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Mirrors > Home > MPE Home > Th. List > xpsms | Structured version Visualization version GIF version |
Description: A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
xpsms.t | β’ π = (π Γs π) |
Ref | Expression |
---|---|
xpsms | β’ ((π β MetSp β§ π β MetSp) β π β MetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsms.t | . . 3 β’ π = (π Γs π) | |
2 | eqid 2724 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
3 | eqid 2724 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
4 | simpl 482 | . . 3 β’ ((π β MetSp β§ π β MetSp) β π β MetSp) | |
5 | simpr 484 | . . 3 β’ ((π β MetSp β§ π β MetSp) β π β MetSp) | |
6 | eqid 2724 | . . 3 β’ (π₯ β (Baseβπ ), π¦ β (Baseβπ) β¦ {β¨β , π₯β©, β¨1o, π¦β©}) = (π₯ β (Baseβπ ), π¦ β (Baseβπ) β¦ {β¨β , π₯β©, β¨1o, π¦β©}) | |
7 | eqid 2724 | . . 3 β’ (Scalarβπ ) = (Scalarβπ ) | |
8 | eqid 2724 | . . 3 β’ ((Scalarβπ )Xs{β¨β , π β©, β¨1o, πβ©}) = ((Scalarβπ )Xs{β¨β , π β©, β¨1o, πβ©}) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsval 17515 | . 2 β’ ((π β MetSp β§ π β MetSp) β π = (β‘(π₯ β (Baseβπ ), π¦ β (Baseβπ) β¦ {β¨β , π₯β©, β¨1o, π¦β©}) βs ((Scalarβπ )Xs{β¨β , π β©, β¨1o, πβ©}))) |
10 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsrnbas 17516 | . 2 β’ ((π β MetSp β§ π β MetSp) β ran (π₯ β (Baseβπ ), π¦ β (Baseβπ) β¦ {β¨β , π₯β©, β¨1o, π¦β©}) = (Baseβ((Scalarβπ )Xs{β¨β , π β©, β¨1o, πβ©}))) |
11 | 6 | xpsff1o2 17514 | . . 3 β’ (π₯ β (Baseβπ ), π¦ β (Baseβπ) β¦ {β¨β , π₯β©, β¨1o, π¦β©}):((Baseβπ ) Γ (Baseβπ))β1-1-ontoβran (π₯ β (Baseβπ ), π¦ β (Baseβπ) β¦ {β¨β , π₯β©, β¨1o, π¦β©}) |
12 | f1ocnv 6835 | . . 3 β’ ((π₯ β (Baseβπ ), π¦ β (Baseβπ) β¦ {β¨β , π₯β©, β¨1o, π¦β©}):((Baseβπ ) Γ (Baseβπ))β1-1-ontoβran (π₯ β (Baseβπ ), π¦ β (Baseβπ) β¦ {β¨β , π₯β©, β¨1o, π¦β©}) β β‘(π₯ β (Baseβπ ), π¦ β (Baseβπ) β¦ {β¨β , π₯β©, β¨1o, π¦β©}):ran (π₯ β (Baseβπ ), π¦ β (Baseβπ) β¦ {β¨β , π₯β©, β¨1o, π¦β©})β1-1-ontoβ((Baseβπ ) Γ (Baseβπ))) | |
13 | 11, 12 | mp1i 13 | . 2 β’ ((π β MetSp β§ π β MetSp) β β‘(π₯ β (Baseβπ ), π¦ β (Baseβπ) β¦ {β¨β , π₯β©, β¨1o, π¦β©}):ran (π₯ β (Baseβπ ), π¦ β (Baseβπ) β¦ {β¨β , π₯β©, β¨1o, π¦β©})β1-1-ontoβ((Baseβπ ) Γ (Baseβπ))) |
14 | fvexd 6896 | . . 3 β’ ((π β MetSp β§ π β MetSp) β (Scalarβπ ) β V) | |
15 | 2onn 8637 | . . . 4 β’ 2o β Ο | |
16 | nnfi 9163 | . . . 4 β’ (2o β Ο β 2o β Fin) | |
17 | 15, 16 | mp1i 13 | . . 3 β’ ((π β MetSp β§ π β MetSp) β 2o β Fin) |
18 | xpscf 17510 | . . . 4 β’ ({β¨β , π β©, β¨1o, πβ©}:2oβΆMetSp β (π β MetSp β§ π β MetSp)) | |
19 | 18 | biimpri 227 | . . 3 β’ ((π β MetSp β§ π β MetSp) β {β¨β , π β©, β¨1o, πβ©}:2oβΆMetSp) |
20 | 8 | prdsms 24362 | . . 3 β’ (((Scalarβπ ) β V β§ 2o β Fin β§ {β¨β , π β©, β¨1o, πβ©}:2oβΆMetSp) β ((Scalarβπ )Xs{β¨β , π β©, β¨1o, πβ©}) β MetSp) |
21 | 14, 17, 19, 20 | syl3anc 1368 | . 2 β’ ((π β MetSp β§ π β MetSp) β ((Scalarβπ )Xs{β¨β , π β©, β¨1o, πβ©}) β MetSp) |
22 | 9, 10, 13, 21 | imasf1oms 24321 | 1 β’ ((π β MetSp β§ π β MetSp) β π β MetSp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3466 β c0 4314 {cpr 4622 β¨cop 4626 Γ cxp 5664 β‘ccnv 5665 ran crn 5667 βΆwf 6529 β1-1-ontoβwf1o 6532 βcfv 6533 (class class class)co 7401 β cmpo 7403 Οcom 7848 1oc1o 8454 2oc2o 8455 Fincfn 8935 Basecbs 17143 Scalarcsca 17199 Xscprds 17390 Γs cxps 17451 MetSpcms 24146 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-icc 13328 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-mulg 18986 df-cntz 19223 df-cmn 19692 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-xms 24148 df-ms 24149 |
This theorem is referenced by: (None) |
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