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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climeldmeq | Structured version Visualization version GIF version |
Description: Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
climeldmeq.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climeldmeq.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
climeldmeq.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
climeldmeq.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climeldmeq.e | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
Ref | Expression |
---|---|
climeldmeq | ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climeldmeq.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
2 | 1 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ 𝑊) |
3 | fvexd 6915 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) ∈ V) | |
4 | climdm 15551 | . . . . . . 7 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹)) | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))) |
6 | 5 | biimpa 475 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘𝐹)) |
7 | climeldmeq.z | . . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
8 | climeldmeq.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
9 | climeldmeq.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
10 | climeldmeq.e | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) | |
11 | 7, 8, 1, 9, 10 | climeq 15564 | . . . . . 6 ⊢ (𝜑 → (𝐹 ⇝ ( ⇝ ‘𝐹) ↔ 𝐺 ⇝ ( ⇝ ‘𝐹))) |
12 | 11 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐹 ⇝ ( ⇝ ‘𝐹) ↔ 𝐺 ⇝ ( ⇝ ‘𝐹))) |
13 | 6, 12 | mpbid 231 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ⇝ ( ⇝ ‘𝐹)) |
14 | breldmg 5915 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ ( ⇝ ‘𝐹) ∈ V ∧ 𝐺 ⇝ ( ⇝ ‘𝐹)) → 𝐺 ∈ dom ⇝ ) | |
15 | 2, 3, 13, 14 | syl3anc 1368 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ dom ⇝ ) |
16 | 15 | ex 411 | . 2 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ → 𝐺 ∈ dom ⇝ )) |
17 | 8 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ dom ⇝ ) → 𝐹 ∈ 𝑉) |
18 | fvexd 6915 | . . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ dom ⇝ ) → ( ⇝ ‘𝐺) ∈ V) | |
19 | climdm 15551 | . . . . . . 7 ⊢ (𝐺 ∈ dom ⇝ ↔ 𝐺 ⇝ ( ⇝ ‘𝐺)) | |
20 | 19 | biimpi 215 | . . . . . 6 ⊢ (𝐺 ∈ dom ⇝ → 𝐺 ⇝ ( ⇝ ‘𝐺)) |
21 | 20 | adantl 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ dom ⇝ ) → 𝐺 ⇝ ( ⇝ ‘𝐺)) |
22 | 10 | eqcomd 2731 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘𝑘)) |
23 | 7, 1, 8, 9, 22 | climeq 15564 | . . . . . 6 ⊢ (𝜑 → (𝐺 ⇝ ( ⇝ ‘𝐺) ↔ 𝐹 ⇝ ( ⇝ ‘𝐺))) |
24 | 23 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ dom ⇝ ) → (𝐺 ⇝ ( ⇝ ‘𝐺) ↔ 𝐹 ⇝ ( ⇝ ‘𝐺))) |
25 | 21, 24 | mpbid 231 | . . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘𝐺)) |
26 | breldmg 5915 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ ( ⇝ ‘𝐺) ∈ V ∧ 𝐹 ⇝ ( ⇝ ‘𝐺)) → 𝐹 ∈ dom ⇝ ) | |
27 | 17, 18, 25, 26 | syl3anc 1368 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ ) |
28 | 27 | ex 411 | . 2 ⊢ (𝜑 → (𝐺 ∈ dom ⇝ → 𝐹 ∈ dom ⇝ )) |
29 | 16, 28 | impbid 211 | 1 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 class class class wbr 5152 dom cdm 5681 ‘cfv 6553 ℤcz 12605 ℤ≥cuz 12869 ⇝ cli 15481 |
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 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 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 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 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 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-sup 9481 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-n0 12520 df-z 12606 df-uz 12870 df-rp 13024 df-seq 14017 df-exp 14077 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 |
This theorem is referenced by: climeldmeqmpt 45226 climfveq 45227 climfveqf 45238 climeldmeqf 45241 climeldmeqmpt3 45247 |
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