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Mirrors > Home > MPE Home > Th. List > istdrg | Structured version Visualization version GIF version |
Description: Express the predicate "π is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
istrg.1 | β’ π = (mulGrpβπ ) |
istdrg.1 | β’ π = (Unitβπ ) |
Ref | Expression |
---|---|
istdrg | β’ (π β TopDRing β (π β TopRing β§ π β DivRing β§ (π βΎs π) β TopGrp)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3963 | . . 3 β’ (π β (TopRing β© DivRing) β (π β TopRing β§ π β DivRing)) | |
2 | 1 | anbi1i 624 | . 2 β’ ((π β (TopRing β© DivRing) β§ (π βΎs π) β TopGrp) β ((π β TopRing β§ π β DivRing) β§ (π βΎs π) β TopGrp)) |
3 | fveq2 6888 | . . . . . 6 β’ (π = π β (mulGrpβπ) = (mulGrpβπ )) | |
4 | istrg.1 | . . . . . 6 β’ π = (mulGrpβπ ) | |
5 | 3, 4 | eqtr4di 2790 | . . . . 5 β’ (π = π β (mulGrpβπ) = π) |
6 | fveq2 6888 | . . . . . 6 β’ (π = π β (Unitβπ) = (Unitβπ )) | |
7 | istdrg.1 | . . . . . 6 β’ π = (Unitβπ ) | |
8 | 6, 7 | eqtr4di 2790 | . . . . 5 β’ (π = π β (Unitβπ) = π) |
9 | 5, 8 | oveq12d 7423 | . . . 4 β’ (π = π β ((mulGrpβπ) βΎs (Unitβπ)) = (π βΎs π)) |
10 | 9 | eleq1d 2818 | . . 3 β’ (π = π β (((mulGrpβπ) βΎs (Unitβπ)) β TopGrp β (π βΎs π) β TopGrp)) |
11 | df-tdrg 23656 | . . 3 β’ TopDRing = {π β (TopRing β© DivRing) β£ ((mulGrpβπ) βΎs (Unitβπ)) β TopGrp} | |
12 | 10, 11 | elrab2 3685 | . 2 β’ (π β TopDRing β (π β (TopRing β© DivRing) β§ (π βΎs π) β TopGrp)) |
13 | df-3an 1089 | . 2 β’ ((π β TopRing β§ π β DivRing β§ (π βΎs π) β TopGrp) β ((π β TopRing β§ π β DivRing) β§ (π βΎs π) β TopGrp)) | |
14 | 2, 12, 13 | 3bitr4i 302 | 1 β’ (π β TopDRing β (π β TopRing β§ π β DivRing β§ (π βΎs π) β TopGrp)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β© cin 3946 βcfv 6540 (class class class)co 7405 βΎs cress 17169 mulGrpcmgp 19981 Unitcui 20161 DivRingcdr 20307 TopGrpctgp 23566 TopRingctrg 23651 TopDRingctdrg 23652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-ov 7408 df-tdrg 23656 |
This theorem is referenced by: tdrgunit 23662 tdrgtrg 23668 tdrgdrng 23669 istdrg2 23673 nrgtdrg 24201 |
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