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Mirrors > Home > MPE Home > Th. List > 2idlridld | Structured version Visualization version GIF version |
Description: A two-sided ideal is a right ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
Ref | Expression |
---|---|
2idllidld.1 | โข (๐ โ ๐ผ โ (2Idealโ๐ )) |
2idlridld.o | โข ๐ = (opprโ๐ ) |
Ref | Expression |
---|---|
2idlridld | โข (๐ โ ๐ผ โ (LIdealโ๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2idllidld.1 | . . 3 โข (๐ โ ๐ผ โ (2Idealโ๐ )) | |
2 | eqid 2724 | . . . 4 โข (LIdealโ๐ ) = (LIdealโ๐ ) | |
3 | 2idlridld.o | . . . 4 โข ๐ = (opprโ๐ ) | |
4 | eqid 2724 | . . . 4 โข (LIdealโ๐) = (LIdealโ๐) | |
5 | eqid 2724 | . . . 4 โข (2Idealโ๐ ) = (2Idealโ๐ ) | |
6 | 2, 3, 4, 5 | 2idlval 21097 | . . 3 โข (2Idealโ๐ ) = ((LIdealโ๐ ) โฉ (LIdealโ๐)) |
7 | 1, 6 | eleqtrdi 2835 | . 2 โข (๐ โ ๐ผ โ ((LIdealโ๐ ) โฉ (LIdealโ๐))) |
8 | 7 | elin2d 4191 | 1 โข (๐ โ ๐ผ โ (LIdealโ๐)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โฉ cin 3939 โcfv 6533 opprcoppr 20224 LIdealclidl 21054 2Idealc2idl 21095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6485 df-fun 6535 df-fv 6541 df-2idl 21096 |
This theorem is referenced by: qsdrng 33046 |
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