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| Mirrors > Home > MPE Home > Th. List > txtop | Structured version Visualization version GIF version | ||
| Description: The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| Ref | Expression |
|---|---|
| txtop | β’ ((π β Top β§ π β Top) β (π Γt π) β Top) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2733 | . . 3 β’ ran (π’ β π , π£ β π β¦ (π’ Γ π£)) = ran (π’ β π , π£ β π β¦ (π’ Γ π£)) | |
| 2 | 1 | txval 23068 | . 2 β’ ((π β Top β§ π β Top) β (π Γt π) = (topGenβran (π’ β π , π£ β π β¦ (π’ Γ π£)))) |
| 3 | topbas 22475 | . . . 4 β’ (π β Top β π β TopBases) | |
| 4 | topbas 22475 | . . . 4 β’ (π β Top β π β TopBases) | |
| 5 | 1 | txbas 23071 | . . . 4 β’ ((π β TopBases β§ π β TopBases) β ran (π’ β π , π£ β π β¦ (π’ Γ π£)) β TopBases) |
| 6 | 3, 4, 5 | syl2an 597 | . . 3 β’ ((π β Top β§ π β Top) β ran (π’ β π , π£ β π β¦ (π’ Γ π£)) β TopBases) |
| 7 | tgcl 22472 | . . 3 β’ (ran (π’ β π , π£ β π β¦ (π’ Γ π£)) β TopBases β (topGenβran (π’ β π , π£ β π β¦ (π’ Γ π£))) β Top) | |
| 8 | 6, 7 | syl 17 | . 2 β’ ((π β Top β§ π β Top) β (topGenβran (π’ β π , π£ β π β¦ (π’ Γ π£))) β Top) |
| 9 | 2, 8 | eqeltrd 2834 | 1 β’ ((π β Top β§ π β Top) β (π Γt π) β Top) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 397 β wcel 2107 Γ cxp 5675 ran crn 5678 βcfv 6544 (class class class)co 7409 β cmpo 7411 topGenctg 17383 Topctop 22395 TopBasesctb 22448 Γt ctx 23064 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-topgen 17389 df-top 22396 df-bases 22449 df-tx 23066 |
| This theorem is referenced by: txtopi 23094 txtopon 23095 txcld 23107 neitx 23111 txlly 23140 txnlly 23141 txcmplem1 23145 txcmp 23147 hausdiag 23149 txhaus 23151 tx1stc 23154 txkgen 23156 xkococn 23164 xkoinjcn 23191 txconn 23193 imasnopn 23194 imasncls 23196 utop2nei 23755 utop3cls 23756 qtophaus 32816 txpconn 34223 |
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