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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 22835 | . . . . 5 β’ (πΎ β (TopOnβπ΅) β πΎ β Top) | |
2 | 1 | ralimi 3080 | . . . 4 β’ (βπ₯ β π΄ πΎ β (TopOnβπ΅) β βπ₯ β π΄ πΎ β Top) |
3 | eqid 2728 | . . . . 5 β’ (π₯ β π΄ β¦ πΎ) = (π₯ β π΄ β¦ πΎ) | |
4 | 3 | fmpt 7125 | . . . 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 23506 | . . . 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 2833 | . . 3 β’ ((π΄ β π β§ (π₯ β π΄ β¦ πΎ):π΄βΆTop) β π½ β Top) |
9 | 5, 8 | sylan2 591 | . 2 β’ ((π΄ β π β§ βπ₯ β π΄ πΎ β (TopOnβπ΅)) β π½ β Top) |
10 | toponuni 22836 | . . . . . 6 β’ (πΎ β (TopOnβπ΅) β π΅ = βͺ πΎ) | |
11 | 10 | ralimi 3080 | . . . . 5 β’ (βπ₯ β π΄ πΎ β (TopOnβπ΅) β βπ₯ β π΄ π΅ = βͺ πΎ) |
12 | ixpeq2 8936 | . . . . 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 480 | . . 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 23519 | . . . 4 β’ ((π΄ β π β§ βπ₯ β π΄ πΎ β Top) β Xπ₯ β π΄ βͺ πΎ = βͺ π½) |
16 | 2, 15 | sylan2 591 | . . 3 β’ ((π΄ β π β§ βπ₯ β π΄ πΎ β (TopOnβπ΅)) β Xπ₯ β π΄ βͺ πΎ = βͺ π½) |
17 | 14, 16 | eqtrd 2768 | . 2 β’ ((π΄ β π β§ βπ₯ β π΄ πΎ β (TopOnβπ΅)) β Xπ₯ β π΄ π΅ = βͺ π½) |
18 | istopon 22834 | . 2 β’ (π½ β (TopOnβXπ₯ β π΄ π΅) β (π½ β Top β§ Xπ₯ β π΄ π΅ = βͺ π½)) | |
19 | 9, 17, 18 | sylanbrc 581 | 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 394 = wceq 1533 β wcel 2098 βwral 3058 βͺ cuni 4912 β¦ cmpt 5235 βΆwf 6549 βcfv 6553 Xcixp 8922 βtcpt 17427 Topctop 22815 TopOnctopon 22832 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-om 7877 df-1o 8493 df-er 8731 df-ixp 8923 df-en 8971 df-fin 8974 df-fi 9442 df-topgen 17432 df-pt 17433 df-top 22816 df-topon 22833 df-bases 22869 |
This theorem is referenced by: pttoponconst 23521 ptclsg 23539 dfac14lem 23541 ptcnp 23546 prdstps 23553 |
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