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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 22750 | . . . . . . 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 5314 | . . . . 5 β’ ((π β§ π β π΄) β π β V) |
8 | 7 | ralrimiva 3138 | . . . 4 β’ (π β βπ β π΄ π β V) |
9 | iunexg 7943 | . . . 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 10455 | . . . 4 β’ CHOICE | |
12 | acacni 10131 | . . . 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 2828 | . 2 β’ (π β βͺ π β π΄ π β AC π΄) |
15 | 1, 2, 3, 4, 14 | ptclsg 23441 | 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 1533 β wcel 2098 βwral 3053 Vcvv 3466 β wss 3940 βͺ ciun 4987 β¦ cmpt 5221 βcfv 6533 Xcixp 8887 AC wacn 9929 CHOICEwac 10106 βtcpt 17383 TopOnctopon 22734 clsccl 22844 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-ac2 10454 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-1o 8461 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-fin 8939 df-fi 9402 df-card 9930 df-acn 9933 df-ac 10107 df-topgen 17388 df-pt 17389 df-top 22718 df-topon 22735 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 |
This theorem is referenced by: (None) |
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