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Mirrors > Home > MPE Home > Th. List > drngunit | Structured version Visualization version GIF version |
Description: Elementhood in the set of units when π is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
isdrng.b | β’ π΅ = (Baseβπ ) |
isdrng.u | β’ π = (Unitβπ ) |
isdrng.z | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
drngunit | β’ (π β DivRing β (π β π β (π β π΅ β§ π β 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isdrng.b | . . . . 5 β’ π΅ = (Baseβπ ) | |
2 | isdrng.u | . . . . 5 β’ π = (Unitβπ ) | |
3 | isdrng.z | . . . . 5 β’ 0 = (0gβπ ) | |
4 | 1, 2, 3 | isdrng 20505 | . . . 4 β’ (π β DivRing β (π β Ring β§ π = (π΅ β { 0 }))) |
5 | 4 | simprbi 496 | . . 3 β’ (π β DivRing β π = (π΅ β { 0 })) |
6 | 5 | eleq2d 2818 | . 2 β’ (π β DivRing β (π β π β π β (π΅ β { 0 }))) |
7 | eldifsn 4790 | . 2 β’ (π β (π΅ β { 0 }) β (π β π΅ β§ π β 0 )) | |
8 | 6, 7 | bitrdi 287 | 1 β’ (π β DivRing β (π β π β (π β π΅ β§ π β 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1540 β wcel 2105 β wne 2939 β cdif 3945 {csn 4628 βcfv 6543 Basecbs 17149 0gc0g 17390 Ringcrg 20128 Unitcui 20247 DivRingcdr 20501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-drng 20503 |
This theorem is referenced by: drngunz 20520 drnginvrcl 20523 drnginvrn0 20524 drnginvrl 20526 drnginvrr 20527 issubdrg 20545 sdrgunit 20556 abvdiv 20589 qsssubdrg 21205 redvr 21390 drnguc1p 25924 lgseisenlem3 27117 sdrgdvcl 32668 sdrginvcl 32669 ornglmullt 32696 orngrmullt 32697 isarchiofld 32706 ply1asclunit 32929 qqhval2lem 33260 qqhf 33265 matunitlindf 36790 fldhmf1 41262 lincreslvec3 47251 isldepslvec2 47254 |
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