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Mirrors > Home > MPE Home > Th. List > prdstps | Structured version Visualization version GIF version |
Description: A structure product of topological spaces is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
prdstopn.y | β’ π = (πXsπ ) |
prdstopn.s | β’ (π β π β π) |
prdstopn.i | β’ (π β πΌ β π) |
prdstps.r | β’ (π β π :πΌβΆTopSp) |
Ref | Expression |
---|---|
prdstps | β’ (π β π β TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdstopn.i | . . . 4 β’ (π β πΌ β π) | |
2 | prdstps.r | . . . . . . 7 β’ (π β π :πΌβΆTopSp) | |
3 | 2 | ffvelcdmda 7079 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β (π βπ₯) β TopSp) |
4 | eqid 2726 | . . . . . . 7 β’ (Baseβ(π βπ₯)) = (Baseβ(π βπ₯)) | |
5 | eqid 2726 | . . . . . . 7 β’ (TopOpenβ(π βπ₯)) = (TopOpenβ(π βπ₯)) | |
6 | 4, 5 | istps 22786 | . . . . . 6 β’ ((π βπ₯) β TopSp β (TopOpenβ(π βπ₯)) β (TopOnβ(Baseβ(π βπ₯)))) |
7 | 3, 6 | sylib 217 | . . . . 5 β’ ((π β§ π₯ β πΌ) β (TopOpenβ(π βπ₯)) β (TopOnβ(Baseβ(π βπ₯)))) |
8 | 7 | ralrimiva 3140 | . . . 4 β’ (π β βπ₯ β πΌ (TopOpenβ(π βπ₯)) β (TopOnβ(Baseβ(π βπ₯)))) |
9 | eqid 2726 | . . . . 5 β’ (βtβ(π₯ β πΌ β¦ (TopOpenβ(π βπ₯)))) = (βtβ(π₯ β πΌ β¦ (TopOpenβ(π βπ₯)))) | |
10 | 9 | pttopon 23450 | . . . 4 β’ ((πΌ β π β§ βπ₯ β πΌ (TopOpenβ(π βπ₯)) β (TopOnβ(Baseβ(π βπ₯)))) β (βtβ(π₯ β πΌ β¦ (TopOpenβ(π βπ₯)))) β (TopOnβXπ₯ β πΌ (Baseβ(π βπ₯)))) |
11 | 1, 8, 10 | syl2anc 583 | . . 3 β’ (π β (βtβ(π₯ β πΌ β¦ (TopOpenβ(π βπ₯)))) β (TopOnβXπ₯ β πΌ (Baseβ(π βπ₯)))) |
12 | prdstopn.y | . . . . 5 β’ π = (πXsπ ) | |
13 | prdstopn.s | . . . . 5 β’ (π β π β π) | |
14 | 2, 1 | fexd 7223 | . . . . 5 β’ (π β π β V) |
15 | eqid 2726 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
16 | 2 | fdmd 6721 | . . . . 5 β’ (π β dom π = πΌ) |
17 | eqid 2726 | . . . . 5 β’ (TopSetβπ) = (TopSetβπ) | |
18 | 12, 13, 14, 15, 16, 17 | prdstset 17418 | . . . 4 β’ (π β (TopSetβπ) = (βtβ(TopOpen β π ))) |
19 | topnfn 17377 | . . . . . . 7 β’ TopOpen Fn V | |
20 | dffn2 6712 | . . . . . . 7 β’ (TopOpen Fn V β TopOpen:VβΆV) | |
21 | 19, 20 | mpbi 229 | . . . . . 6 β’ TopOpen:VβΆV |
22 | ssv 4001 | . . . . . . 7 β’ TopSp β V | |
23 | fss 6727 | . . . . . . 7 β’ ((π :πΌβΆTopSp β§ TopSp β V) β π :πΌβΆV) | |
24 | 2, 22, 23 | sylancl 585 | . . . . . 6 β’ (π β π :πΌβΆV) |
25 | fcompt 7126 | . . . . . 6 β’ ((TopOpen:VβΆV β§ π :πΌβΆV) β (TopOpen β π ) = (π₯ β πΌ β¦ (TopOpenβ(π βπ₯)))) | |
26 | 21, 24, 25 | sylancr 586 | . . . . 5 β’ (π β (TopOpen β π ) = (π₯ β πΌ β¦ (TopOpenβ(π βπ₯)))) |
27 | 26 | fveq2d 6888 | . . . 4 β’ (π β (βtβ(TopOpen β π )) = (βtβ(π₯ β πΌ β¦ (TopOpenβ(π βπ₯))))) |
28 | 18, 27 | eqtrd 2766 | . . 3 β’ (π β (TopSetβπ) = (βtβ(π₯ β πΌ β¦ (TopOpenβ(π βπ₯))))) |
29 | 12, 13, 14, 15, 16 | prdsbas 17409 | . . . 4 β’ (π β (Baseβπ) = Xπ₯ β πΌ (Baseβ(π βπ₯))) |
30 | 29 | fveq2d 6888 | . . 3 β’ (π β (TopOnβ(Baseβπ)) = (TopOnβXπ₯ β πΌ (Baseβ(π βπ₯)))) |
31 | 11, 28, 30 | 3eltr4d 2842 | . 2 β’ (π β (TopSetβπ) β (TopOnβ(Baseβπ))) |
32 | 15, 17 | tsettps 22793 | . 2 β’ ((TopSetβπ) β (TopOnβ(Baseβπ)) β π β TopSp) |
33 | 31, 32 | syl 17 | 1 β’ (π β π β TopSp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 Vcvv 3468 β wss 3943 β¦ cmpt 5224 β ccom 5673 Fn wfn 6531 βΆwf 6532 βcfv 6536 (class class class)co 7404 Xcixp 8890 Basecbs 17150 TopSetcts 17209 TopOpenctopn 17373 βtcpt 17390 Xscprds 17397 TopOnctopon 22762 TopSpctps 22784 |
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 |
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-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-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-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-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 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-fz 13488 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-hom 17227 df-cco 17228 df-rest 17374 df-topn 17375 df-topgen 17395 df-pt 17396 df-prds 17399 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 |
This theorem is referenced by: pwstps 23484 xpstps 23664 prdstmdd 23978 |
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