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Theorem igenmin 38050
Description: The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
igenmin ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)

Proof of Theorem igenmin
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 eqid 2734 . . . . 5 (1st𝑅) = (1st𝑅)
2 eqid 2734 . . . . 5 ran (1st𝑅) = ran (1st𝑅)
31, 2idlss 38002 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1st𝑅))
4 sstr 4003 . . . . . . 7 ((𝑆𝐼𝐼 ⊆ ran (1st𝑅)) → 𝑆 ⊆ ran (1st𝑅))
54ancoms 458 . . . . . 6 ((𝐼 ⊆ ran (1st𝑅) ∧ 𝑆𝐼) → 𝑆 ⊆ ran (1st𝑅))
61, 2igenval 38047 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ ran (1st𝑅)) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
75, 6sylan2 593 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran (1st𝑅) ∧ 𝑆𝐼)) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
87anassrs 467 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1st𝑅)) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
93, 8syldanl 602 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
1093impa 1109 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
11 sseq2 4021 . . . 4 (𝑗 = 𝐼 → (𝑆𝑗𝑆𝐼))
1211intminss 4978 . . 3 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ 𝐼)
13123adant1 1129 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ 𝐼)
1410, 13eqsstrd 4033 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  {crab 3432  wss 3962   cint 4950  ran crn 5689  cfv 6562  (class class class)co 7430  1st c1st 8010  RingOpscrngo 37880  Idlcidl 37993   IdlGen cigen 38045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-grpo 30521  df-gid 30522  df-ablo 30573  df-rngo 37881  df-idl 37996  df-igen 38046
This theorem is referenced by:  igenval2  38052
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