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Theorem 2idllidld 21100
Description: A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
Hypothesis
Ref Expression
2idllidld.1 (๐œ‘ โ†’ ๐ผ โˆˆ (2Idealโ€˜๐‘…))
Assertion
Ref Expression
2idllidld (๐œ‘ โ†’ ๐ผ โˆˆ (LIdealโ€˜๐‘…))

Proof of Theorem 2idllidld
StepHypRef Expression
1 2idllidld.1 . . 3 (๐œ‘ โ†’ ๐ผ โˆˆ (2Idealโ€˜๐‘…))
2 eqid 2724 . . . 4 (LIdealโ€˜๐‘…) = (LIdealโ€˜๐‘…)
3 eqid 2724 . . . 4 (opprโ€˜๐‘…) = (opprโ€˜๐‘…)
4 eqid 2724 . . . 4 (LIdealโ€˜(opprโ€˜๐‘…)) = (LIdealโ€˜(opprโ€˜๐‘…))
5 eqid 2724 . . . 4 (2Idealโ€˜๐‘…) = (2Idealโ€˜๐‘…)
62, 3, 4, 52idlval 21097 . . 3 (2Idealโ€˜๐‘…) = ((LIdealโ€˜๐‘…) โˆฉ (LIdealโ€˜(opprโ€˜๐‘…)))
71, 6eleqtrdi 2835 . 2 (๐œ‘ โ†’ ๐ผ โˆˆ ((LIdealโ€˜๐‘…) โˆฉ (LIdealโ€˜(opprโ€˜๐‘…))))
87elin1d 4190 1 (๐œ‘ โ†’ ๐ผ โˆˆ (LIdealโ€˜๐‘…))
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆˆ wcel 2098   โˆฉ cin 3939  โ€˜cfv 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:  df2idl2  21103  2idlss  21108  qusmul2  21123  rng2idl1cntr  21147  qsnzr  33009  opprqusmulr  33040  opprqus1r  33041  opprqusdrng  33042  qsdrnglem2  33045  qsdrng  33046
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