Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  igenval Structured version   Visualization version   GIF version

Theorem igenval 36924
Description: The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)
Hypotheses
Ref Expression
igenval.1 𝐺 = (1st𝑅)
igenval.2 𝑋 = ran 𝐺
Assertion
Ref Expression
igenval ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
Distinct variable groups:   𝑅,𝑗   𝑆,𝑗   𝑗,𝑋
Allowed substitution hint:   𝐺(𝑗)

Proof of Theorem igenval
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 igenval.1 . . . . . 6 𝐺 = (1st𝑅)
2 igenval.2 . . . . . 6 𝑋 = ran 𝐺
31, 2rngoidl 36887 . . . . 5 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
4 sseq2 4008 . . . . . 6 (𝑗 = 𝑋 → (𝑆𝑗𝑆𝑋))
54rspcev 3612 . . . . 5 ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
63, 5sylan 580 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
7 rabn0 4385 . . . 4 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
86, 7sylibr 233 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅)
9 intex 5337 . . 3 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V)
108, 9sylib 217 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V)
111fvexi 6905 . . . . . 6 𝐺 ∈ V
1211rnex 7902 . . . . 5 ran 𝐺 ∈ V
132, 12eqeltri 2829 . . . 4 𝑋 ∈ V
1413elpw2 5345 . . 3 (𝑆 ∈ 𝒫 𝑋𝑆𝑋)
15 simpl 483 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑟 = 𝑅)
1615fveq2d 6895 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (Idl‘𝑟) = (Idl‘𝑅))
17 sseq1 4007 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑗𝑆𝑗))
1817adantl 482 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠𝑗𝑆𝑗))
1916, 18rabeqbidv 3449 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
2019inteqd 4955 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
21 fveq2 6891 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
2221, 1eqtr4di 2790 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
2322rneqd 5937 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
2423, 2eqtr4di 2790 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
2524pweqd 4619 . . . 4 (𝑟 = 𝑅 → 𝒫 ran (1st𝑟) = 𝒫 𝑋)
26 df-igen 36923 . . . 4 IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st𝑟) ↦ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗})
2720, 25, 26ovmpox 7560 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ 𝒫 𝑋 {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
2814, 27syl3an2br 1407 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋 {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
2910, 28mpd3an3 1462 1 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940  wrex 3070  {crab 3432  Vcvv 3474  wss 3948  c0 4322  𝒫 cpw 4602   cint 4950  ran crn 5677  cfv 6543  (class class class)co 7408  1st c1st 7972  RingOpscrngo 36757  Idlcidl 36870   IdlGen cigen 36922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-grpo 29741  df-gid 29742  df-ablo 29793  df-rngo 36758  df-idl 36873  df-igen 36923
This theorem is referenced by:  igenss  36925  igenidl  36926  igenmin  36927  igenidl2  36928  igenval2  36929
  Copyright terms: Public domain W3C validator