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 36929
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 36892 . . . . 5 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
4 sseq2 4009 . . . . . 6 (𝑗 = 𝑋 → (𝑆𝑗𝑆𝑋))
54rspcev 3613 . . . . 5 ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
63, 5sylan 581 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
7 rabn0 4386 . . . 4 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
86, 7sylibr 233 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅)
9 intex 5338 . . 3 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V)
108, 9sylib 217 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V)
111fvexi 6906 . . . . . 6 𝐺 ∈ V
1211rnex 7903 . . . . 5 ran 𝐺 ∈ V
132, 12eqeltri 2830 . . . 4 𝑋 ∈ V
1413elpw2 5346 . . 3 (𝑆 ∈ 𝒫 𝑋𝑆𝑋)
15 simpl 484 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑟 = 𝑅)
1615fveq2d 6896 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (Idl‘𝑟) = (Idl‘𝑅))
17 sseq1 4008 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑗𝑆𝑗))
1817adantl 483 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠𝑗𝑆𝑗))
1916, 18rabeqbidv 3450 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
2019inteqd 4956 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
21 fveq2 6892 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
2221, 1eqtr4di 2791 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
2322rneqd 5938 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
2423, 2eqtr4di 2791 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
2524pweqd 4620 . . . 4 (𝑟 = 𝑅 → 𝒫 ran (1st𝑟) = 𝒫 𝑋)
26 df-igen 36928 . . . 4 IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st𝑟) ↦ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗})
2720, 25, 26ovmpox 7561 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ 𝒫 𝑋 {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
2814, 27syl3an2br 1408 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋 {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
2910, 28mpd3an3 1463 1 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2941  wrex 3071  {crab 3433  Vcvv 3475  wss 3949  c0 4323  𝒫 cpw 4603   cint 4951  ran crn 5678  cfv 6544  (class class class)co 7409  1st c1st 7973  RingOpscrngo 36762  Idlcidl 36875   IdlGen cigen 36927
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-grpo 29746  df-gid 29747  df-ablo 29798  df-rngo 36763  df-idl 36878  df-igen 36928
This theorem is referenced by:  igenss  36930  igenidl  36931  igenmin  36932  igenidl2  36933  igenval2  36934
  Copyright terms: Public domain W3C validator