Theorem rlimdiv 14996

Description: Limit of the quotient 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)
rlimdiv.7	⊢ (𝜑 → 𝐸 ≠ 0)
rlimdiv.8	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 0)

Assertion

Ref	Expression
rlimdiv	⊢ (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 rlimdiv

Dummy variables 𝑤 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rlimadd.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
2		rlimadd.5	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷)
3	1, 2	rlimmptrcl 14958	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
4		rlimadd.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
5		rlimadd.6	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸)
6	4, 5	rlimmptrcl 14958	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
7		rlimdiv.8	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 0)
8	6, 7	reccld 11403	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1 / 𝐶) ∈ ℂ)
9		eldifsn 4712	. . . . . . 7 ⊢ (𝐶 ∈ (ℂ ∖ {0}) ↔ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
10	6, 7, 9	sylanbrc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (ℂ ∖ {0}))
11	10	fmpttd 6873	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶(ℂ ∖ {0}))
12		rlimcl 14854	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸 → 𝐸 ∈ ℂ)
13	5, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℂ)
14		rlimdiv.7	. . . . . 6 ⊢ (𝜑 → 𝐸 ≠ 0)
15		eldifsn 4712	. . . . . 6 ⊢ (𝐸 ∈ (ℂ ∖ {0}) ↔ (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
16	13, 14, 15	sylanbrc 585	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (ℂ ∖ {0}))
17		eldifsn 4712	. . . . . . . 8 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
18		reccl 11299	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / 𝑦) ∈ ℂ)
19	17, 18	sylbi 219	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → (1 / 𝑦) ∈ ℂ)
20	19	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
21	20	fmpttd 6873	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)):(ℂ ∖ {0})⟶ℂ)
22		eqid 2821	. . . . . . . 8 ⊢ (if(1 ≤ ((abs‘𝐸) · 𝑧), 1, ((abs‘𝐸) · 𝑧)) · ((abs‘𝐸) / 2)) = (if(1 ≤ ((abs‘𝐸) · 𝑧), 1, ((abs‘𝐸) · 𝑧)) · ((abs‘𝐸) / 2))
23	22	reccn2 14947	. . . . . . 7 ⊢ ((𝐸 ∈ (ℂ ∖ {0}) ∧ 𝑧 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (ℂ ∖ {0})((abs‘(𝑣 − 𝐸)) < 𝑤 → (abs‘((1 / 𝑣) − (1 / 𝐸))) < 𝑧))
24	16, 23	sylan 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (ℂ ∖ {0})((abs‘(𝑣 − 𝐸)) < 𝑤 → (abs‘((1 / 𝑣) − (1 / 𝐸))) < 𝑧))
25		oveq2 7158	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → (1 / 𝑦) = (1 / 𝑣))
26		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) = (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))
27		ovex 7183	. . . . . . . . . . . . . 14 ⊢ (1 / 𝑣) ∈ V
28	25, 26, 27	fvmpt 6762	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (ℂ ∖ {0}) → ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝑣) = (1 / 𝑣))
29		oveq2 7158	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐸 → (1 / 𝑦) = (1 / 𝐸))
30		ovex 7183	. . . . . . . . . . . . . . 15 ⊢ (1 / 𝐸) ∈ V
31	29, 26, 30	fvmpt 6762	. . . . . . . . . . . . . 14 ⊢ (𝐸 ∈ (ℂ ∖ {0}) → ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝐸) = (1 / 𝐸))
32	16, 31	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝐸) = (1 / 𝐸))
33	28, 32	oveqan12rd 7170	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℂ ∖ {0})) → (((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝑣) − ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝐸)) = ((1 / 𝑣) − (1 / 𝐸)))
34	33	fveq2d 6668	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℂ ∖ {0})) → (abs‘(((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝑣) − ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝐸))) = (abs‘((1 / 𝑣) − (1 / 𝐸))))
35	34	breq1d 5068	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℂ ∖ {0})) → ((abs‘(((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝑣) − ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝐸))) < 𝑧 ↔ (abs‘((1 / 𝑣) − (1 / 𝐸))) < 𝑧))
