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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 15442 | . . . . 5 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹)) | |
2 | 1 | biimpi 215 | . . . 4 ⊢ (𝐹 ∈ dom ⇝ → 𝐹 ⇝ ( ⇝ ‘𝐹)) |
3 | 2 | adantl 483 | . . 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 43992 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
11 | 10 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
12 | 4, 11 | mpbid 231 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ dom ⇝ ) |
13 | climdm 15442 | . . . . 5 ⊢ (𝐺 ∈ dom ⇝ ↔ 𝐺 ⇝ ( ⇝ ‘𝐺)) | |
14 | 12, 13 | sylib 217 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ⇝ ( ⇝ ‘𝐺)) |
15 | 7 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ 𝑊) |
16 | 6 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ 𝑉) |
17 | 8 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ) |
18 | 9 | eqcomd 2739 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘𝑘)) |
19 | 18 | adantlr 714 | . . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘𝑘)) |
20 | 5, 15, 16, 17, 19 | climeq 15455 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐺 ⇝ ( ⇝ ‘𝐺) ↔ 𝐹 ⇝ ( ⇝ ‘𝐺))) |
21 | 14, 20 | mpbid 231 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘𝐺)) |
22 | climuni 15440 | . . 3 ⊢ ((𝐹 ⇝ ( ⇝ ‘𝐹) ∧ 𝐹 ⇝ ( ⇝ ‘𝐺)) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) | |
23 | 3, 21, 22 | syl2anc 585 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
24 | ndmfv 6878 | . . . 4 ⊢ (¬ 𝐹 ∈ dom ⇝ → ( ⇝ ‘𝐹) = ∅) | |
25 | 24 | adantl 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ∅) |
26 | simpr 486 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐹 ∈ dom ⇝ ) | |
27 | 10 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
28 | 26, 27 | mtbid 324 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐺 ∈ dom ⇝ ) |
29 | ndmfv 6878 | . . . 4 ⊢ (¬ 𝐺 ∈ dom ⇝ → ( ⇝ ‘𝐺) = ∅) | |
30 | 28, 29 | syl 17 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐺) = ∅) |
31 | 25, 30 | eqtr4d 2776 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
32 | 23, 31 | pm2.61dan 812 | 1 ⊢ (𝜑 → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∅c0 4283 class class class wbr 5106 dom cdm 5634 ‘cfv 6497 ℤcz 12504 ℤ≥cuz 12768 ⇝ cli 15372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 |
This theorem is referenced by: climfveqmpt 43998 climfveqmpt3 44009 |
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