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Mirrors > Home > MPE Home > Th. List > isrrg | Structured version Visualization version GIF version |
Description: Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
Ref | Expression |
---|---|
rrgval.e | β’ πΈ = (RLRegβπ ) |
rrgval.b | β’ π΅ = (Baseβπ ) |
rrgval.t | β’ Β· = (.rβπ ) |
rrgval.z | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
isrrg | β’ (π β πΈ β (π β π΅ β§ βπ¦ β π΅ ((π Β· π¦) = 0 β π¦ = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7365 | . . . . 5 β’ (π₯ = π β (π₯ Β· π¦) = (π Β· π¦)) | |
2 | 1 | eqeq1d 2735 | . . . 4 β’ (π₯ = π β ((π₯ Β· π¦) = 0 β (π Β· π¦) = 0 )) |
3 | 2 | imbi1d 342 | . . 3 β’ (π₯ = π β (((π₯ Β· π¦) = 0 β π¦ = 0 ) β ((π Β· π¦) = 0 β π¦ = 0 ))) |
4 | 3 | ralbidv 3171 | . 2 β’ (π₯ = π β (βπ¦ β π΅ ((π₯ Β· π¦) = 0 β π¦ = 0 ) β βπ¦ β π΅ ((π Β· π¦) = 0 β π¦ = 0 ))) |
5 | rrgval.e | . . 3 β’ πΈ = (RLRegβπ ) | |
6 | rrgval.b | . . 3 β’ π΅ = (Baseβπ ) | |
7 | rrgval.t | . . 3 β’ Β· = (.rβπ ) | |
8 | rrgval.z | . . 3 β’ 0 = (0gβπ ) | |
9 | 5, 6, 7, 8 | rrgval 20773 | . 2 β’ πΈ = {π₯ β π΅ β£ βπ¦ β π΅ ((π₯ Β· π¦) = 0 β π¦ = 0 )} |
10 | 4, 9 | elrab2 3649 | 1 β’ (π β πΈ β (π β π΅ β§ βπ¦ β π΅ ((π Β· π¦) = 0 β π¦ = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 βcfv 6497 (class class class)co 7358 Basecbs 17088 .rcmulr 17139 0gc0g 17326 RLRegcrlreg 20765 |
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 5257 ax-nul 5264 ax-pr 5385 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-rlreg 20769 |
This theorem is referenced by: rrgeq0i 20775 unitrrg 20779 isdomn2 20785 |
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