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Mirrors > Home > MPE Home > Th. List > pttopon | Structured version Visualization version GIF version |
Description: The base set for the product topology. (Contributed by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
ptunimpt.j | β’ π½ = (βtβ(π₯ β π΄ β¦ πΎ)) |
Ref | Expression |
---|---|
pttopon | β’ ((π΄ β π β§ βπ₯ β π΄ πΎ β (TopOnβπ΅)) β π½ β (TopOnβXπ₯ β π΄ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 22765 | . . . . 5 β’ (πΎ β (TopOnβπ΅) β πΎ β Top) | |
2 | 1 | ralimi 3077 | . . . 4 β’ (βπ₯ β π΄ πΎ β (TopOnβπ΅) β βπ₯ β π΄ πΎ β Top) |
3 | eqid 2726 | . . . . 5 β’ (π₯ β π΄ β¦ πΎ) = (π₯ β π΄ β¦ πΎ) | |
4 | 3 | fmpt 7104 | . . . 4 β’ (βπ₯ β π΄ πΎ β Top β (π₯ β π΄ β¦ πΎ):π΄βΆTop) |
5 | 2, 4 | sylib 217 | . . 3 β’ (βπ₯ β π΄ πΎ β (TopOnβπ΅) β (π₯ β π΄ β¦ πΎ):π΄βΆTop) |
6 | ptunimpt.j | . . . 4 β’ π½ = (βtβ(π₯ β π΄ β¦ πΎ)) | |
7 | pttop 23436 | . . . 4 β’ ((π΄ β π β§ (π₯ β π΄ β¦ πΎ):π΄βΆTop) β (βtβ(π₯ β π΄ β¦ πΎ)) β Top) | |
8 | 6, 7 | eqeltrid 2831 | . . 3 β’ ((π΄ β π β§ (π₯ β π΄ β¦ πΎ):π΄βΆTop) β π½ β Top) |
9 | 5, 8 | sylan2 592 | . 2 β’ ((π΄ β π β§ βπ₯ β π΄ πΎ β (TopOnβπ΅)) β π½ β Top) |
10 | toponuni 22766 | . . . . . 6 β’ (πΎ β (TopOnβπ΅) β π΅ = βͺ πΎ) | |
11 | 10 | ralimi 3077 | . . . . 5 β’ (βπ₯ β π΄ πΎ β (TopOnβπ΅) β βπ₯ β π΄ π΅ = βͺ πΎ) |
12 | ixpeq2 8904 | . . . . 5 β’ (βπ₯ β π΄ π΅ = βͺ πΎ β Xπ₯ β π΄ π΅ = Xπ₯ β π΄ βͺ πΎ) | |
13 | 11, 12 | syl 17 | . . . 4 β’ (βπ₯ β π΄ πΎ β (TopOnβπ΅) β Xπ₯ β π΄ π΅ = Xπ₯ β π΄ βͺ πΎ) |
14 | 13 | adantl 481 | . . 3 β’ ((π΄ β π β§ βπ₯ β π΄ πΎ β (TopOnβπ΅)) β Xπ₯ β π΄ π΅ = Xπ₯ β π΄ βͺ πΎ) |
15 | 6 | ptunimpt 23449 | . . . 4 β’ ((π΄ β π β§ βπ₯ β π΄ πΎ β Top) β Xπ₯ β π΄ βͺ πΎ = βͺ π½) |
16 | 2, 15 | sylan2 592 | . . 3 β’ ((π΄ β π β§ βπ₯ β π΄ πΎ β (TopOnβπ΅)) β Xπ₯ β π΄ βͺ πΎ = βͺ π½) |
17 | 14, 16 | eqtrd 2766 | . 2 β’ ((π΄ β π β§ βπ₯ β π΄ πΎ β (TopOnβπ΅)) β Xπ₯ β π΄ π΅ = βͺ π½) |
18 | istopon 22764 | . 2 β’ (π½ β (TopOnβXπ₯ β π΄ π΅) β (π½ β Top β§ Xπ₯ β π΄ π΅ = βͺ π½)) | |
19 | 9, 17, 18 | sylanbrc 582 | 1 β’ ((π΄ β π β§ βπ₯ β π΄ πΎ β (TopOnβπ΅)) β π½ β (TopOnβXπ₯ β π΄ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 βͺ cuni 4902 β¦ cmpt 5224 βΆwf 6532 βcfv 6536 Xcixp 8890 βtcpt 17390 Topctop 22745 TopOnctopon 22762 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-om 7852 df-1o 8464 df-er 8702 df-ixp 8891 df-en 8939 df-fin 8942 df-fi 9405 df-topgen 17395 df-pt 17396 df-top 22746 df-topon 22763 df-bases 22799 |
This theorem is referenced by: pttoponconst 23451 ptclsg 23469 dfac14lem 23471 ptcnp 23476 prdstps 23483 |
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