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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 15540 | . . . . 5 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹)) | |
2 | 1 | biimpi 215 | . . . 4 ⊢ (𝐹 ∈ dom ⇝ → 𝐹 ⇝ ( ⇝ ‘𝐹)) |
3 | 2 | adantl 480 | . . 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 45100 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
11 | 10 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
12 | 4, 11 | mpbid 231 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ dom ⇝ ) |
13 | climdm 15540 | . . . . 5 ⊢ (𝐺 ∈ dom ⇝ ↔ 𝐺 ⇝ ( ⇝ ‘𝐺)) | |
14 | 12, 13 | sylib 217 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ⇝ ( ⇝ ‘𝐺)) |
15 | 7 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ 𝑊) |
16 | 6 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ 𝑉) |
17 | 8 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ) |
18 | 9 | eqcomd 2734 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘𝑘)) |
19 | 18 | adantlr 713 | . . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘𝑘)) |
20 | 5, 15, 16, 17, 19 | climeq 15553 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐺 ⇝ ( ⇝ ‘𝐺) ↔ 𝐹 ⇝ ( ⇝ ‘𝐺))) |
21 | 14, 20 | mpbid 231 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘𝐺)) |
22 | climuni 15538 | . . 3 ⊢ ((𝐹 ⇝ ( ⇝ ‘𝐹) ∧ 𝐹 ⇝ ( ⇝ ‘𝐺)) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) | |
23 | 3, 21, 22 | syl2anc 582 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
24 | ndmfv 6937 | . . . 4 ⊢ (¬ 𝐹 ∈ dom ⇝ → ( ⇝ ‘𝐹) = ∅) | |
25 | 24 | adantl 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ∅) |
26 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐹 ∈ dom ⇝ ) | |
27 | 10 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
28 | 26, 27 | mtbid 323 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐺 ∈ dom ⇝ ) |
29 | ndmfv 6937 | . . . 4 ⊢ (¬ 𝐺 ∈ dom ⇝ → ( ⇝ ‘𝐺) = ∅) | |
30 | 28, 29 | syl 17 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐺) = ∅) |
31 | 25, 30 | eqtr4d 2771 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
32 | 23, 31 | pm2.61dan 811 | 1 ⊢ (𝜑 → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∅c0 4326 class class class wbr 5152 dom cdm 5682 ‘cfv 6553 ℤcz 12598 ℤ≥cuz 12862 ⇝ cli 15470 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-sup 9475 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-z 12599 df-uz 12863 df-rp 13017 df-seq 14009 df-exp 14069 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-clim 15474 |
This theorem is referenced by: climfveqmpt 45106 climfveqmpt3 45117 |
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