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Mirrors > Home > MPE Home > Th. List > prdsxms | Structured version Visualization version GIF version |
Description: The indexed product structure is an extended metric space when the index set is finite. (Although the extended metric is still valid when the index set is infinite, it no longer agrees with the product topology, which is not metrizable in any case.) (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
prdsxms.y | β’ π = (πXsπ ) |
Ref | Expression |
---|---|
prdsxms | β’ ((π β π β§ πΌ β Fin β§ π :πΌβΆβMetSp) β π β βMetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsxms.y | . . . 4 β’ π = (πXsπ ) | |
2 | simp1 1134 | . . . 4 β’ ((π β π β§ πΌ β Fin β§ π :πΌβΆβMetSp) β π β π) | |
3 | simp2 1135 | . . . 4 β’ ((π β π β§ πΌ β Fin β§ π :πΌβΆβMetSp) β πΌ β Fin) | |
4 | eqid 2730 | . . . 4 β’ (distβπ) = (distβπ) | |
5 | eqid 2730 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
6 | simp3 1136 | . . . 4 β’ ((π β π β§ πΌ β Fin β§ π :πΌβΆβMetSp) β π :πΌβΆβMetSp) | |
7 | 1, 2, 3, 4, 5, 6 | prdsxmslem1 24257 | . . 3 β’ ((π β π β§ πΌ β Fin β§ π :πΌβΆβMetSp) β (distβπ) β (βMetβ(Baseβπ))) |
8 | ssid 4003 | . . 3 β’ (Baseβπ) β (Baseβπ) | |
9 | xmetres2 24087 | . . 3 β’ (((distβπ) β (βMetβ(Baseβπ)) β§ (Baseβπ) β (Baseβπ)) β ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) β (βMetβ(Baseβπ))) | |
10 | 7, 8, 9 | sylancl 584 | . 2 β’ ((π β π β§ πΌ β Fin β§ π :πΌβΆβMetSp) β ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) β (βMetβ(Baseβπ))) |
11 | eqid 2730 | . . . 4 β’ (TopOpenβπ) = (TopOpenβπ) | |
12 | eqid 2730 | . . . 4 β’ (Baseβ(π βπ)) = (Baseβ(π βπ)) | |
13 | eqid 2730 | . . . 4 β’ ((distβ(π βπ)) βΎ ((Baseβ(π βπ)) Γ (Baseβ(π βπ)))) = ((distβ(π βπ)) βΎ ((Baseβ(π βπ)) Γ (Baseβ(π βπ)))) | |
14 | eqid 2730 | . . . 4 β’ (TopOpenβ(π βπ)) = (TopOpenβ(π βπ)) | |
15 | eqid 2730 | . . . 4 β’ {π₯ β£ βπ((π Fn πΌ β§ βπ β πΌ (πβπ) β ((TopOpen β π )βπ) β§ βπ§ β Fin βπ β (πΌ β π§)(πβπ) = βͺ ((TopOpen β π )βπ)) β§ π₯ = Xπ β πΌ (πβπ))} = {π₯ β£ βπ((π Fn πΌ β§ βπ β πΌ (πβπ) β ((TopOpen β π )βπ) β§ βπ§ β Fin βπ β (πΌ β π§)(πβπ) = βͺ ((TopOpen β π )βπ)) β§ π₯ = Xπ β πΌ (πβπ))} | |
16 | 1, 2, 3, 4, 5, 6, 11, 12, 13, 14, 15 | prdsxmslem2 24258 | . . 3 β’ ((π β π β§ πΌ β Fin β§ π :πΌβΆβMetSp) β (TopOpenβπ) = (MetOpenβ(distβπ))) |
17 | xmetf 24055 | . . . . 5 β’ ((distβπ) β (βMetβ(Baseβπ)) β (distβπ):((Baseβπ) Γ (Baseβπ))βΆβ*) | |
18 | ffn 6716 | . . . . 5 β’ ((distβπ):((Baseβπ) Γ (Baseβπ))βΆβ* β (distβπ) Fn ((Baseβπ) Γ (Baseβπ))) | |
19 | fnresdm 6668 | . . . . 5 β’ ((distβπ) Fn ((Baseβπ) Γ (Baseβπ)) β ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) = (distβπ)) | |
20 | 7, 17, 18, 19 | 4syl 19 | . . . 4 β’ ((π β π β§ πΌ β Fin β§ π :πΌβΆβMetSp) β ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) = (distβπ)) |
21 | 20 | fveq2d 6894 | . . 3 β’ ((π β π β§ πΌ β Fin β§ π :πΌβΆβMetSp) β (MetOpenβ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ)))) = (MetOpenβ(distβπ))) |
22 | 16, 21 | eqtr4d 2773 | . 2 β’ ((π β π β§ πΌ β Fin β§ π :πΌβΆβMetSp) β (TopOpenβπ) = (MetOpenβ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))))) |
23 | eqid 2730 | . . 3 β’ ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) = ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) | |
24 | 11, 5, 23 | isxms2 24174 | . 2 β’ (π β βMetSp β (((distβπ) βΎ ((Baseβπ) Γ (Baseβπ))) β (βMetβ(Baseβπ)) β§ (TopOpenβπ) = (MetOpenβ((distβπ) βΎ ((Baseβπ) Γ (Baseβπ)))))) |
25 | 10, 22, 24 | sylanbrc 581 | 1 β’ ((π β π β§ πΌ β Fin β§ π :πΌβΆβMetSp) β π β βMetSp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1085 = wceq 1539 βwex 1779 β wcel 2104 {cab 2707 βwral 3059 βwrex 3068 β cdif 3944 β wss 3947 βͺ cuni 4907 Γ cxp 5673 βΎ cres 5677 β ccom 5679 Fn wfn 6537 βΆwf 6538 βcfv 6542 (class class class)co 7411 Xcixp 8893 Fincfn 8941 β*cxr 11251 Basecbs 17148 distcds 17210 TopOpenctopn 17371 Xscprds 17395 βMetcxmet 21129 MetOpencmopn 21134 βMetSpcxms 24043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 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-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-icc 13335 df-fz 13489 df-struct 17084 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-topgen 17393 df-pt 17394 df-prds 17397 df-psmet 21136 df-xmet 21137 df-bl 21139 df-mopn 21140 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-xms 24046 |
This theorem is referenced by: prdsms 24260 pwsxms 24261 xpsxms 24263 |
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