Theorem 2idlval 21544

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 6885	. . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3		2idlval.i	. . . . . 6 ⊢ 𝐼 = (LIdeal‘𝑅)
4	2, 3	eqtr4di 2813	. . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼)
5		fveq2 6885	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = (oppr‘𝑅))
6		2idlval.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
7	5, 6	eqtr4di 2813	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (oppr‘𝑟) = 𝑂)
8	7	fveq2d 6889	. . . . . 6 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = (LIdeal‘𝑂))
9		2idlval.j	. . . . . 6 ⊢ 𝐽 = (LIdeal‘𝑂)
10	8, 9	eqtr4di 2813	. . . . 5 ⊢ (𝑟 = 𝑅 → (LIdeal‘(oppr‘𝑟)) = 𝐽)
11	4, 10	ineq12d 4118	. . . 4 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))) = (𝐼 ∩ 𝐽))
12		df-2idl 21543	. . . 4 ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))))
13	3	fvexi 6899	. . . . 5 ⊢ 𝐼 ∈ V
14	13	inex1 5276	. . . 4 ⊢ (𝐼 ∩ 𝐽) ∈ V
15	11, 12, 14	fvmpt 6993	. . 3 ⊢ (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽))
16		fvprc 6877	. . . 4 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = ∅)
17		inss1 4133	. . . . 5 ⊢ (𝐼 ∩ 𝐽) ⊆ 𝐼
18		fvprc 6877	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (LIdeal‘𝑅) = ∅)
19	3, 18	eqtrid 2807	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐼 = ∅)
20		sseq0 4353	. . . . 5 ⊢ (((𝐼 ∩ 𝐽) ⊆ 𝐼 ∧ 𝐼 = ∅) → (𝐼 ∩ 𝐽) = ∅)
21	17, 19, 20	sylancr 599	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝐼 ∩ 𝐽) = ∅)
22	16, 21	eqtr4d 2798	. . 3 ⊢ (¬ 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼 ∩ 𝐽))
23	15, 22	pm2.61i 184	. 2 ⊢ (2Ideal‘𝑅) = (𝐼 ∩ 𝐽)
24	1, 23	eqtri 2783	1 ⊢ 𝑇 = (𝐼 ∩ 𝐽)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  ‘cfv 6538  opp_rcoppr 20566  LIdealclidl 21484  2Idealc2idl 21542

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-2idl 21543

This theorem is used by:  2idlelb  21546  2idllidld  21547  2idlridld  21548  2idl0  21554  2idl1  21555  qus1  21568  qusrhm  21570  crng2idl  21576  oppr2idl  34052  qsdrngilem  34060  qsdrngi  34061