36	35	imbi2d 343	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℂ ∖ {0})) → (((abs‘(𝑣 − 𝐸)) < 𝑤 → (abs‘(((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝑣) − ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝐸))) < 𝑧) ↔ ((abs‘(𝑣 − 𝐸)) < 𝑤 → (abs‘((1 / 𝑣) − (1 / 𝐸))) < 𝑧)))
37	36	ralbidva 3196	. . . . . . . 8 ⊢ (𝜑 → (∀𝑣 ∈ (ℂ ∖ {0})((abs‘(𝑣 − 𝐸)) < 𝑤 → (abs‘(((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝑣) − ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝐸))) < 𝑧) ↔ ∀𝑣 ∈ (ℂ ∖ {0})((abs‘(𝑣 − 𝐸)) < 𝑤 → (abs‘((1 / 𝑣) − (1 / 𝐸))) < 𝑧)))
38	37	rexbidv 3297	. . . . . . 7 ⊢ (𝜑 → (∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (ℂ ∖ {0})((abs‘(𝑣 − 𝐸)) < 𝑤 → (abs‘(((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝑣) − ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝐸))) < 𝑧) ↔ ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (ℂ ∖ {0})((abs‘(𝑣 − 𝐸)) < 𝑤 → (abs‘((1 / 𝑣) − (1 / 𝐸))) < 𝑧)))
39	38	biimpar 480	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (ℂ ∖ {0})((abs‘(𝑣 − 𝐸)) < 𝑤 → (abs‘((1 / 𝑣) − (1 / 𝐸))) < 𝑧)) → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (ℂ ∖ {0})((abs‘(𝑣 − 𝐸)) < 𝑤 → (abs‘(((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝑣) − ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝐸))) < 𝑧))
40	24, 39	syldan 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (ℂ ∖ {0})((abs‘(𝑣 − 𝐸)) < 𝑤 → (abs‘(((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝑣) − ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝐸))) < 𝑧))
41	11, 16, 5, 21, 40	rlimcn1 14939	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∘ (𝑥 ∈ 𝐴 ↦ 𝐶)) ⇝𝑟 ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))‘𝐸))
42		eqidd 2822	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶))
43		eqidd 2822	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) = (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)))
44		oveq2 7158	. . . . 5 ⊢ (𝑦 = 𝐶 → (1 / 𝑦) = (1 / 𝐶))
45	10, 42, 43, 44	fmptco 6885	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∘ (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ (1 / 𝐶)))
46	41, 45, 32	3brtr3d 5089	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (1 / 𝐶)) ⇝𝑟 (1 / 𝐸))
47	3, 8, 2, 46	rlimmul 14995	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (1 / 𝐶))) ⇝𝑟 (𝐷 · (1 / 𝐸)))
48	3, 6, 7	divrecd 11413	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 / 𝐶) = (𝐵 · (1 / 𝐶)))
49	48	mpteq2dva 5153	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 / 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (1 / 𝐶))))
50		rlimcl 14854	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷 → 𝐷 ∈ ℂ)
51	2, 50	syl 17	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ)
52	51, 13, 14	divrecd 11413	. 2 ⊢ (𝜑 → (𝐷 / 𝐸) = (𝐷 · (1 / 𝐸)))
53	47, 49, 52	3brtr4d 5090	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 / 𝐶)) ⇝𝑟 (𝐷 / 𝐸))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   ∖ cdif 3932  ifcif 4466  {csn 4560   class class class wbr 5058   ↦ cmpt 5138   ∘ ccom 5553  ‘cfv 6349  (class class class)co 7150  ℂcc 10529  0cc0 10531  1c1 10532   · cmul 10536   < clt 10669   ≤ cle 10670   − cmin 10864   / cdiv 11291  2c2 11686  ℝ⁺crp 12383  abscabs 14587   ⇝_𝑟 crli 14836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609  ax-mulf 10611

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  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 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-pm 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-sup 8900  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12384  df-seq 13364  df-exp 13424  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-rlim 14840

This theorem is referenced by:  logexprlim  25795  chebbnd2  26047  chto1lb  26048  pnt2  26183  pnt  26184