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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climfveq | Structured version Visualization version GIF version |
Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
climfveq.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climfveq.2 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
climfveq.3 | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
climfveq.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climfveq.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
Ref | Expression |
---|---|
climfveq | ⊢ (𝜑 → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climdm 15504 | . . . . 5 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹)) | |
2 | 1 | biimpi 215 | . . . 4 ⊢ (𝐹 ∈ dom ⇝ → 𝐹 ⇝ ( ⇝ ‘𝐹)) |
3 | 2 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘𝐹)) |
4 | 3, 1 | sylibr 233 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ ) |
5 | climfveq.1 | . . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | climfveq.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
7 | climfveq.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
8 | climfveq.4 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
9 | climfveq.5 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) | |
10 | 5, 6, 7, 8, 9 | climeldmeq 44953 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
12 | 4, 11 | mpbid 231 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ dom ⇝ ) |
13 | climdm 15504 | . . . . 5 ⊢ (𝐺 ∈ dom ⇝ ↔ 𝐺 ⇝ ( ⇝ ‘𝐺)) | |
14 | 12, 13 | sylib 217 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ⇝ ( ⇝ ‘𝐺)) |
15 | 7 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ 𝑊) |
16 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ 𝑉) |
17 | 8 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ) |
18 | 9 | eqcomd 2732 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘𝑘)) |
19 | 18 | adantlr 712 | . . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘𝑘)) |
20 | 5, 15, 16, 17, 19 | climeq 15517 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐺 ⇝ ( ⇝ ‘𝐺) ↔ 𝐹 ⇝ ( ⇝ ‘𝐺))) |
21 | 14, 20 | mpbid 231 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘𝐺)) |
22 | climuni 15502 | . . 3 ⊢ ((𝐹 ⇝ ( ⇝ ‘𝐹) ∧ 𝐹 ⇝ ( ⇝ ‘𝐺)) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) | |
23 | 3, 21, 22 | syl2anc 583 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
24 | ndmfv 6920 | . . . 4 ⊢ (¬ 𝐹 ∈ dom ⇝ → ( ⇝ ‘𝐹) = ∅) | |
25 | 24 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ∅) |
26 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐹 ∈ dom ⇝ ) | |
27 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
28 | 26, 27 | mtbid 324 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐺 ∈ dom ⇝ ) |
29 | ndmfv 6920 | . . . 4 ⊢ (¬ 𝐺 ∈ dom ⇝ → ( ⇝ ‘𝐺) = ∅) | |
30 | 28, 29 | syl 17 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐺) = ∅) |
31 | 25, 30 | eqtr4d 2769 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
32 | 23, 31 | pm2.61dan 810 | 1 ⊢ (𝜑 → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∅c0 4317 class class class wbr 5141 dom cdm 5669 ‘cfv 6537 ℤcz 12562 ℤ≥cuz 12826 ⇝ cli 15434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 |
This theorem is referenced by: climfveqmpt 44959 climfveqmpt3 44970 |
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