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Theorem 2idlval 21261
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 6906 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3 2idlval.i . . . . . 6 𝐼 = (LIdeal‘𝑅)
42, 3eqtr4di 2795 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼)
5 fveq2 6906 . . . . . . . 8 (𝑟 = 𝑅 → (oppr𝑟) = (oppr𝑅))
6 2idlval.o . . . . . . . 8 𝑂 = (oppr𝑅)
75, 6eqtr4di 2795 . . . . . . 7 (𝑟 = 𝑅 → (oppr𝑟) = 𝑂)
87fveq2d 6910 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = (LIdeal‘𝑂))
9 2idlval.j . . . . . 6 𝐽 = (LIdeal‘𝑂)
108, 9eqtr4di 2795 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = 𝐽)
114, 10ineq12d 4221 . . . 4 (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))) = (𝐼𝐽))
12 df-2idl 21260 . . . 4 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))
133fvexi 6920 . . . . 5 𝐼 ∈ V
1413inex1 5317 . . . 4 (𝐼𝐽) ∈ V
1511, 12, 14fvmpt 7016 . . 3 (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
16 fvprc 6898 . . . 4 𝑅 ∈ V → (2Ideal‘𝑅) = ∅)
17 inss1 4237 . . . . 5 (𝐼𝐽) ⊆ 𝐼
18 fvprc 6898 . . . . . 6 𝑅 ∈ V → (LIdeal‘𝑅) = ∅)
193, 18eqtrid 2789 . . . . 5 𝑅 ∈ V → 𝐼 = ∅)
20 sseq0 4403 . . . . 5 (((𝐼𝐽) ⊆ 𝐼𝐼 = ∅) → (𝐼𝐽) = ∅)
2117, 19, 20sylancr 587 . . . 4 𝑅 ∈ V → (𝐼𝐽) = ∅)
2216, 21eqtr4d 2780 . . 3 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
2315, 22pm2.61i 182 . 2 (2Ideal‘𝑅) = (𝐼𝐽)
241, 23eqtri 2765 1 𝑇 = (𝐼𝐽)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  wss 3951  c0 4333  cfv 6561  opprcoppr 20333  LIdealclidl 21216  2Idealc2idl 21259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-2idl 21260
This theorem is referenced by:  2idlelb  21263  2idllidld  21264  2idlridld  21265  2idl0  21270  2idl1  21271  qus1  21284  qusrhm  21286  crng2idl  21291  oppr2idl  33514  qsdrngilem  33522  qsdrngi  33523
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