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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 22427 | . . . . . . 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 5324 | . . . . 5 β’ ((π β§ π β π΄) β π β V) |
8 | 7 | ralrimiva 3146 | . . . 4 β’ (π β βπ β π΄ π β V) |
9 | iunexg 7949 | . . . 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 584 | . . 3 β’ (π β βͺ π β π΄ π β V) |
11 | axac3 10458 | . . . 4 β’ CHOICE | |
12 | acacni 10134 | . . . 4 β’ ((CHOICE β§ π΄ β π) β AC π΄ = V) | |
13 | 11, 2, 12 | sylancr 587 | . . 3 β’ (π β AC π΄ = V) |
14 | 10, 13 | eleqtrrd 2836 | . 2 β’ (π β βͺ π β π΄ π β AC π΄) |
15 | 1, 2, 3, 4, 14 | ptclsg 23118 | 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 396 = wceq 1541 β wcel 2106 βwral 3061 Vcvv 3474 β wss 3948 βͺ ciun 4997 β¦ cmpt 5231 βcfv 6543 Xcixp 8890 AC wacn 9932 CHOICEwac 10109 βtcpt 17383 TopOnctopon 22411 clsccl 22521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-ac2 10457 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-1o 8465 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-fin 8942 df-fi 9405 df-card 9933 df-acn 9936 df-ac 10110 df-topgen 17388 df-pt 17389 df-top 22395 df-topon 22412 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 |
This theorem is referenced by: (None) |
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