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Theorem igenmin 38071
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 2737 . . . . 5 (1st𝑅) = (1st𝑅)
2 eqid 2737 . . . . 5 ran (1st𝑅) = ran (1st𝑅)
31, 2idlss 38023 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1st𝑅))
4 sstr 3992 . . . . . . 7 ((𝑆𝐼𝐼 ⊆ ran (1st𝑅)) → 𝑆 ⊆ ran (1st𝑅))
54ancoms 458 . . . . . 6 ((𝐼 ⊆ ran (1st𝑅) ∧ 𝑆𝐼) → 𝑆 ⊆ ran (1st𝑅))
61, 2igenval 38068 . . . . . 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 1110 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
11 sseq2 4010 . . . 4 (𝑗 = 𝐼 → (𝑆𝑗𝑆𝐼))
1211intminss 4974 . . 3 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ 𝐼)
13123adant1 1131 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ 𝐼)
1410, 13eqsstrd 4018 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  wss 3951   cint 4946  ran crn 5686  cfv 6561  (class class class)co 7431  1st c1st 8012  RingOpscrngo 37901  Idlcidl 38014   IdlGen cigen 38066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-grpo 30512  df-gid 30513  df-ablo 30564  df-rngo 37902  df-idl 38017  df-igen 38067
This theorem is referenced by:  igenval2  38073
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