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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrncms | Structured version Visualization version GIF version |
Description: Euclidean space is complete. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) |
Ref | Expression |
---|---|
rrncms.1 | ⊢ 𝑋 = (ℝ ↑m 𝐼) |
Ref | Expression |
---|---|
rrncms | ⊢ (𝐼 ∈ Fin → (ℝn‘𝐼) ∈ (CMet‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrncms.1 | . . . . 5 ⊢ 𝑋 = (ℝ ↑m 𝐼) | |
2 | eqid 2736 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
3 | eqid 2736 | . . . . 5 ⊢ (MetOpen‘(ℝn‘𝐼)) = (MetOpen‘(ℝn‘𝐼)) | |
4 | simpll 765 | . . . . 5 ⊢ (((𝐼 ∈ Fin ∧ 𝑓 ∈ (Cau‘(ℝn‘𝐼))) ∧ 𝑓:ℕ⟶𝑋) → 𝐼 ∈ Fin) | |
5 | simplr 767 | . . . . 5 ⊢ (((𝐼 ∈ Fin ∧ 𝑓 ∈ (Cau‘(ℝn‘𝐼))) ∧ 𝑓:ℕ⟶𝑋) → 𝑓 ∈ (Cau‘(ℝn‘𝐼))) | |
6 | simpr 485 | . . . . 5 ⊢ (((𝐼 ∈ Fin ∧ 𝑓 ∈ (Cau‘(ℝn‘𝐼))) ∧ 𝑓:ℕ⟶𝑋) → 𝑓:ℕ⟶𝑋) | |
7 | eqid 2736 | . . . . 5 ⊢ (𝑚 ∈ 𝐼 ↦ ( ⇝ ‘(𝑡 ∈ ℕ ↦ ((𝑓‘𝑡)‘𝑚)))) = (𝑚 ∈ 𝐼 ↦ ( ⇝ ‘(𝑡 ∈ ℕ ↦ ((𝑓‘𝑡)‘𝑚)))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | rrncmslem 36282 | . . . 4 ⊢ (((𝐼 ∈ Fin ∧ 𝑓 ∈ (Cau‘(ℝn‘𝐼))) ∧ 𝑓:ℕ⟶𝑋) → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘(ℝn‘𝐼)))) |
9 | 8 | ex 413 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ (Cau‘(ℝn‘𝐼))) → (𝑓:ℕ⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘(ℝn‘𝐼))))) |
10 | 9 | ralrimiva 3143 | . 2 ⊢ (𝐼 ∈ Fin → ∀𝑓 ∈ (Cau‘(ℝn‘𝐼))(𝑓:ℕ⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘(ℝn‘𝐼))))) |
11 | nnuz 12805 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
12 | 1zzd 12533 | . . 3 ⊢ (𝐼 ∈ Fin → 1 ∈ ℤ) | |
13 | 1 | rrnmet 36279 | . . 3 ⊢ (𝐼 ∈ Fin → (ℝn‘𝐼) ∈ (Met‘𝑋)) |
14 | 11, 3, 12, 13 | iscmet3 24655 | . 2 ⊢ (𝐼 ∈ Fin → ((ℝn‘𝐼) ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘(ℝn‘𝐼))(𝑓:ℕ⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘(ℝn‘𝐼)))))) |
15 | 10, 14 | mpbird 256 | 1 ⊢ (𝐼 ∈ Fin → (ℝn‘𝐼) ∈ (CMet‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ↦ cmpt 5188 × cxp 5631 dom cdm 5633 ↾ cres 5635 ∘ ccom 5637 ⟶wf 6492 ‘cfv 6496 (class class class)co 7356 ↑m cmap 8764 Fincfn 8882 ℝcr 11049 1c1 11051 − cmin 11384 ℕcn 12152 abscabs 15118 ⇝ cli 15365 MetOpencmopn 20784 ⇝𝑡clm 22575 Cauccau 24615 CMetccmet 24616 ℝncrrn 36275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 ax-cc 10370 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-oadd 8415 df-omul 8416 df-er 8647 df-map 8766 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fi 9346 df-sup 9377 df-inf 9378 df-oi 9445 df-card 9874 df-acn 9877 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-n0 12413 df-z 12499 df-uz 12763 df-q 12873 df-rp 12915 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ico 13269 df-fz 13424 df-fzo 13567 df-fl 13696 df-seq 13906 df-exp 13967 df-hash 14230 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-limsup 15352 df-clim 15369 df-rlim 15370 df-sum 15570 df-rest 17303 df-topgen 17324 df-psmet 20786 df-xmet 20787 df-met 20788 df-bl 20789 df-mopn 20790 df-fbas 20791 df-fg 20792 df-top 22241 df-topon 22258 df-bases 22294 df-ntr 22369 df-nei 22447 df-lm 22578 df-fil 23195 df-fm 23287 df-flim 23288 df-flf 23289 df-cfil 24617 df-cau 24618 df-cmet 24619 df-rrn 36276 |
This theorem is referenced by: rrnheibor 36287 |
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