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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climsubmpt | Structured version Visualization version GIF version |
Description: Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
climsubmpt.k | ⊢ Ⅎ𝑘𝜑 |
climsubmpt.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climsubmpt.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climsubmpt.a | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
climsubmpt.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) |
climsubmpt.c | ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴) ⇝ 𝐶) |
climsubmpt.d | ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐷) |
Ref | Expression |
---|---|
climsubmpt | ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐴 − 𝐵)) ⇝ (𝐶 − 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climsubmpt.z | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climsubmpt.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climsubmpt.c | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴) ⇝ 𝐶) | |
4 | 1 | fvexi 6910 | . . . 4 ⊢ 𝑍 ∈ V |
5 | 4 | mptex 7235 | . . 3 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐴 − 𝐵)) ∈ V |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐴 − 𝐵)) ∈ V) |
7 | climsubmpt.d | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐷) | |
8 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
9 | climsubmpt.k | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
10 | nfv 1909 | . . . . . . 7 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
11 | 9, 10 | nfan 1894 | . . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
12 | nfcv 2891 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑗 | |
13 | 12 | nfcsb1 3913 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 |
14 | 13 | nfel1 2908 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ |
15 | 11, 14 | nfim 1891 | . . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) |
16 | eleq1w 2808 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
17 | 16 | anbi2d 628 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
18 | csbeq1a 3903 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴) | |
19 | 18 | eleq1d 2810 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℂ ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ)) |
20 | 17, 19 | imbi12d 343 | . . . . 5 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ))) |
21 | climsubmpt.a | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
22 | 15, 20, 21 | chvarfv 2228 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) |
23 | eqid 2725 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐴) = (𝑘 ∈ 𝑍 ↦ 𝐴) | |
24 | 12, 13, 18, 23 | fvmptf 7025 | . . . 4 ⊢ ((𝑗 ∈ 𝑍 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
25 | 8, 22, 24 | syl2anc 582 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
26 | 25, 22 | eqeltrd 2825 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) ∈ ℂ) |
27 | 12 | nfcsb1 3913 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 |
28 | nfcv 2891 | . . . . . . 7 ⊢ Ⅎ𝑘ℂ | |
29 | 27, 28 | nfel 2906 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ |
30 | 11, 29 | nfim 1891 | . . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ) |
31 | csbeq1a 3903 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) | |
32 | 31 | eleq1d 2810 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ)) |
33 | 17, 32 | imbi12d 343 | . . . . 5 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ))) |
34 | climsubmpt.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) | |
35 | 30, 33, 34 | chvarfv 2228 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ) |
36 | eqid 2725 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐵) = (𝑘 ∈ 𝑍 ↦ 𝐵) | |
37 | 12, 27, 31, 36 | fvmptf 7025 | . . . 4 ⊢ ((𝑗 ∈ 𝑍 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
38 | 8, 35, 37 | syl2anc 582 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
39 | 38, 35 | eqeltrd 2825 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑗) ∈ ℂ) |
40 | ovexd 7454 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (⦋𝑗 / 𝑘⦌𝐴 − ⦋𝑗 / 𝑘⦌𝐵) ∈ V) | |
41 | nfcv 2891 | . . . . . 6 ⊢ Ⅎ𝑘 − | |
42 | 13, 41, 27 | nfov 7449 | . . . . 5 ⊢ Ⅎ𝑘(⦋𝑗 / 𝑘⦌𝐴 − ⦋𝑗 / 𝑘⦌𝐵) |
43 | 18, 31 | oveq12d 7437 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐴 − 𝐵) = (⦋𝑗 / 𝑘⦌𝐴 − ⦋𝑗 / 𝑘⦌𝐵)) |
44 | eqid 2725 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐴 − 𝐵)) = (𝑘 ∈ 𝑍 ↦ (𝐴 − 𝐵)) | |
45 | 12, 42, 43, 44 | fvmptf 7025 | . . . 4 ⊢ ((𝑗 ∈ 𝑍 ∧ (⦋𝑗 / 𝑘⦌𝐴 − ⦋𝑗 / 𝑘⦌𝐵) ∈ V) → ((𝑘 ∈ 𝑍 ↦ (𝐴 − 𝐵))‘𝑗) = (⦋𝑗 / 𝑘⦌𝐴 − ⦋𝑗 / 𝑘⦌𝐵)) |
46 | 8, 40, 45 | syl2anc 582 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐴 − 𝐵))‘𝑗) = (⦋𝑗 / 𝑘⦌𝐴 − ⦋𝑗 / 𝑘⦌𝐵)) |
47 | 25, 38 | oveq12d 7437 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) − ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑗)) = (⦋𝑗 / 𝑘⦌𝐴 − ⦋𝑗 / 𝑘⦌𝐵)) |
48 | 46, 47 | eqtr4d 2768 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐴 − 𝐵))‘𝑗) = (((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) − ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑗))) |
49 | 1, 2, 3, 6, 7, 26, 39, 48 | climsub 15614 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐴 − 𝐵)) ⇝ (𝐶 − 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Vcvv 3461 ⦋csb 3889 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 − cmin 11476 ℤcz 12591 ℤ≥cuz 12855 ⇝ cli 15464 |
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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
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 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 |
This theorem is referenced by: climsubc2mpt 45187 climsubc1mpt 45188 |
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