igenidl2 - Mathbox for Jeff Madsen

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Theorem igenidl2 38528

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 2761	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2761	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3	1, 2	idlss 38479	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1^st ‘𝑅))
4	1, 2	igenval 38524	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1^st ‘𝑅)) → (𝑅 IdlGen 𝐼) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗})
5	3, 4	syldan 600	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗})
6		intmin 4925	. . 3 ⊢ (𝐼 ∈ (Idl‘𝑅) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝐼 ⊆ 𝑗} = 𝐼)
7	6	adantl 485	. 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

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 {crab 3413 ⊆ wss 3904 ∩ cint 4904 ran crn 5646 ‘cfv 6517 (class class class)co 7392 1^st c1st 7964 RingOpscrngo 38357 Idlcidl 38470 IdlGen cigen 38522

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-fo 6523 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-1st 7966 df-2nd 7967 df-grpo 30642 df-gid 30643 df-ablo 30694 df-rngo 38358 df-idl 38473 df-igen 38523

This theorem is referenced by: (None)

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