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Theorem igenmin 35501
 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 2801 . . . . 5 (1st𝑅) = (1st𝑅)
2 eqid 2801 . . . . 5 ran (1st𝑅) = ran (1st𝑅)
31, 2idlss 35453 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1st𝑅))
4 sstr 3926 . . . . . . 7 ((𝑆𝐼𝐼 ⊆ ran (1st𝑅)) → 𝑆 ⊆ ran (1st𝑅))
54ancoms 462 . . . . . 6 ((𝐼 ⊆ ran (1st𝑅) ∧ 𝑆𝐼) → 𝑆 ⊆ ran (1st𝑅))
61, 2igenval 35498 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ ran (1st𝑅)) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
75, 6sylan2 595 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran (1st𝑅) ∧ 𝑆𝐼)) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
87anassrs 471 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1st𝑅)) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
93, 8syldanl 604 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
1093impa 1107 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
11 sseq2 3944 . . . 4 (𝑗 = 𝐼 → (𝑆𝑗𝑆𝐼))
1211intminss 4867 . . 3 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ 𝐼)
13123adant1 1127 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ⊆ 𝐼)
1410, 13eqsstrd 3956 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  {crab 3113   ⊆ wss 3884  ∩ cint 4841  ran crn 5524  ‘cfv 6328  (class class class)co 7139  1st c1st 7673  RingOpscrngo 35331  Idlcidl 35444   IdlGen cigen 35496 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-grpo 28280  df-gid 28281  df-ablo 28332  df-rngo 35332  df-idl 35447  df-igen 35497 This theorem is referenced by:  igenval2  35503
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