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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 2727 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
3 | eqid 2727 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
4 | simpl 481 | . . 3 β’ ((π β βMetSp β§ π β βMetSp) β π β βMetSp) | |
5 | simpr 483 | . . 3 β’ ((π β βMetSp β§ π β βMetSp) β π β βMetSp) | |
6 | eqid 2727 | . . 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 2727 | . . 3 β’ (Scalarβπ ) = (Scalarβπ ) | |
8 | eqid 2727 | . . 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 17557 | . 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 17558 | . 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 17556 | . . 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 6854 | . . 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 6915 | . . 3 β’ ((π β βMetSp β§ π β βMetSp) β (Scalarβπ ) β V) | |
15 | 2onn 8667 | . . . 4 β’ 2o β Ο | |
16 | nnfi 9196 | . . . 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 17552 | . . . 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 24457 | . . 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 1368 | . 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 24416 | 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 394 = wceq 1533 β wcel 2098 Vcvv 3471 β c0 4324 {cpr 4632 β¨cop 4636 Γ cxp 5678 β‘ccnv 5679 ran crn 5681 βΆwf 6547 β1-1-ontoβwf1o 6550 βcfv 6551 (class class class)co 7424 β cmpo 7426 Οcom 7874 1oc1o 8484 2oc2o 8485 Fincfn 8968 Basecbs 17185 Scalarcsca 17241 Xscprds 17432 Γs cxps 17493 βMetSpcxms 24241 |
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-rep 5287 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-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 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-se 5636 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-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-er 8729 df-map 8851 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9539 df-card 9968 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-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-icc 13369 df-fz 13523 df-fzo 13666 df-seq 14005 df-hash 14328 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-hom 17262 df-cco 17263 df-rest 17409 df-topn 17410 df-0g 17428 df-gsum 17429 df-topgen 17430 df-pt 17431 df-prds 17434 df-xrs 17489 df-qtop 17494 df-imas 17495 df-xps 17497 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-mulg 19029 df-cntz 19273 df-cmn 19742 df-psmet 21276 df-xmet 21277 df-bl 21279 df-mopn 21280 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-xms 24244 |
This theorem is referenced by: tmsxps 24463 tmsxpsmopn 24464 |
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