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Mirrors > Home > MPE Home > Th. List > ptcls | Structured version Visualization version GIF version |
Description: The closure of a box in the product topology is the box formed from the closures of the factors. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
ptcls.2 | β’ π½ = (βtβ(π β π΄ β¦ π )) |
ptcls.a | β’ (π β π΄ β π) |
ptcls.j | β’ ((π β§ π β π΄) β π β (TopOnβπ)) |
ptcls.c | β’ ((π β§ π β π΄) β π β π) |
Ref | Expression |
---|---|
ptcls | β’ (π β ((clsβπ½)βXπ β π΄ π) = Xπ β π΄ ((clsβπ )βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ptcls.2 | . 2 β’ π½ = (βtβ(π β π΄ β¦ π )) | |
2 | ptcls.a | . 2 β’ (π β π΄ β π) | |
3 | ptcls.j | . 2 β’ ((π β§ π β π΄) β π β (TopOnβπ)) | |
4 | ptcls.c | . 2 β’ ((π β§ π β π΄) β π β π) | |
5 | toponmax 22821 | . . . . . . 7 β’ (π β (TopOnβπ) β π β π ) | |
6 | 3, 5 | syl 17 | . . . . . 6 β’ ((π β§ π β π΄) β π β π ) |
7 | 6, 4 | ssexd 5318 | . . . . 5 β’ ((π β§ π β π΄) β π β V) |
8 | 7 | ralrimiva 3141 | . . . 4 β’ (π β βπ β π΄ π β V) |
9 | iunexg 7961 | . . . 4 β’ ((π΄ β π β§ βπ β π΄ π β V) β βͺ π β π΄ π β V) | |
10 | 2, 8, 9 | syl2anc 583 | . . 3 β’ (π β βͺ π β π΄ π β V) |
11 | axac3 10481 | . . . 4 β’ CHOICE | |
12 | acacni 10157 | . . . 4 β’ ((CHOICE β§ π΄ β π) β AC π΄ = V) | |
13 | 11, 2, 12 | sylancr 586 | . . 3 β’ (π β AC π΄ = V) |
14 | 10, 13 | eleqtrrd 2831 | . 2 β’ (π β βͺ π β π΄ π β AC π΄) |
15 | 1, 2, 3, 4, 14 | ptclsg 23512 | 1 β’ (π β ((clsβπ½)βXπ β π΄ π) = Xπ β π΄ ((clsβπ )βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βwral 3056 Vcvv 3469 β wss 3944 βͺ ciun 4991 β¦ cmpt 5225 βcfv 6542 Xcixp 8909 AC wacn 9955 CHOICEwac 10132 βtcpt 17413 TopOnctopon 22805 clsccl 22915 |
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-ac2 10480 |
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-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-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-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-1o 8480 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-fin 8961 df-fi 9428 df-card 9956 df-acn 9959 df-ac 10133 df-topgen 17418 df-pt 17419 df-top 22789 df-topon 22806 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 |
This theorem is referenced by: (None) |
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