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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 15587 | . . . . 5 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹)) | |
2 | 1 | biimpi 216 | . . . 4 ⊢ (𝐹 ∈ dom ⇝ → 𝐹 ⇝ ( ⇝ ‘𝐹)) |
3 | 2 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘𝐹)) |
4 | 3, 1 | sylibr 234 | . . . . . 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 45621 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
12 | 4, 11 | mpbid 232 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ dom ⇝ ) |
13 | climdm 15587 | . . . . 5 ⊢ (𝐺 ∈ dom ⇝ ↔ 𝐺 ⇝ ( ⇝ ‘𝐺)) | |
14 | 12, 13 | sylib 218 | . . . 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 2741 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘𝑘)) |
19 | 18 | adantlr 715 | . . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘𝑘)) |
20 | 5, 15, 16, 17, 19 | climeq 15600 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐺 ⇝ ( ⇝ ‘𝐺) ↔ 𝐹 ⇝ ( ⇝ ‘𝐺))) |
21 | 14, 20 | mpbid 232 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘𝐺)) |
22 | climuni 15585 | . . 3 ⊢ ((𝐹 ⇝ ( ⇝ ‘𝐹) ∧ 𝐹 ⇝ ( ⇝ ‘𝐺)) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) | |
23 | 3, 21, 22 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
24 | ndmfv 6942 | . . . 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 6942 | . . . 4 ⊢ (¬ 𝐺 ∈ dom ⇝ → ( ⇝ ‘𝐺) = ∅) | |
30 | 28, 29 | syl 17 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐺) = ∅) |
31 | 25, 30 | eqtr4d 2778 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
32 | 23, 31 | pm2.61dan 813 | 1 ⊢ (𝜑 → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∅c0 4339 class class class wbr 5148 dom cdm 5689 ‘cfv 6563 ℤcz 12611 ℤ≥cuz 12876 ⇝ 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-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-clim 15521 |
This theorem is referenced by: climfveqmpt 45627 climfveqmpt3 45638 |
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