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| Mirrors > Home > MPE Home > Th. List > ptuniconst | Structured version Visualization version GIF version | ||
| Description: The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 3-Feb-2015.) |
| Ref | Expression |
|---|---|
| ptuniconst.2 | β’ π½ = (βtβ(π΄ Γ {π })) |
| ptuniconst.1 | β’ π = βͺ π |
| Ref | Expression |
|---|---|
| ptuniconst | β’ ((π΄ β π β§ π β Top) β (π βm π΄) = βͺ π½) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ptuniconst.1 | . . . 4 β’ π = βͺ π | |
| 2 | 1 | toptopon 22741 | . . 3 β’ (π β Top β π β (TopOnβπ)) |
| 3 | ptuniconst.2 | . . . 4 β’ π½ = (βtβ(π΄ Γ {π })) | |
| 4 | 3 | pttoponconst 23423 | . . 3 β’ ((π΄ β π β§ π β (TopOnβπ)) β π½ β (TopOnβ(π βm π΄))) |
| 5 | 2, 4 | sylan2b 593 | . 2 β’ ((π΄ β π β§ π β Top) β π½ β (TopOnβ(π βm π΄))) |
| 6 | toponuni 22738 | . 2 β’ (π½ β (TopOnβ(π βm π΄)) β (π βm π΄) = βͺ π½) | |
| 7 | 5, 6 | syl 17 | 1 β’ ((π΄ β π β§ π β Top) β (π βm π΄) = βͺ π½) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {csn 4620 βͺ cuni 4899 Γ cxp 5664 βcfv 6533 (class class class)co 7401 βm cmap 8816 βtcpt 17383 Topctop 22717 TopOnctopon 22734 |
| 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 |
| 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-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-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-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-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-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1o 8461 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-fin 8939 df-fi 9402 df-topgen 17388 df-pt 17389 df-top 22718 df-topon 22735 df-bases 22771 |
| This theorem is referenced by: xkopt 23481 xkopjcn 23482 poimirlem29 37007 poimirlem30 37008 poimirlem31 37009 |
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