Theorem 2idlval 21210

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 6835	. . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3		2idlval.i	. . . . . 6 ⊢ 𝐼 = (LIdeal‘𝑅)
4	2, 3	eqtr4di 2790	. . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼)
5		fveq2 6835	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅))
6		2idlval.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
7	5, 6	eqtr4di 2790	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑂)
8	7	fveq2d 6839	. . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = (LIdeal‘𝑂))
9		2idlval.j	. . . . . 6 ⊢ 𝐽 = (LIdeal‘𝑂)
10	8, 9	eqtr4di 2790	. . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = 𝐽)
11	4, 10	ineq12d 4174	. . . 4 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))) = (𝐼 ∩ 𝐽))
12		df-2idl 21209	. . . 4 ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))))
13	3	fvexi 6849	. . . . 5 ⊢ 𝐼 ∈ V
14	13	inex1 5263	. . . 4 ⊢ (𝐼 ∩ 𝐽) ∈ V
15	11, 12, 14	fvmpt 6942	. . 3 ⊢ (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽))
16		fvprc 6827	. . . 4 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = ∅)
17		inss1 4190	. . . . 5 ⊢ (𝐼 ∩ 𝐽) ⊆ 𝐼
18		fvprc 6827	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (LIdeal‘𝑅) = ∅)
19	3, 18	eqtrid 2784	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐼 = ∅)
20		sseq0 4356	. . . . 5 ⊢ (((𝐼 ∩ 𝐽) ⊆ 𝐼 ∧ 𝐼 = ∅) → (𝐼 ∩ 𝐽) = ∅)
21	17, 19, 20	sylancr 588	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝐼 ∩ 𝐽) = ∅)
22	16, 21	eqtr4d 2775	. . 3 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽))
23	15, 22	pm2.61i 182	. 2 ⊢ (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)
24	1, 23	eqtri 2760	1 ⊢ 𝑇 = (𝐼 ∩ 𝐽)

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286  ‘cfv 6493  opp_rcoppr 20276  LIdealclidl 21165  2Idealc2idl 21208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-2idl 21209

This theorem is referenced by:  2idlelb  21212  2idllidld  21213  2idlridld  21214  2idl0  21219  2idl1  21220  qus1  21233  qusrhm  21235  crng2idl  21240  oppr2idl  33548  qsdrngilem  33556  qsdrngi  33557