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 14913 | . . . . 5 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹)) | |
2 | 1 | biimpi 218 | . . . 4 ⊢ (𝐹 ∈ dom ⇝ → 𝐹 ⇝ ( ⇝ ‘𝐹)) |
3 | 2 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘𝐹)) |
4 | 3, 1 | sylibr 236 | . . . . . 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 41953 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
11 | 10 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
12 | 4, 11 | mpbid 234 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ dom ⇝ ) |
13 | climdm 14913 | . . . . 5 ⊢ (𝐺 ∈ dom ⇝ ↔ 𝐺 ⇝ ( ⇝ ‘𝐺)) | |
14 | 12, 13 | sylib 220 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ⇝ ( ⇝ ‘𝐺)) |
15 | 7 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ 𝑊) |
16 | 6 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ 𝑉) |
17 | 8 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ) |
18 | 9 | eqcomd 2829 | . . . . . 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 14926 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐺 ⇝ ( ⇝ ‘𝐺) ↔ 𝐹 ⇝ ( ⇝ ‘𝐺))) |
21 | 14, 20 | mpbid 234 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘𝐺)) |
22 | climuni 14911 | . . 3 ⊢ ((𝐹 ⇝ ( ⇝ ‘𝐹) ∧ 𝐹 ⇝ ( ⇝ ‘𝐺)) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) | |
23 | 3, 21, 22 | syl2anc 586 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
24 | ndmfv 6702 | . . . 4 ⊢ (¬ 𝐹 ∈ dom ⇝ → ( ⇝ ‘𝐹) = ∅) | |
25 | 24 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ∅) |
26 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐹 ∈ dom ⇝ ) | |
27 | 10 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
28 | 26, 27 | mtbid 326 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐺 ∈ dom ⇝ ) |
29 | ndmfv 6702 | . . . 4 ⊢ (¬ 𝐺 ∈ dom ⇝ → ( ⇝ ‘𝐺) = ∅) | |
30 | 28, 29 | syl 17 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐺) = ∅) |
31 | 25, 30 | eqtr4d 2861 | . 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∅c0 4293 class class class wbr 5068 dom cdm 5557 ‘cfv 6357 ℤcz 11984 ℤ≥cuz 12246 ⇝ cli 14843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 |
This theorem is referenced by: climfveqmpt 41959 climfveqmpt3 41970 |
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