Theorem 2idlval 20009

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 6673	. . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3		2idlval.i	. . . . . 6 ⊢ 𝐼 = (LIdeal‘𝑅)
4	2, 3	syl6eqr 2877	. . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼)
5		fveq2 6673	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅))
6		2idlval.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
7	5, 6	syl6eqr 2877	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑂)
8	7	fveq2d 6677	. . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = (LIdeal‘𝑂))
9		2idlval.j	. . . . . 6 ⊢ 𝐽 = (LIdeal‘𝑂)
10	8, 9	syl6eqr 2877	. . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = 𝐽)
11	4, 10	ineq12d 4193	. . . 4 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))) = (𝐼 ∩ 𝐽))
12		df-2idl 20008	. . . 4 ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))))
13	3	fvexi 6687	. . . . 5 ⊢ 𝐼 ∈ V
14	13	inex1 5224	. . . 4 ⊢ (𝐼 ∩ 𝐽) ∈ V
15	11, 12, 14	fvmpt 6771	. . 3 ⊢ (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽))
16		fvprc 6666	. . . 4 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = ∅)
17		inss1 4208	. . . . 5 ⊢ (𝐼 ∩ 𝐽) ⊆ 𝐼
18		fvprc 6666	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (LIdeal‘𝑅) = ∅)
19	3, 18	syl5eq 2871	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐼 = ∅)
20		sseq0 4356	. . . . 5 ⊢ (((𝐼 ∩ 𝐽) ⊆ 𝐼 ∧ 𝐼 = ∅) → (𝐼 ∩ 𝐽) = ∅)
21	17, 19, 20	sylancr 589	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝐼 ∩ 𝐽) = ∅)
22	16, 21	eqtr4d 2862	. . 3 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽))
23	15, 22	pm2.61i 184	. 2 ⊢ (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)
24	1, 23	eqtri 2847	1 ⊢ 𝑇 = (𝐼 ∩ 𝐽)

Syntax hints:  ¬ wn 3   = wceq 1536   ∈ wcel 2113  Vcvv 3497   ∩ cin 3938   ⊆ wss 3939  ∅c0 4294  ‘cfv 6358  opp_rcoppr 19375  LIdealclidl 19945  2Idealc2idl 20007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fv 6366  df-2idl 20008

This theorem is referenced by:  2idlcpbl  20010  qus1  20011  qusrhm  20013  crng2idl  20015