Theorem 2idlval 21181

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 6817	. . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3		2idlval.i	. . . . . 6 ⊢ 𝐼 = (LIdeal‘𝑅)
4	2, 3	eqtr4di 2783	. . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼)
5		fveq2 6817	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅))
6		2idlval.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
7	5, 6	eqtr4di 2783	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑂)
8	7	fveq2d 6821	. . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = (LIdeal‘𝑂))
9		2idlval.j	. . . . . 6 ⊢ 𝐽 = (LIdeal‘𝑂)
10	8, 9	eqtr4di 2783	. . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = 𝐽)
11	4, 10	ineq12d 4169	. . . 4 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))) = (𝐼 ∩ 𝐽))
12		df-2idl 21180	. . . 4 ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))))
13	3	fvexi 6831	. . . . 5 ⊢ 𝐼 ∈ V
14	13	inex1 5253	. . . 4 ⊢ (𝐼 ∩ 𝐽) ∈ V
15	11, 12, 14	fvmpt 6924	. . 3 ⊢ (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽))
16		fvprc 6809	. . . 4 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = ∅)
17		inss1 4185	. . . . 5 ⊢ (𝐼 ∩ 𝐽) ⊆ 𝐼
18		fvprc 6809	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (LIdeal‘𝑅) = ∅)
19	3, 18	eqtrid 2777	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐼 = ∅)
20		sseq0 4351	. . . . 5 ⊢ (((𝐼 ∩ 𝐽) ⊆ 𝐼 ∧ 𝐼 = ∅) → (𝐼 ∩ 𝐽) = ∅)
21	17, 19, 20	sylancr 587	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝐼 ∩ 𝐽) = ∅)
22	16, 21	eqtr4d 2768	. . 3 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽))
23	15, 22	pm2.61i 182	. 2 ⊢ (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)
24	1, 23	eqtri 2753	1 ⊢ 𝑇 = (𝐼 ∩ 𝐽)

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2110  Vcvv 3434   ∩ cin 3899   ⊆ wss 3900  ∅c0 4281  ‘cfv 6477  opp_rcoppr 20247  LIdealclidl 21136  2Idealc2idl 21179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6433  df-fun 6479  df-fv 6485  df-2idl 21180

This theorem is referenced by:  2idlelb  21183  2idllidld  21184  2idlridld  21185  2idl0  21190  2idl1  21191  qus1  21204  qusrhm  21206  crng2idl  21211  oppr2idl  33441  qsdrngilem  33449  qsdrngi  33450