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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 2727 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
3 | eqid 2727 | . . 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 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 17545 | . 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 17546 | . 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 17544 | . . 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 6845 | . . 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 6906 | . . 3 β’ ((π β MetSp β§ π β MetSp) β (Scalarβπ ) β V) | |
15 | 2onn 8656 | . . . 4 β’ 2o β Ο | |
16 | nnfi 9185 | . . . 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 17540 | . . . 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 24433 | . . 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 1369 | . 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 24392 | 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 1534 β wcel 2099 Vcvv 3469 β c0 4318 {cpr 4626 β¨cop 4630 Γ cxp 5670 β‘ccnv 5671 ran crn 5673 βΆwf 6538 β1-1-ontoβwf1o 6541 βcfv 6542 (class class class)co 7414 β cmpo 7416 Οcom 7864 1oc1o 8473 2oc2o 8474 Fincfn 8957 Basecbs 17173 Scalarcsca 17229 Xscprds 17420 Γs cxps 17481 MetSpcms 24217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-icc 13357 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-hom 17250 df-cco 17251 df-rest 17397 df-topn 17398 df-0g 17416 df-gsum 17417 df-topgen 17418 df-pt 17419 df-prds 17422 df-xrs 17477 df-qtop 17482 df-imas 17483 df-xps 17485 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-mulg 19017 df-cntz 19261 df-cmn 19730 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-xms 24219 df-ms 24220 |
This theorem is referenced by: (None) |
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