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Mirrors > Home > MPE Home > Th. List > dvrcn | Structured version Visualization version GIF version |
Description: The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
dvrcn.j | β’ π½ = (TopOpenβπ ) |
dvrcn.d | β’ / = (/rβπ ) |
dvrcn.u | β’ π = (Unitβπ ) |
Ref | Expression |
---|---|
dvrcn | β’ (π β TopDRing β / β ((π½ Γt (π½ βΎt π)) Cn π½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
2 | eqid 2733 | . . 3 β’ (.rβπ ) = (.rβπ ) | |
3 | dvrcn.u | . . 3 β’ π = (Unitβπ ) | |
4 | eqid 2733 | . . 3 β’ (invrβπ ) = (invrβπ ) | |
5 | dvrcn.d | . . 3 β’ / = (/rβπ ) | |
6 | 1, 2, 3, 4, 5 | dvrfval 20118 | . 2 β’ / = (π₯ β (Baseβπ ), π¦ β π β¦ (π₯(.rβπ )((invrβπ )βπ¦))) |
7 | dvrcn.j | . . 3 β’ π½ = (TopOpenβπ ) | |
8 | tdrgtrg 23540 | . . 3 β’ (π β TopDRing β π β TopRing) | |
9 | tdrgtps 23544 | . . . 4 β’ (π β TopDRing β π β TopSp) | |
10 | 1, 7 | istps 22299 | . . . 4 β’ (π β TopSp β π½ β (TopOnβ(Baseβπ ))) |
11 | 9, 10 | sylib 217 | . . 3 β’ (π β TopDRing β π½ β (TopOnβ(Baseβπ ))) |
12 | 1, 3 | unitss 20094 | . . . 4 β’ π β (Baseβπ ) |
13 | resttopon 22528 | . . . 4 β’ ((π½ β (TopOnβ(Baseβπ )) β§ π β (Baseβπ )) β (π½ βΎt π) β (TopOnβπ)) | |
14 | 11, 12, 13 | sylancl 587 | . . 3 β’ (π β TopDRing β (π½ βΎt π) β (TopOnβπ)) |
15 | 11, 14 | cnmpt1st 23035 | . . 3 β’ (π β TopDRing β (π₯ β (Baseβπ ), π¦ β π β¦ π₯) β ((π½ Γt (π½ βΎt π)) Cn π½)) |
16 | 11, 14 | cnmpt2nd 23036 | . . . 4 β’ (π β TopDRing β (π₯ β (Baseβπ ), π¦ β π β¦ π¦) β ((π½ Γt (π½ βΎt π)) Cn (π½ βΎt π))) |
17 | 7, 4, 3 | invrcn 23548 | . . . 4 β’ (π β TopDRing β (invrβπ ) β ((π½ βΎt π) Cn π½)) |
18 | 11, 14, 16, 17 | cnmpt21f 23039 | . . 3 β’ (π β TopDRing β (π₯ β (Baseβπ ), π¦ β π β¦ ((invrβπ )βπ¦)) β ((π½ Γt (π½ βΎt π)) Cn π½)) |
19 | 7, 2, 8, 11, 14, 15, 18 | cnmpt2mulr 23550 | . 2 β’ (π β TopDRing β (π₯ β (Baseβπ ), π¦ β π β¦ (π₯(.rβπ )((invrβπ )βπ¦))) β ((π½ Γt (π½ βΎt π)) Cn π½)) |
20 | 6, 19 | eqeltrid 2838 | 1 β’ (π β TopDRing β / β ((π½ Γt (π½ βΎt π)) Cn π½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3911 βcfv 6497 (class class class)co 7358 β cmpo 7360 Basecbs 17088 .rcmulr 17139 βΎt crest 17307 TopOpenctopn 17308 Unitcui 20073 invrcinvr 20105 /rcdvr 20116 TopOnctopon 22275 TopSpctps 22297 Cn ccn 22591 Γt ctx 22927 TopDRingctdrg 23524 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9352 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-tset 17157 df-rest 17309 df-topn 17310 df-topgen 17330 df-plusf 18501 df-minusg 18757 df-mgp 19902 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cn 22594 df-tx 22929 df-tmd 23439 df-tgp 23440 df-trg 23527 df-tdrg 23528 |
This theorem is referenced by: (None) |
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