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Theorem igenval2 36224
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 36221 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
41, 2igenss 36220 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))
5 igenmin 36222 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅) ∧ 𝑆𝑗) → (𝑅 IdlGen 𝑆) ⊆ 𝑗)
653expia 1120 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → (𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
76ralrimiva 3103 . . . . 5 (𝑅 ∈ RingOps → ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
87adantr 481 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
93, 4, 83jca 1127 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ (𝑅 IdlGen 𝑆) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗)))
10 eleq1 2826 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ↔ 𝐼 ∈ (Idl‘𝑅)))
11 sseq2 3947 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → (𝑆 ⊆ (𝑅 IdlGen 𝑆) ↔ 𝑆𝐼))
12 sseq1 3946 . . . . . 6 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ⊆ 𝑗𝐼𝑗))
1312imbi2d 341 . . . . 5 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ (𝑆𝑗𝐼𝑗)))
1413ralbidv 3112 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → (∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗)))
1510, 11, 143anbi123d 1435 . . 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 244 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 → (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))))
17 igenmin 36222 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
18173adant3r3 1183 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
1918adantlr 712 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
20 ssint 4895 . . . . . . . 8 (𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖} ↔ ∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖}𝐼𝑗)
21 sseq2 3947 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑆𝑖𝑆𝑗))
2221ralrab 3630 . . . . . . . 8 (∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖}𝐼𝑗 ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))
2320, 22sylbbr 235 . . . . . . 7 (∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
24233ad2ant3 1134 . . . . . 6 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗)) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
2524adantl 482 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
261, 2igenval 36219 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
2726adantr 481 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) = {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
2825, 27sseqtrrd 3962 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → 𝐼 ⊆ (𝑅 IdlGen 𝑆))
2919, 28eqssd 3938 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) = 𝐼)
3029ex 413 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗)) → (𝑅 IdlGen 𝑆) = 𝐼))
3116, 30impbid 211 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 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  {crab 3068  wss 3887   cint 4879  ran crn 5590  cfv 6433  (class class class)co 7275  1st c1st 7829  RingOpscrngo 36052  Idlcidl 36165   IdlGen cigen 36217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-grpo 28855  df-gid 28856  df-ablo 28907  df-rngo 36053  df-idl 36168  df-igen 36218
This theorem is referenced by:  prnc  36225
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