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Theorem igenval 36570
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 36533 . . . . 5 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
4 sseq2 3974 . . . . . 6 (𝑗 = 𝑋 → (𝑆𝑗𝑆𝑋))
54rspcev 3583 . . . . 5 ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
63, 5sylan 581 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
7 rabn0 4349 . . . 4 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆𝑗)
86, 7sylibr 233 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅)
9 intex 5298 . . 3 ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ≠ ∅ ↔ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V)
108, 9sylib 217 . 2 ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗} ∈ V)
111fvexi 6860 . . . . . 6 𝐺 ∈ V
1211rnex 7853 . . . . 5 ran 𝐺 ∈ V
132, 12eqeltri 2830 . . . 4 𝑋 ∈ V
1413elpw2 5306 . . 3 (𝑆 ∈ 𝒫 𝑋𝑆𝑋)
15 simpl 484 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑟 = 𝑅)
1615fveq2d 6850 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (Idl‘𝑟) = (Idl‘𝑅))
17 sseq1 3973 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑗𝑆𝑗))
1817adantl 483 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠𝑗𝑆𝑗))
1916, 18rabeqbidv 3423 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
2019inteqd 4916 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})
21 fveq2 6846 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
2221, 1eqtr4di 2791 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
2322rneqd 5897 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
2423, 2eqtr4di 2791 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
2524pweqd 4581 . . . 4 (𝑟 = 𝑅 → 𝒫 ran (1st𝑟) = 𝒫 𝑋)
26 df-igen 36569 . . . 4 IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st𝑟) ↦ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗})
2720, 25, 26ovmpox 7512 . . 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 2940  wrex 3070  {crab 3406  Vcvv 3447  wss 3914  c0 4286  𝒫 cpw 4564   cint 4911  ran crn 5638  cfv 6500  (class class class)co 7361  1st c1st 7923  RingOpscrngo 36403  Idlcidl 36516   IdlGen cigen 36568
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-grpo 29484  df-gid 29485  df-ablo 29536  df-rngo 36404  df-idl 36519  df-igen 36569
This theorem is referenced by:  igenss  36571  igenidl  36572  igenmin  36573  igenidl2  36574  igenval2  36575
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