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Mirrors > Home > MPE Home > Th. List > xpsxms | 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 |
---|---|
xpsxms | β’ ((π β βMetSp β§ π β βMetSp) β π β βMetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsms.t | . . 3 β’ π = (π Γs π) | |
2 | eqid 2736 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
3 | eqid 2736 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
4 | simpl 483 | . . 3 β’ ((π β βMetSp β§ π β βMetSp) β π β βMetSp) | |
5 | simpr 485 | . . 3 β’ ((π β βMetSp β§ π β βMetSp) β π β βMetSp) | |
6 | eqid 2736 | . . 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 2736 | . . 3 β’ (Scalarβπ ) = (Scalarβπ ) | |
8 | eqid 2736 | . . 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 17452 | . 2 β’ ((π β βMetSp β§ π β βMetSp) 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 17453 | . 2 β’ ((π β βMetSp β§ π β βMetSp) 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 17451 | . . 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 6796 | . . 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 βMetSp β§ π β βMetSp) 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 6857 | . . 3 β’ ((π β βMetSp β§ π β βMetSp) β (Scalarβπ ) β V) | |
15 | 2onn 8588 | . . . 4 β’ 2o β Ο | |
16 | nnfi 9111 | . . . 4 β’ (2o β Ο β 2o β Fin) | |
17 | 15, 16 | mp1i 13 | . . 3 β’ ((π β βMetSp β§ π β βMetSp) β 2o β Fin) |
18 | xpscf 17447 | . . . 4 β’ ({β¨β , π β©, β¨1o, πβ©}:2oβΆβMetSp β (π β βMetSp β§ π β βMetSp)) | |
19 | 18 | biimpri 227 | . . 3 β’ ((π β βMetSp β§ π β βMetSp) β {β¨β , π β©, β¨1o, πβ©}:2oβΆβMetSp) |
20 | 8 | prdsxms 23886 | . . 3 β’ (((Scalarβπ ) β V β§ 2o β Fin β§ {β¨β , π β©, β¨1o, πβ©}:2oβΆβMetSp) β ((Scalarβπ )Xs{β¨β , π β©, β¨1o, πβ©}) β βMetSp) |
21 | 14, 17, 19, 20 | syl3anc 1371 | . 2 β’ ((π β βMetSp β§ π β βMetSp) β ((Scalarβπ )Xs{β¨β , π β©, β¨1o, πβ©}) β βMetSp) |
22 | 9, 10, 13, 21 | imasf1oxms 23845 | 1 β’ ((π β βMetSp β§ π β βMetSp) β π β βMetSp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 Vcvv 3445 β c0 4282 {cpr 4588 β¨cop 4592 Γ cxp 5631 β‘ccnv 5632 ran crn 5634 βΆwf 6492 β1-1-ontoβwf1o 6495 βcfv 6496 (class class class)co 7357 β cmpo 7359 Οcom 7802 1oc1o 8405 2oc2o 8406 Fincfn 8883 Basecbs 17083 Scalarcsca 17136 Xscprds 17327 Γs cxps 17388 βMetSpcxms 23670 |
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 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
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-tp 4591 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 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-bl 20791 df-mopn 20792 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-xms 23673 |
This theorem is referenced by: tmsxps 23892 tmsxpsmopn 23893 |
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