MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2idlval Structured version   Visualization version   GIF version

Theorem 2idlval 19695
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 6541 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3 2idlval.i . . . . . 6 𝐼 = (LIdeal‘𝑅)
42, 3syl6eqr 2848 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼)
5 fveq2 6541 . . . . . . . 8 (𝑟 = 𝑅 → (oppr𝑟) = (oppr𝑅))
6 2idlval.o . . . . . . . 8 𝑂 = (oppr𝑅)
75, 6syl6eqr 2848 . . . . . . 7 (𝑟 = 𝑅 → (oppr𝑟) = 𝑂)
87fveq2d 6545 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = (LIdeal‘𝑂))
9 2idlval.j . . . . . 6 𝐽 = (LIdeal‘𝑂)
108, 9syl6eqr 2848 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = 𝐽)
114, 10ineq12d 4112 . . . 4 (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))) = (𝐼𝐽))
12 df-2idl 19694 . . . 4 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))
133fvexi 6555 . . . . 5 𝐼 ∈ V
1413inex1 5115 . . . 4 (𝐼𝐽) ∈ V
1511, 12, 14fvmpt 6638 . . 3 (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
16 fvprc 6534 . . . 4 𝑅 ∈ V → (2Ideal‘𝑅) = ∅)
17 inss1 4127 . . . . 5 (𝐼𝐽) ⊆ 𝐼
18 fvprc 6534 . . . . . 6 𝑅 ∈ V → (LIdeal‘𝑅) = ∅)
193, 18syl5eq 2842 . . . . 5 𝑅 ∈ V → 𝐼 = ∅)
20 sseq0 4275 . . . . 5 (((𝐼𝐽) ⊆ 𝐼𝐼 = ∅) → (𝐼𝐽) = ∅)
2117, 19, 20sylancr 587 . . . 4 𝑅 ∈ V → (𝐼𝐽) = ∅)
2216, 21eqtr4d 2833 . . 3 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
2315, 22pm2.61i 183 . 2 (2Ideal‘𝑅) = (𝐼𝐽)
241, 23eqtri 2818 1 𝑇 = (𝐼𝐽)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1522  wcel 2080  Vcvv 3436  cin 3860  wss 3861  c0 4213  cfv 6228  opprcoppr 19062  LIdealclidl 19632  2Idealc2idl 19693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-iota 6192  df-fun 6230  df-fv 6236  df-2idl 19694
This theorem is referenced by:  2idlcpbl  19696  qus1  19697  qusrhm  19699  crng2idl  19701
  Copyright terms: Public domain W3C validator