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Theorem igenval2 35879
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 35876 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
41, 2igenss 35875 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))
5 igenmin 35877 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅) ∧ 𝑆𝑗) → (𝑅 IdlGen 𝑆) ⊆ 𝑗)
653expia 1122 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → (𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
76ralrimiva 3097 . . . . 5 (𝑅 ∈ RingOps → ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
87adantr 484 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
93, 4, 83jca 1129 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ (𝑅 IdlGen 𝑆) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗)))
10 eleq1 2821 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ↔ 𝐼 ∈ (Idl‘𝑅)))
11 sseq2 3913 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → (𝑆 ⊆ (𝑅 IdlGen 𝑆) ↔ 𝑆𝐼))
12 sseq1 3912 . . . . . 6 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ⊆ 𝑗𝐼𝑗))
1312imbi2d 344 . . . . 5 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ (𝑆𝑗𝐼𝑗)))
1413ralbidv 3110 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → (∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗)))
1510, 11, 143anbi123d 1437 . . 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 248 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 → (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))))
17 igenmin 35877 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
18173adant3r3 1185 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
1918adantlr 715 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
20 ssint 4862 . . . . . . . 8 (𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖} ↔ ∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖}𝐼𝑗)
21 sseq2 3913 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑆𝑖𝑆𝑗))
2221ralrab 3598 . . . . . . . 8 (∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖}𝐼𝑗 ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))
2320, 22sylbbr 239 . . . . . . 7 (∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
24233ad2ant3 1136 . . . . . 6 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗)) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
2524adantl 485 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
261, 2igenval 35874 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
2726adantr 484 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) = {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
2825, 27sseqtrrd 3928 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → 𝐼 ⊆ (𝑅 IdlGen 𝑆))
2919, 28eqssd 3904 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) = 𝐼)
3029ex 416 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗)) → (𝑅 IdlGen 𝑆) = 𝐼))
3116, 30impbid 215 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  {crab 3058  wss 3853   cint 4846  ran crn 5536  cfv 6349  (class class class)co 7182  1st c1st 7724  RingOpscrngo 35707  Idlcidl 35820   IdlGen cigen 35872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-fo 6355  df-fv 6357  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-1st 7726  df-2nd 7727  df-grpo 28440  df-gid 28441  df-ablo 28492  df-rngo 35708  df-idl 35823  df-igen 35873
This theorem is referenced by:  prnc  35880
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