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Theorem igenval2 34178
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 34175 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
41, 2igenss 34174 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))
5 igenmin 34176 . . . . . . 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 3104 . . . . 5 (𝑅 ∈ RingOps → ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
87adantr 472 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
93, 4, 83jca 1123 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ (𝑅 IdlGen 𝑆) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗)))
10 eleq1 2827 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ↔ 𝐼 ∈ (Idl‘𝑅)))
11 sseq2 3768 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → (𝑆 ⊆ (𝑅 IdlGen 𝑆) ↔ 𝑆𝐼))
12 sseq1 3767 . . . . . 6 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ⊆ 𝑗𝐼𝑗))
1312imbi2d 329 . . . . 5 ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ (𝑆𝑗𝐼𝑗)))
1413ralbidv 3124 . . . 4 ((𝑅 IdlGen 𝑆) = 𝐼 → (∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗)))
1510, 11, 143anbi123d 1548 . . 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 235 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 → (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))))
17 igenmin 34176 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
18173adant3r3 1200 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
1918adantlr 753 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
20 ssint 4645 . . . . . . . 8 (𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖} ↔ ∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖}𝐼𝑗)
21 sseq2 3768 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑆𝑖𝑆𝑗))
2221ralrab 3509 . . . . . . . 8 (∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖}𝐼𝑗 ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))
2320, 22sylbbr 226 . . . . . . 7 (∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
24233ad2ant3 1130 . . . . . 6 ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗)) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
2524adantl 473 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → 𝐼 {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
261, 2igenval 34173 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
2726adantr 472 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → (𝑅 IdlGen 𝑆) = {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆𝑖})
2825, 27sseqtr4d 3783 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))) → 𝐼 ⊆ (𝑅 IdlGen 𝑆))
2919, 28eqssd 3761 . . 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 202 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  {crab 3054  wss 3715   cint 4627  ran crn 5267  cfv 6049  (class class class)co 6813  1st c1st 7331  RingOpscrngo 34006  Idlcidl 34119   IdlGen cigen 34171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-grpo 27656  df-gid 27657  df-ablo 27708  df-rngo 34007  df-idl 34122  df-igen 34172
This theorem is referenced by:  prnc  34179
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