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Theorem igenval 32830
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 32793 . . . . 5 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
4 sseq2 3586 . . . . . 6 (𝑗 = 𝑋 → (𝑆𝑗𝑆𝑋))
54rspcev 3278 . . . . 5 ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
63, 5sylan 486 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
7 rabn0 3908 . . . 4 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
86, 7sylibr 222 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅)
9 intex 4739 . . 3 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V)
108, 9sylib 206 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V)
11 fvex 6095 . . . . . . 7 (1st𝑅) ∈ V
121, 11eqeltri 2680 . . . . . 6 𝐺 ∈ V
1312rnex 6966 . . . . 5 ran 𝐺 ∈ V
142, 13eqeltri 2680 . . . 4 𝑋 ∈ V
1514elpw2 4747 . . 3 (𝑆 ∈ 𝒫 𝑋𝑆𝑋)
16 simpl 471 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑟 = 𝑅)
1716fveq2d 6089 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (Idl‘𝑟) = (Idl‘𝑅))
18 sseq1 3585 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑗𝑆𝑗))
1918adantl 480 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠𝑗𝑆𝑗))
2017, 19rabeqbidv 3164 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
2120inteqd 4406 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
22 fveq2 6085 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
2322, 1syl6eqr 2658 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
2423rneqd 5258 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
2524, 2syl6eqr 2658 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
2625pweqd 4109 . . . 4 (𝑟 = 𝑅 → 𝒫 ran (1st𝑟) = 𝒫 𝑋)
27 df-igen 32829 . . . 4 IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st𝑟) ↦ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗})
2821, 26, 27ovmpt2x 6662 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ 𝒫 𝑋 {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
2915, 28syl3an2br 1357 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋 {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
3010, 29mpd3an3 1416 1 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2776  wrex 2893  {crab 2896  Vcvv 3169  wss 3536  c0 3870  𝒫 cpw 4104   cint 4401  ran crn 5026  cfv 5787  (class class class)co 6524  1st c1st 7031  RingOpscrngo 32663  Idlcidl 32776   IdlGen cigen 32828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-fo 5793  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-1st 7033  df-2nd 7034  df-grpo 26494  df-gid 26495  df-ablo 26549  df-rngo 32664  df-idl 32779  df-igen 32829
This theorem is referenced by:  igenss  32831  igenidl  32832  igenmin  32833  igenidl2  32834  igenval2  32835
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