![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2726 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
3 | eqid 2726 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
4 | simpl 482 | . . 3 β’ ((π β βMetSp β§ π β βMetSp) β π β βMetSp) | |
5 | simpr 484 | . . 3 β’ ((π β βMetSp β§ π β βMetSp) β π β βMetSp) | |
6 | eqid 2726 | . . 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 2726 | . . 3 β’ (Scalarβπ ) = (Scalarβπ ) | |
8 | eqid 2726 | . . 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 17523 | . 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 17524 | . 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 17522 | . . 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 6838 | . . 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 6899 | . . 3 β’ ((π β βMetSp β§ π β βMetSp) β (Scalarβπ ) β V) | |
15 | 2onn 8640 | . . . 4 β’ 2o β Ο | |
16 | nnfi 9166 | . . . 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 17518 | . . . 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 24390 | . . 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 24349 | 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 395 = wceq 1533 β wcel 2098 Vcvv 3468 β c0 4317 {cpr 4625 β¨cop 4629 Γ cxp 5667 β‘ccnv 5668 ran crn 5670 βΆwf 6532 β1-1-ontoβwf1o 6535 βcfv 6536 (class class class)co 7404 β cmpo 7406 Οcom 7851 1oc1o 8457 2oc2o 8458 Fincfn 8938 Basecbs 17151 Scalarcsca 17207 Xscprds 17398 Γs cxps 17459 βMetSpcxms 24174 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-icc 13334 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14294 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19231 df-cmn 19700 df-psmet 21228 df-xmet 21229 df-bl 21231 df-mopn 21232 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-xms 24177 |
This theorem is referenced by: tmsxps 24396 tmsxpsmopn 24397 |
Copyright terms: Public domain | W3C validator |