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Mirrors > Home > MPE Home > Th. List > caucvgr | Structured version Visualization version GIF version |
Description: A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Revised by Mario Carneiro, 8-May-2016.) |
Ref | Expression |
---|---|
caucvgr.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
caucvgr.2 | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
caucvgr.3 | ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) |
caucvgr.4 | ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) |
Ref | Expression |
---|---|
caucvgr | ⊢ (𝜑 → 𝐹 ∈ dom ⇝𝑟 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgr.2 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
2 | 1 | feqmptd 6970 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑛 ∈ 𝐴 ↦ (𝐹‘𝑛))) |
3 | 1 | ffvelcdmda 7097 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐹‘𝑛) ∈ ℂ) |
4 | 3 | replimd 15182 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐹‘𝑛) = ((ℜ‘(𝐹‘𝑛)) + (i · (ℑ‘(𝐹‘𝑛))))) |
5 | 4 | mpteq2dva 5250 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ (𝐹‘𝑛)) = (𝑛 ∈ 𝐴 ↦ ((ℜ‘(𝐹‘𝑛)) + (i · (ℑ‘(𝐹‘𝑛)))))) |
6 | 2, 5 | eqtrd 2767 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑛 ∈ 𝐴 ↦ ((ℜ‘(𝐹‘𝑛)) + (i · (ℑ‘(𝐹‘𝑛)))))) |
7 | fvexd 6915 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (ℜ‘(𝐹‘𝑛)) ∈ V) | |
8 | ovexd 7459 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (i · (ℑ‘(𝐹‘𝑛))) ∈ V) | |
9 | caucvgr.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
10 | caucvgr.3 | . . . . 5 ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) | |
11 | caucvgr.4 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) | |
12 | ref 15097 | . . . . 5 ⊢ ℜ:ℂ⟶ℝ | |
13 | resub 15112 | . . . . . . 7 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (ℜ‘((𝐹‘𝑘) − (𝐹‘𝑗))) = ((ℜ‘(𝐹‘𝑘)) − (ℜ‘(𝐹‘𝑗)))) | |
14 | 13 | fveq2d 6904 | . . . . . 6 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘(ℜ‘((𝐹‘𝑘) − (𝐹‘𝑗)))) = (abs‘((ℜ‘(𝐹‘𝑘)) − (ℜ‘(𝐹‘𝑗))))) |
15 | subcl 11495 | . . . . . . 7 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → ((𝐹‘𝑘) − (𝐹‘𝑗)) ∈ ℂ) | |
16 | absrele 15293 | . . . . . . 7 ⊢ (((𝐹‘𝑘) − (𝐹‘𝑗)) ∈ ℂ → (abs‘(ℜ‘((𝐹‘𝑘) − (𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗)))) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘(ℜ‘((𝐹‘𝑘) − (𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗)))) |
18 | 14, 17 | eqbrtrrd 5174 | . . . . 5 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘((ℜ‘(𝐹‘𝑘)) − (ℜ‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗)))) |
19 | 9, 1, 10, 11, 12, 18 | caucvgrlem2 15659 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ (ℜ‘(𝐹‘𝑛))) ⇝𝑟 ( ⇝𝑟 ‘(ℜ ∘ 𝐹))) |
20 | ax-icn 11203 | . . . . . . 7 ⊢ i ∈ ℂ | |
21 | 20 | elexi 3491 | . . . . . 6 ⊢ i ∈ V |
22 | 21 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → i ∈ V) |
23 | fvexd 6915 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (ℑ‘(𝐹‘𝑛)) ∈ V) | |
24 | rlimconst 15526 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ i ∈ ℂ) → (𝑛 ∈ 𝐴 ↦ i) ⇝𝑟 i) | |
25 | 9, 20, 24 | sylancl 584 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ i) ⇝𝑟 i) |
26 | imf 15098 | . . . . . 6 ⊢ ℑ:ℂ⟶ℝ | |
27 | imsub 15120 | . . . . . . . 8 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (ℑ‘((𝐹‘𝑘) − (𝐹‘𝑗))) = ((ℑ‘(𝐹‘𝑘)) − (ℑ‘(𝐹‘𝑗)))) | |
28 | 27 | fveq2d 6904 | . . . . . . 7 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘(ℑ‘((𝐹‘𝑘) − (𝐹‘𝑗)))) = (abs‘((ℑ‘(𝐹‘𝑘)) − (ℑ‘(𝐹‘𝑗))))) |
29 | absimle 15294 | . . . . . . . 8 ⊢ (((𝐹‘𝑘) − (𝐹‘𝑗)) ∈ ℂ → (abs‘(ℑ‘((𝐹‘𝑘) − (𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗)))) | |
30 | 15, 29 | syl 17 | . . . . . . 7 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘(ℑ‘((𝐹‘𝑘) − (𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗)))) |
31 | 28, 30 | eqbrtrrd 5174 | . . . . . 6 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘((ℑ‘(𝐹‘𝑘)) − (ℑ‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗)))) |
32 | 9, 1, 10, 11, 26, 31 | caucvgrlem2 15659 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ (ℑ‘(𝐹‘𝑛))) ⇝𝑟 ( ⇝𝑟 ‘(ℑ ∘ 𝐹))) |
33 | 22, 23, 25, 32 | rlimmul 15628 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ (i · (ℑ‘(𝐹‘𝑛)))) ⇝𝑟 (i · ( ⇝𝑟 ‘(ℑ ∘ 𝐹)))) |
34 | 7, 8, 19, 33 | rlimadd 15625 | . . 3 ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ ((ℜ‘(𝐹‘𝑛)) + (i · (ℑ‘(𝐹‘𝑛))))) ⇝𝑟 (( ⇝𝑟 ‘(ℜ ∘ 𝐹)) + (i · ( ⇝𝑟 ‘(ℑ ∘ 𝐹))))) |
35 | 6, 34 | eqbrtrd 5172 | . 2 ⊢ (𝜑 → 𝐹 ⇝𝑟 (( ⇝𝑟 ‘(ℜ ∘ 𝐹)) + (i · ( ⇝𝑟 ‘(ℑ ∘ 𝐹))))) |
36 | rlimrel 15475 | . . 3 ⊢ Rel ⇝𝑟 | |
37 | 36 | releldmi 5952 | . 2 ⊢ (𝐹 ⇝𝑟 (( ⇝𝑟 ‘(ℜ ∘ 𝐹)) + (i · ( ⇝𝑟 ‘(ℑ ∘ 𝐹)))) → 𝐹 ∈ dom ⇝𝑟 ) |
38 | 35, 37 | syl 17 | 1 ⊢ (𝜑 → 𝐹 ∈ dom ⇝𝑟 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3057 ∃wrex 3066 Vcvv 3471 ⊆ wss 3947 class class class wbr 5150 ↦ cmpt 5233 dom cdm 5680 ∘ ccom 5684 ⟶wf 6547 ‘cfv 6551 (class class class)co 7424 supcsup 9469 ℂcc 11142 ℝcr 11143 ici 11146 + caddc 11147 · cmul 11149 +∞cpnf 11281 ℝ*cxr 11283 < clt 11284 ≤ cle 11285 − cmin 11480 ℝ+crp 13012 ℜcre 15082 ℑcim 15083 abscabs 15219 ⇝𝑟 crli 15467 |
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 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-pm 8852 df-en 8969 df-dom 8970 df-sdom 8971 df-sup 9471 df-inf 9472 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-ico 13368 df-seq 14005 df-exp 14065 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-limsup 15453 df-rlim 15471 |
This theorem is referenced by: caucvg 15663 dvfsumrlim 25984 |
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