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Theorem igenval2 33035
Description: The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
igenval2.1 𝐺 = (1st𝑅)
igenval2.2 𝑋 = ran 𝐺
Assertion
Ref Expression
igenval2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))))
Distinct variable groups:   𝑅,𝑗   𝑆,𝑗   𝑗,𝐼
Allowed substitution hints:   𝐺(𝑗)   𝑋(𝑗)

Proof of Theorem igenval2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 igenval2.1 . . . . 5 𝐺 = (1st𝑅)
2 igenval2.2 . . . . 5 𝑋 = ran 𝐺
31, 2igenidl 33032 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
41, 2igenss 33031 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))
5 igenmin 33033 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅) ∧ 𝑆𝑗) → (𝑅 IdlGen 𝑆) ⊆ 𝑗)
653expia 1259 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → (𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
76ralrimiva 2949 . . . . 5 (𝑅 ∈ RingOps → ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
87adantr 480 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
93, 4, 83jca 1235 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ (𝑅 IdlGen 𝑆) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗)))
10 eleq1 2676 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ↔ 𝐼 ∈ (Idl‘𝑅)))
11 sseq2 3590 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → (𝑆 ⊆ (𝑅 IdlGen 𝑆) ↔ 𝑆𝐼))
12 sseq1 3589 . . . . . 6 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ⊆ 𝑗𝐼𝑗))
1312imbi2d 329 . . . . 5 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ (𝑆𝑗𝐼𝑗)))
1413ralbidv 2969 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → (∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗)))
1510, 11, 143anbi123d 1391 . . 3 ((𝑅 IdlGen 𝑆) = 𝐼 → (((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ (𝑅 IdlGen 𝑆) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗)) ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))))
169, 15syl5ibcom 234 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 → (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))))
17 igenmin 33033 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
18173adant3r3 1268 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
1918adantlr 747 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
20 ssint 4428 . . . . . . . 8 (𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖} ↔ ∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖}𝐼𝑗)
21 sseq2 3590 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑆𝑖𝑆𝑗))
2221ralrab 3335 . . . . . . . 8 (∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖}𝐼𝑗 ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))
2320, 22sylbbr 225 . . . . . . 7 (∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
24233ad2ant3 1077 . . . . . 6 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗)) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
2524adantl 481 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
261, 2igenval 33030 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
2726adantr 480 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) = {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
2825, 27sseqtr4d 3605 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → 𝐼 ⊆ (𝑅 IdlGen 𝑆))
2919, 28eqssd 3585 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) = 𝐼)
3029ex 449 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗)) → (𝑅 IdlGen 𝑆) = 𝐼))
3116, 30impbid 201 1 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  {crab 2900  wss 3540   cint 4410  ran crn 5039  cfv 5804  (class class class)co 6549  1st c1st 7057  RingOpscrngo 32863  Idlcidl 32976   IdlGen cigen 33028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-grpo 26731  df-gid 26732  df-ablo 26783  df-rngo 32864  df-idl 32979  df-igen 33029
This theorem is referenced by:  prnc  33036
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