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Mirrors > Home > MPE Home > Th. List > climsub | Structured version Visualization version GIF version |
Description: Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.) |
Ref | Expression |
---|---|
climadd.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climadd.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climadd.4 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climadd.6 | ⊢ (𝜑 → 𝐻 ∈ 𝑋) |
climadd.7 | ⊢ (𝜑 → 𝐺 ⇝ 𝐵) |
climadd.8 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
climadd.9 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
climsub.h | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘))) |
Ref | Expression |
---|---|
climsub | ⊢ (𝜑 → 𝐻 ⇝ (𝐴 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climadd.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climadd.2 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climadd.4 | . . 3 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
4 | climcl 14841 | . . 3 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
6 | climadd.7 | . . 3 ⊢ (𝜑 → 𝐺 ⇝ 𝐵) | |
7 | climcl 14841 | . . 3 ⊢ (𝐺 ⇝ 𝐵 → 𝐵 ∈ ℂ) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
9 | subcl 10871 | . . 3 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 − 𝑣) ∈ ℂ) | |
10 | 9 | adantl 484 | . 2 ⊢ ((𝜑 ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 − 𝑣) ∈ ℂ) |
11 | climadd.6 | . 2 ⊢ (𝜑 → 𝐻 ∈ 𝑋) | |
12 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+) | |
13 | 5 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝐴 ∈ ℂ) |
14 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝐵 ∈ ℂ) |
15 | subcn2 14936 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢 − 𝑣) − (𝐴 − 𝐵))) < 𝑥)) | |
16 | 12, 13, 14, 15 | syl3anc 1367 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢 − 𝑣) − (𝐴 − 𝐵))) < 𝑥)) |
17 | climadd.8 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
18 | climadd.9 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) | |
19 | climsub.h | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘))) | |
20 | 1, 2, 5, 8, 10, 3, 6, 11, 16, 17, 18, 19 | climcn2 14934 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 class class class wbr 5052 ‘cfv 6341 (class class class)co 7142 ℂcc 10521 < clt 10661 − cmin 10856 ℤcz 11968 ℤ≥cuz 12230 ℝ+crp 12376 abscabs 14578 ⇝ cli 14826 |
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 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-sup 8892 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-n0 11885 df-z 11969 df-uz 12231 df-rp 12377 df-seq 13360 df-exp 13420 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-clim 14830 |
This theorem is referenced by: climsubc1 14979 climsubc2 14980 climle 14981 supcvg 15196 mbfi1flimlem 24306 ulmdvlem1 24974 abelthlem6 25010 atantayl 25501 lgamcvg2 25618 hashnzfzclim 40744 binomcxplemrat 40772 climsubmpt 42031 ioodvbdlimc2lem 42309 |
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