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Mirrors > Home > MPE Home > Th. List > rlimadd | Structured version Visualization version GIF version |
Description: Limit of the sum of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Ref | Expression |
---|---|
rlimadd.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
rlimadd.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) |
rlimadd.5 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷) |
rlimadd.6 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸) |
Ref | Expression |
---|---|
rlimadd | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ⇝𝑟 (𝐷 + 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimadd.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
2 | rlimadd.5 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷) | |
3 | 1, 2 | rlimmptrcl 15056 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
4 | rlimadd.4 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) | |
5 | rlimadd.6 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸) | |
6 | 4, 5 | rlimmptrcl 15056 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
7 | 3, 6 | addcld 10739 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℂ) |
8 | rlimcl 14951 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷 → 𝐷 ∈ ℂ) | |
9 | 2, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
10 | rlimcl 14951 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸 → 𝐸 ∈ ℂ) | |
11 | 5, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
12 | 9, 11 | addcld 10739 | . 2 ⊢ (𝜑 → (𝐷 + 𝐸) ∈ ℂ) |
13 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+) | |
14 | 9 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝐷 ∈ ℂ) |
15 | 11 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝐸 ∈ ℂ) |
16 | addcn2 15042 | . . 3 ⊢ ((𝑦 ∈ ℝ+ ∧ 𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ) → ∃𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝐷)) < 𝑧 ∧ (abs‘(𝑣 − 𝐸)) < 𝑤) → (abs‘((𝑢 + 𝑣) − (𝐷 + 𝐸))) < 𝑦)) | |
17 | 13, 14, 15, 16 | syl3anc 1372 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝐷)) < 𝑧 ∧ (abs‘(𝑣 − 𝐸)) < 𝑤) → (abs‘((𝑢 + 𝑣) − (𝐷 + 𝐸))) < 𝑦)) |
18 | 3, 6, 7, 12, 2, 5, 17 | rlimcn3 15038 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ⇝𝑟 (𝐷 + 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2113 ∀wral 3053 ∃wrex 3054 class class class wbr 5031 ↦ cmpt 5111 ‘cfv 6340 (class class class)co 7171 ℂcc 10614 + caddc 10619 < clt 10754 − cmin 10949 ℝ+crp 12473 abscabs 14684 ⇝𝑟 crli 14933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 ax-pre-sup 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-2nd 7716 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-er 8321 df-pm 8441 df-en 8557 df-dom 8558 df-sdom 8559 df-sup 8980 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-div 11377 df-nn 11718 df-2 11780 df-3 11781 df-n0 11978 df-z 12064 df-uz 12326 df-rp 12474 df-seq 13462 df-exp 13523 df-cj 14549 df-re 14550 df-im 14551 df-sqrt 14685 df-abs 14686 df-rlim 14937 |
This theorem is referenced by: caucvgr 15126 fsumrlim 15260 logfacrlim 25960 logexprlim 25961 chpchtlim 26215 selberglem2 26282 signsplypnf 32099 |
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