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Theorem rlimadd 15591

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.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
rlimadd	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ⇝_𝑟 (𝐷 + 𝐸))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐷 𝜑,𝑥 𝑥,𝐸 Allowed substitution hints: 𝐵(𝑥) 𝐶(𝑥) 𝑉(𝑥)

Proof of Theorem rlimadd Dummy variables 𝑤 𝑣 𝑦 𝑧 𝑢 are mutually distinct and distinct from all other variables.

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 15556	. 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 15556	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
7	3, 6	addcld 11234	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℂ)
8		rlimcl 15451	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ⇝_𝑟 𝐷 → 𝐷 ∈ ℂ)
9	2, 8	syl 17	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ)
10		rlimcl 15451	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝_𝑟 𝐸 → 𝐸 ∈ ℂ)
11	5, 10	syl 17	. . 3 ⊢ (𝜑 → 𝐸 ∈ ℂ)
12	9, 11	addcld 11234	. 2 ⊢ (𝜑 → (𝐷 + 𝐸) ∈ ℂ)
13		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
14	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐷 ∈ ℂ)
15	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐸 ∈ ℂ)
16		addcn2 15542	. . 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 1368	. 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 15538	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ⇝_𝑟 (𝐷 + 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 class class class wbr 5141 ↦ cmpt 5224 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 + caddc 11112 < clt 11249 − cmin 11445 ℝ⁺crp 12977 abscabs 15185 ⇝_𝑟 crli 15433

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-pm 8822 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-rlim 15437

This theorem is referenced by: caucvgr 15626 fsumrlim 15761 logfacrlim 27108 logexprlim 27109 chpchtlim 27363 selberglem2 27430 signsplypnf 34091

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