Theorem 2idlval 21152

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 = (oppr‘𝑅)
2idlval.j	⊢ 𝐽 = (LIdeal‘𝑂)
2idlval.t	⊢ 𝑇 = (2Ideal‘𝑅)

Assertion

Ref	Expression
2idlval	⊢ 𝑇 = (𝐼 ∩ 𝐽)

Proof of Theorem 2idlval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2idlval.t	. 2 ⊢ 𝑇 = (2Ideal‘𝑅)
2		fveq2 6902	. . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3		2idlval.i	. . . . . 6 ⊢ 𝐼 = (LIdeal‘𝑅)
4	2, 3	eqtr4di 2786	. . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼)
5		fveq2 6902	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅))
6		2idlval.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
7	5, 6	eqtr4di 2786	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑂)
8	7	fveq2d 6906	. . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = (LIdeal‘𝑂))
9		2idlval.j	. . . . . 6 ⊢ 𝐽 = (LIdeal‘𝑂)
10	8, 9	eqtr4di 2786	. . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = 𝐽)
11	4, 10	ineq12d 4215	. . . 4 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))) = (𝐼 ∩ 𝐽))
12		df-2idl 21151	. . . 4 ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))))
13	3	fvexi 6916	. . . . 5 ⊢ 𝐼 ∈ V
14	13	inex1 5321	. . . 4 ⊢ (𝐼 ∩ 𝐽) ∈ V
15	11, 12, 14	fvmpt 7010	. . 3 ⊢ (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽))
16		fvprc 6894	. . . 4 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = ∅)
17		inss1 4231	. . . . 5 ⊢ (𝐼 ∩ 𝐽) ⊆ 𝐼
18		fvprc 6894	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (LIdeal‘𝑅) = ∅)
19	3, 18	eqtrid 2780	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐼 = ∅)
20		sseq0 4403	. . . . 5 ⊢ (((𝐼 ∩ 𝐽) ⊆ 𝐼 ∧ 𝐼 = ∅) → (𝐼 ∩ 𝐽) = ∅)
21	17, 19, 20	sylancr 585	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝐼 ∩ 𝐽) = ∅)
22	16, 21	eqtr4d 2771	. . 3 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽))
23	15, 22	pm2.61i 182	. 2 ⊢ (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)
24	1, 23	eqtri 2756	1 ⊢ 𝑇 = (𝐼 ∩ 𝐽)

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098  Vcvv 3473   ∩ cin 3948   ⊆ wss 3949  ∅c0 4326  ‘cfv 6553  opp_rcoppr 20279  LIdealclidl 21109  2Idealc2idl 21150

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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-2idl 21151

This theorem is referenced by:  2idlelb  21154  2idllidld  21155  2idlridld  21156  2idl0  21161  2idl1  21162  qus1  21175  qusrhm  21177  crng2idl  21180  oppr2idl  33222  qsdrngilem  33230  qsdrngi  33231