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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 15447 | . . 3 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
6 | climadd.7 | . . 3 ⊢ (𝜑 → 𝐺 ⇝ 𝐵) | |
7 | climcl 15447 | . . 3 ⊢ (𝐺 ⇝ 𝐵 → 𝐵 ∈ ℂ) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
9 | subcl 11460 | . . 3 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 − 𝑣) ∈ ℂ) | |
10 | 9 | adantl 481 | . 2 ⊢ ((𝜑 ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 − 𝑣) ∈ ℂ) |
11 | climadd.6 | . 2 ⊢ (𝜑 → 𝐻 ∈ 𝑋) | |
12 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+) | |
13 | 5 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝐴 ∈ ℂ) |
14 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝐵 ∈ ℂ) |
15 | subcn2 15543 | . . 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 1368 | . 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 15541 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 < clt 11249 − cmin 11445 ℤcz 12559 ℤ≥cuz 12823 ℝ+crp 12977 abscabs 15185 ⇝ cli 15432 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 |
This theorem is referenced by: climsubc1 15586 climsubc2 15587 climle 15588 supcvg 15806 mbfi1flimlem 25603 ulmdvlem1 26287 abelthlem6 26324 atantayl 26820 lgamcvg2 26938 hashnzfzclim 43638 binomcxplemrat 43666 climsubmpt 44929 ioodvbdlimc2lem 45203 |
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