igenidl2 - Mathbox for Jeff Madsen

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Theorem igenidl2 38979

Description: Obsolete theorem, use rsp2idlid 33924 instead. The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

**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 2761	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2761	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3	1, 2	idlss 38930	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1^st ‘𝑅))
4	1, 2	igenval 38975	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1^st ‘𝑅)) → (𝑅 IdlGen 𝐼) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗})
5	3, 4	syldan 603	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗})
6		intmin 4928	. . 3 ⊢ (𝐼 ∈ (Idl‘𝑅) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗} = 𝐼)
7	6	adantl 487	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗} = 𝐼)
8	5, 7	eqtrd 2796	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {crab 3413 ⊆ wss 3899 ∩ cint 4907 ran crn 5652 ‘cfv 6537 (class class class)co 7418 1^st c1st 7997 RingOpscrngo 38808 Idlcidl 38921 IdlGen cigen 38973

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-1st 7999 df-2nd 8000 df-grpo 31088 df-gid 31089 df-ablo 31140 df-rngo 38809 df-idl 38924 df-igen 38974

This theorem is used by: (None)

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