igenidl2 - Mathbox for Jeff Madsen

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Theorem igenidl2 38439

Description: The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

**Assertion**
Ref	Expression
igenidl2	⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼)

Proof of Theorem igenidl2 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2740	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2740	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3	1, 2	idlss 38390	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1^st ‘𝑅))
4	1, 2	igenval 38435	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1^st ‘𝑅)) → (𝑅 IdlGen 𝐼) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗})
5	3, 4	syldan 597	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗})
6		intmin 4905	. . 3 ⊢ (𝐼 ∈ (Idl‘𝑅) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗} = 𝐼)
7	6	adantl 482	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗} = 𝐼)
8	5, 7	eqtrd 2775	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 {crab 3392 ⊆ wss 3890 ∩ cint 4884 ran crn 5626 ‘cfv 6492 (class class class)co 7363 1^st c1st 7936 RingOpscrngo 38268 Idlcidl 38381 IdlGen cigen 38433

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fo 6498 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-grpo 30589 df-gid 30590 df-ablo 30641 df-rngo 38269 df-idl 38384 df-igen 38434

This theorem is referenced by: (None)

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