = (𝐺‘𝑗)) | 
| 17 | eleq1w 2824 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (ℤ≥‘𝐾) ↔ 𝑗 ∈ (ℤ≥‘𝐾))) | |
| 18 | 17 | anbi2d 630 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ↔ (𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)))) | 
| 19 | fveq2 6906 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
| 20 | fveq2 6906 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
| 21 | 19, 20 | eqeq12d 2753 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑗) = (𝐺‘𝑗))) | 
| 22 | 18, 21 | imbi12d 344 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘)) ↔ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑗) = (𝐺‘𝑗)))) | 
| 23 | limsupequz.9 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘)) | |
| 24 | 16, 22, 23 | chvarfv 2240 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑗) = (𝐺‘𝑗)) | 
| 25 | 1, 2, 3, 4, 5, 6, 24 | limsupequzlem 45737 | 1 ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 Fn wfn 6556 ‘cfv 6561 ℤcz 12613 ℤ≥cuz 12878 lim supclsp 15506 | 
| 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-rep 5279 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-tp 4631 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-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 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-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-ico 13393 df-limsup 15507 | 
| This theorem is referenced by: limsupequzmptlem 45743 smflimsuplem2 46836 | 
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