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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 7099 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β (π βπ₯) β TopSp) |
4 | eqid 2728 | . . . . . . 7 β’ (Baseβ(π βπ₯)) = (Baseβ(π βπ₯)) | |
5 | eqid 2728 | . . . . . . 7 β’ (TopOpenβ(π βπ₯)) = (TopOpenβ(π βπ₯)) | |
6 | 4, 5 | istps 22856 | . . . . . 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 3143 | . . . 4 β’ (π β βπ₯ β πΌ (TopOpenβ(π βπ₯)) β (TopOnβ(Baseβ(π βπ₯)))) |
9 | eqid 2728 | . . . . 5 β’ (βtβ(π₯ β πΌ β¦ (TopOpenβ(π βπ₯)))) = (βtβ(π₯ β πΌ β¦ (TopOpenβ(π βπ₯)))) | |
10 | 9 | pttopon 23520 | . . . 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 582 | . . 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 7245 | . . . . 5 β’ (π β π β V) |
15 | eqid 2728 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
16 | 2 | fdmd 6738 | . . . . 5 β’ (π β dom π = πΌ) |
17 | eqid 2728 | . . . . 5 β’ (TopSetβπ) = (TopSetβπ) | |
18 | 12, 13, 14, 15, 16, 17 | prdstset 17455 | . . . 4 β’ (π β (TopSetβπ) = (βtβ(TopOpen β π ))) |
19 | topnfn 17414 | . . . . . . 7 β’ TopOpen Fn V | |
20 | dffn2 6729 | . . . . . . 7 β’ (TopOpen Fn V β TopOpen:VβΆV) | |
21 | 19, 20 | mpbi 229 | . . . . . 6 β’ TopOpen:VβΆV |
22 | ssv 4006 | . . . . . . 7 β’ TopSp β V | |
23 | fss 6744 | . . . . . . 7 β’ ((π :πΌβΆTopSp β§ TopSp β V) β π :πΌβΆV) | |
24 | 2, 22, 23 | sylancl 584 | . . . . . 6 β’ (π β π :πΌβΆV) |
25 | fcompt 7148 | . . . . . 6 β’ ((TopOpen:VβΆV β§ π :πΌβΆV) β (TopOpen β π ) = (π₯ β πΌ β¦ (TopOpenβ(π βπ₯)))) | |
26 | 21, 24, 25 | sylancr 585 | . . . . 5 β’ (π β (TopOpen β π ) = (π₯ β πΌ β¦ (TopOpenβ(π βπ₯)))) |
27 | 26 | fveq2d 6906 | . . . 4 β’ (π β (βtβ(TopOpen β π )) = (βtβ(π₯ β πΌ β¦ (TopOpenβ(π βπ₯))))) |
28 | 18, 27 | eqtrd 2768 | . . 3 β’ (π β (TopSetβπ) = (βtβ(π₯ β πΌ β¦ (TopOpenβ(π βπ₯))))) |
29 | 12, 13, 14, 15, 16 | prdsbas 17446 | . . . 4 β’ (π β (Baseβπ) = Xπ₯ β πΌ (Baseβ(π βπ₯))) |
30 | 29 | fveq2d 6906 | . . 3 β’ (π β (TopOnβ(Baseβπ)) = (TopOnβXπ₯ β πΌ (Baseβ(π βπ₯)))) |
31 | 11, 28, 30 | 3eltr4d 2844 | . 2 β’ (π β (TopSetβπ) β (TopOnβ(Baseβπ))) |
32 | 15, 17 | tsettps 22863 | . 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 394 = wceq 1533 β wcel 2098 βwral 3058 Vcvv 3473 β wss 3949 β¦ cmpt 5235 β ccom 5686 Fn wfn 6548 βΆwf 6549 βcfv 6553 (class class class)co 7426 Xcixp 8922 Basecbs 17187 TopSetcts 17246 TopOpenctopn 17410 βtcpt 17427 Xscprds 17434 TopOnctopon 22832 TopSpctps 22854 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fi 9442 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-topgen 17432 df-pt 17433 df-prds 17436 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 |
This theorem is referenced by: pwstps 23554 xpstps 23734 prdstmdd 24048 |
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