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Theorem 2idlval 21105
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 6884 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜π‘Ÿ) = (LIdealβ€˜π‘…))
3 2idlval.i . . . . . 6 𝐼 = (LIdealβ€˜π‘…)
42, 3eqtr4di 2784 . . . . 5 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜π‘Ÿ) = 𝐼)
5 fveq2 6884 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (opprβ€˜π‘Ÿ) = (opprβ€˜π‘…))
6 2idlval.o . . . . . . . 8 𝑂 = (opprβ€˜π‘…)
75, 6eqtr4di 2784 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (opprβ€˜π‘Ÿ) = 𝑂)
87fveq2d 6888 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜(opprβ€˜π‘Ÿ)) = (LIdealβ€˜π‘‚))
9 2idlval.j . . . . . 6 𝐽 = (LIdealβ€˜π‘‚)
108, 9eqtr4di 2784 . . . . 5 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜(opprβ€˜π‘Ÿ)) = 𝐽)
114, 10ineq12d 4208 . . . 4 (π‘Ÿ = 𝑅 β†’ ((LIdealβ€˜π‘Ÿ) ∩ (LIdealβ€˜(opprβ€˜π‘Ÿ))) = (𝐼 ∩ 𝐽))
12 df-2idl 21104 . . . 4 2Ideal = (π‘Ÿ ∈ V ↦ ((LIdealβ€˜π‘Ÿ) ∩ (LIdealβ€˜(opprβ€˜π‘Ÿ))))
133fvexi 6898 . . . . 5 𝐼 ∈ V
1413inex1 5310 . . . 4 (𝐼 ∩ 𝐽) ∈ V
1511, 12, 14fvmpt 6991 . . 3 (𝑅 ∈ V β†’ (2Idealβ€˜π‘…) = (𝐼 ∩ 𝐽))
16 fvprc 6876 . . . 4 (Β¬ 𝑅 ∈ V β†’ (2Idealβ€˜π‘…) = βˆ…)
17 inss1 4223 . . . . 5 (𝐼 ∩ 𝐽) βŠ† 𝐼
18 fvprc 6876 . . . . . 6 (Β¬ 𝑅 ∈ V β†’ (LIdealβ€˜π‘…) = βˆ…)
193, 18eqtrid 2778 . . . . 5 (Β¬ 𝑅 ∈ V β†’ 𝐼 = βˆ…)
20 sseq0 4394 . . . . 5 (((𝐼 ∩ 𝐽) βŠ† 𝐼 ∧ 𝐼 = βˆ…) β†’ (𝐼 ∩ 𝐽) = βˆ…)
2117, 19, 20sylancr 586 . . . 4 (Β¬ 𝑅 ∈ V β†’ (𝐼 ∩ 𝐽) = βˆ…)
2216, 21eqtr4d 2769 . . 3 (Β¬ 𝑅 ∈ V β†’ (2Idealβ€˜π‘…) = (𝐼 ∩ 𝐽))
2315, 22pm2.61i 182 . 2 (2Idealβ€˜π‘…) = (𝐼 ∩ 𝐽)
241, 23eqtri 2754 1 𝑇 = (𝐼 ∩ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ∩ cin 3942   βŠ† wss 3943  βˆ…c0 4317  β€˜cfv 6536  opprcoppr 20232  LIdealclidl 21062  2Idealc2idl 21103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-2idl 21104
This theorem is referenced by:  2idlelb  21107  2idllidld  21108  2idlridld  21109  2idl0  21114  2idl1  21115  qus1  21128  qusrhm  21130  crng2idl  21133  oppr2idl  33105  qsdrngilem  33113  qsdrngi  33114
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