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Theorem 2idlval 20719
Description: Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
2idlval.i 𝐼 = (LIdealβ€˜π‘…)
2idlval.o 𝑂 = (opprβ€˜π‘…)
2idlval.j 𝐽 = (LIdealβ€˜π‘‚)
2idlval.t 𝑇 = (2Idealβ€˜π‘…)
Assertion
Ref Expression
2idlval 𝑇 = (𝐼 ∩ 𝐽)

Proof of Theorem 2idlval
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 2idlval.t . 2 𝑇 = (2Idealβ€˜π‘…)
2 fveq2 6843 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜π‘Ÿ) = (LIdealβ€˜π‘…))
3 2idlval.i . . . . . 6 𝐼 = (LIdealβ€˜π‘…)
42, 3eqtr4di 2791 . . . . 5 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜π‘Ÿ) = 𝐼)
5 fveq2 6843 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (opprβ€˜π‘Ÿ) = (opprβ€˜π‘…))
6 2idlval.o . . . . . . . 8 𝑂 = (opprβ€˜π‘…)
75, 6eqtr4di 2791 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (opprβ€˜π‘Ÿ) = 𝑂)
87fveq2d 6847 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜(opprβ€˜π‘Ÿ)) = (LIdealβ€˜π‘‚))
9 2idlval.j . . . . . 6 𝐽 = (LIdealβ€˜π‘‚)
108, 9eqtr4di 2791 . . . . 5 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜(opprβ€˜π‘Ÿ)) = 𝐽)
114, 10ineq12d 4174 . . . 4 (π‘Ÿ = 𝑅 β†’ ((LIdealβ€˜π‘Ÿ) ∩ (LIdealβ€˜(opprβ€˜π‘Ÿ))) = (𝐼 ∩ 𝐽))
12 df-2idl 20718 . . . 4 2Ideal = (π‘Ÿ ∈ V ↦ ((LIdealβ€˜π‘Ÿ) ∩ (LIdealβ€˜(opprβ€˜π‘Ÿ))))
133fvexi 6857 . . . . 5 𝐼 ∈ V
1413inex1 5275 . . . 4 (𝐼 ∩ 𝐽) ∈ V
1511, 12, 14fvmpt 6949 . . 3 (𝑅 ∈ V β†’ (2Idealβ€˜π‘…) = (𝐼 ∩ 𝐽))
16 fvprc 6835 . . . 4 (Β¬ 𝑅 ∈ V β†’ (2Idealβ€˜π‘…) = βˆ…)
17 inss1 4189 . . . . 5 (𝐼 ∩ 𝐽) βŠ† 𝐼
18 fvprc 6835 . . . . . 6 (Β¬ 𝑅 ∈ V β†’ (LIdealβ€˜π‘…) = βˆ…)
193, 18eqtrid 2785 . . . . 5 (Β¬ 𝑅 ∈ V β†’ 𝐼 = βˆ…)
20 sseq0 4360 . . . . 5 (((𝐼 ∩ 𝐽) βŠ† 𝐼 ∧ 𝐼 = βˆ…) β†’ (𝐼 ∩ 𝐽) = βˆ…)
2117, 19, 20sylancr 588 . . . 4 (Β¬ 𝑅 ∈ V β†’ (𝐼 ∩ 𝐽) = βˆ…)
2216, 21eqtr4d 2776 . . 3 (Β¬ 𝑅 ∈ V β†’ (2Idealβ€˜π‘…) = (𝐼 ∩ 𝐽))
2315, 22pm2.61i 182 . 2 (2Idealβ€˜π‘…) = (𝐼 ∩ 𝐽)
241, 23eqtri 2761 1 𝑇 = (𝐼 ∩ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1542   ∈ wcel 2107  Vcvv 3444   ∩ cin 3910   βŠ† wss 3911  βˆ…c0 4283  β€˜cfv 6497  opprcoppr 20053  LIdealclidl 20647  2Idealc2idl 20717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-2idl 20718
This theorem is referenced by:  2idlcpbl  20720  qus1  20721  qusrhm  20723  crng2idl  20725
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