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Theorem 2idlval 19456
Description: Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
2idlval.i 𝐼 = (LIdeal‘𝑅)
2idlval.o 𝑂 = (oppr𝑅)
2idlval.j 𝐽 = (LIdeal‘𝑂)
2idlval.t 𝑇 = (2Ideal‘𝑅)
Assertion
Ref Expression
2idlval 𝑇 = (𝐼𝐽)

Proof of Theorem 2idlval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 2idlval.t . 2 𝑇 = (2Ideal‘𝑅)
2 fveq2 6354 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3 2idlval.i . . . . . 6 𝐼 = (LIdeal‘𝑅)
42, 3syl6eqr 2813 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼)
5 fveq2 6354 . . . . . . . 8 (𝑟 = 𝑅 → (oppr𝑟) = (oppr𝑅))
6 2idlval.o . . . . . . . 8 𝑂 = (oppr𝑅)
75, 6syl6eqr 2813 . . . . . . 7 (𝑟 = 𝑅 → (oppr𝑟) = 𝑂)
87fveq2d 6358 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = (LIdeal‘𝑂))
9 2idlval.j . . . . . 6 𝐽 = (LIdeal‘𝑂)
108, 9syl6eqr 2813 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = 𝐽)
114, 10ineq12d 3959 . . . 4 (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))) = (𝐼𝐽))
12 df-2idl 19455 . . . 4 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))
13 fvex 6364 . . . . . 6 (LIdeal‘𝑅) ∈ V
143, 13eqeltri 2836 . . . . 5 𝐼 ∈ V
1514inex1 4952 . . . 4 (𝐼𝐽) ∈ V
1611, 12, 15fvmpt 6446 . . 3 (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
17 fvprc 6348 . . . 4 𝑅 ∈ V → (2Ideal‘𝑅) = ∅)
18 inss1 3977 . . . . 5 (𝐼𝐽) ⊆ 𝐼
19 fvprc 6348 . . . . . 6 𝑅 ∈ V → (LIdeal‘𝑅) = ∅)
203, 19syl5eq 2807 . . . . 5 𝑅 ∈ V → 𝐼 = ∅)
21 sseq0 4119 . . . . 5 (((𝐼𝐽) ⊆ 𝐼𝐼 = ∅) → (𝐼𝐽) = ∅)
2218, 20, 21sylancr 698 . . . 4 𝑅 ∈ V → (𝐼𝐽) = ∅)
2317, 22eqtr4d 2798 . . 3 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
2416, 23pm2.61i 176 . 2 (2Ideal‘𝑅) = (𝐼𝐽)
251, 24eqtri 2783 1 𝑇 = (𝐼𝐽)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2140  Vcvv 3341  cin 3715  wss 3716  c0 4059  cfv 6050  opprcoppr 18843  LIdealclidl 19393  2Idealc2idl 19454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-iota 6013  df-fun 6052  df-fv 6058  df-2idl 19455
This theorem is referenced by:  2idlcpbl  19457  qus1  19458  qusrhm  19460  crng2idl  19462
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