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Theorem igenmin 32833
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 2606 . . . . 5 (1st𝑅) = (1st𝑅)
2 eqid 2606 . . . . 5 ran (1st𝑅) = ran (1st𝑅)
31, 2idlss 32785 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1st𝑅))
4 sstr 3572 . . . . . . 7 ((𝑆𝐼𝐼 ⊆ ran (1st𝑅)) → 𝑆 ⊆ ran (1st𝑅))
54ancoms 467 . . . . . 6 ((𝐼 ⊆ ran (1st𝑅) ∧ 𝑆𝐼) → 𝑆 ⊆ ran (1st𝑅))
61, 2igenval 32830 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ ran (1st𝑅)) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
75, 6sylan2 489 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran (1st𝑅) ∧ 𝑆𝐼)) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
87anassrs 677 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1st𝑅)) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
93, 8syldanl 730 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
1093impa 1250 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
11 sseq2 3586 . . . 4 (𝑗 = 𝐼 → (𝑆𝑗𝑆𝐼))
1211intminss 4429 . . 3 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ 𝐼)
13123adant1 1071 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ 𝐼)
1410, 13eqsstrd 3598 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  {crab 2896  wss 3536   cint 4401  ran crn 5026  cfv 5787  (class class class)co 6524  1st c1st 7031  RingOpscrngo 32663  Idlcidl 32776   IdlGen cigen 32828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-fo 5793  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-1st 7033  df-2nd 7034  df-grpo 26494  df-gid 26495  df-ablo 26549  df-rngo 32664  df-idl 32779  df-igen 32829
This theorem is referenced by:  igenval2  32835
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