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Theorem 2idlval 19227
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 6189 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3 2idlval.i . . . . . 6 𝐼 = (LIdeal‘𝑅)
42, 3syl6eqr 2673 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼)
5 fveq2 6189 . . . . . . . 8 (𝑟 = 𝑅 → (oppr𝑟) = (oppr𝑅))
6 2idlval.o . . . . . . . 8 𝑂 = (oppr𝑅)
75, 6syl6eqr 2673 . . . . . . 7 (𝑟 = 𝑅 → (oppr𝑟) = 𝑂)
87fveq2d 6193 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = (LIdeal‘𝑂))
9 2idlval.j . . . . . 6 𝐽 = (LIdeal‘𝑂)
108, 9syl6eqr 2673 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = 𝐽)
114, 10ineq12d 3813 . . . 4 (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))) = (𝐼𝐽))
12 df-2idl 19226 . . . 4 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))
13 fvex 6199 . . . . . 6 (LIdeal‘𝑅) ∈ V
143, 13eqeltri 2696 . . . . 5 𝐼 ∈ V
1514inex1 4797 . . . 4 (𝐼𝐽) ∈ V
1611, 12, 15fvmpt 6280 . . 3 (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
17 fvprc 6183 . . . 4 𝑅 ∈ V → (2Ideal‘𝑅) = ∅)
18 inss1 3831 . . . . 5 (𝐼𝐽) ⊆ 𝐼
19 fvprc 6183 . . . . . 6 𝑅 ∈ V → (LIdeal‘𝑅) = ∅)
203, 19syl5eq 2667 . . . . 5 𝑅 ∈ V → 𝐼 = ∅)
21 sseq0 3973 . . . . 5 (((𝐼𝐽) ⊆ 𝐼𝐼 = ∅) → (𝐼𝐽) = ∅)
2218, 20, 21sylancr 695 . . . 4 𝑅 ∈ V → (𝐼𝐽) = ∅)
2317, 22eqtr4d 2658 . . 3 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
2416, 23pm2.61i 176 . 2 (2Ideal‘𝑅) = (𝐼𝐽)
251, 24eqtri 2643 1 𝑇 = (𝐼𝐽)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1482  wcel 1989  Vcvv 3198  cin 3571  wss 3572  c0 3913  cfv 5886  opprcoppr 18616  LIdealclidl 19164  2Idealc2idl 19225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-2idl 19226
This theorem is referenced by:  2idlcpbl  19228  qus1  19229  qusrhm  19231  crng2idl  19233
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