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Theorem igenval2 35213
 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 35210 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
41, 2igenss 35209 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))
5 igenmin 35211 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅) ∧ 𝑆𝑗) → (𝑅 IdlGen 𝑆) ⊆ 𝑗)
653expia 1115 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → (𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
76ralrimiva 3186 . . . . 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 1122 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ (𝑅 IdlGen 𝑆) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗)))
10 eleq1 2904 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ↔ 𝐼 ∈ (Idl‘𝑅)))
11 sseq2 3996 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → (𝑆 ⊆ (𝑅 IdlGen 𝑆) ↔ 𝑆𝐼))
12 sseq1 3995 . . . . . 6 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ⊆ 𝑗𝐼𝑗))
1312imbi2d 342 . . . . 5 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ (𝑆𝑗𝐼𝑗)))
1413ralbidv 3201 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → (∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗)))
1510, 11, 143anbi123d 1429 . . 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 246 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 → (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))))
17 igenmin 35211 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
18173adant3r3 1178 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
1918adantlr 711 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
20 ssint 4889 . . . . . . . 8 (𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖} ↔ ∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖}𝐼𝑗)
21 sseq2 3996 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑆𝑖𝑆𝑗))
2221ralrab 3688 . . . . . . . 8 (∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖}𝐼𝑗 ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))
2320, 22sylbbr 237 . . . . . . 7 (∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
24233ad2ant3 1129 . . . . . 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 35208 . . . . . 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 4011 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → 𝐼 ⊆ (𝑅 IdlGen 𝑆))
2919, 28eqssd 3987 . . 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 213 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 207   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2106  ∀wral 3142  {crab 3146   ⊆ wss 3939  ∩ cint 4873  ran crn 5554  ‘cfv 6351  (class class class)co 7151  1st c1st 7681  RingOpscrngo 35041  Idlcidl 35154   IdlGen cigen 35206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-13 2385  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fo 6357  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-grpo 28185  df-gid 28186  df-ablo 28237  df-rngo 35042  df-idl 35157  df-igen 35207 This theorem is referenced by:  prnc  35214
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