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Theorem ig1pval 23977
Description: Substitutions for the polynomial ideal generator function. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.)
Hypotheses
Ref Expression
ig1pval.p 𝑃 = (Poly1𝑅)
ig1pval.g 𝐺 = (idlGen1p𝑅)
ig1pval.z 0 = (0g𝑃)
ig1pval.u 𝑈 = (LIdeal‘𝑃)
ig1pval.d 𝐷 = ( deg1𝑅)
ig1pval.m 𝑀 = (Monic1p𝑅)
Assertion
Ref Expression
ig1pval ((𝑅𝑉𝐼𝑈) → (𝐺𝐼) = if(𝐼 = { 0 }, 0 , (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))))
Distinct variable groups:   𝑔,𝐼   𝑔,𝑀   𝑅,𝑔
Allowed substitution hints:   𝐷(𝑔)   𝑃(𝑔)   𝑈(𝑔)   𝐺(𝑔)   𝑉(𝑔)   0 (𝑔)

Proof of Theorem ig1pval
Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ig1pval.g . . . 4 𝐺 = (idlGen1p𝑅)
2 elex 3243 . . . . 5 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6229 . . . . . . . . . 10 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
4 ig1pval.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
53, 4syl6eqr 2703 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
65fveq2d 6233 . . . . . . . 8 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
7 ig1pval.u . . . . . . . 8 𝑈 = (LIdeal‘𝑃)
86, 7syl6eqr 2703 . . . . . . 7 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
95fveq2d 6233 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = (0g𝑃))
10 ig1pval.z . . . . . . . . . . 11 0 = (0g𝑃)
119, 10syl6eqr 2703 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = 0 )
1211sneqd 4222 . . . . . . . . 9 (𝑟 = 𝑅 → {(0g‘(Poly1𝑟))} = { 0 })
1312eqeq2d 2661 . . . . . . . 8 (𝑟 = 𝑅 → (𝑖 = {(0g‘(Poly1𝑟))} ↔ 𝑖 = { 0 }))
14 fveq2 6229 . . . . . . . . . . 11 (𝑟 = 𝑅 → (Monic1p𝑟) = (Monic1p𝑅))
15 ig1pval.m . . . . . . . . . . 11 𝑀 = (Monic1p𝑅)
1614, 15syl6eqr 2703 . . . . . . . . . 10 (𝑟 = 𝑅 → (Monic1p𝑟) = 𝑀)
1716ineq2d 3847 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑖 ∩ (Monic1p𝑟)) = (𝑖𝑀))
18 fveq2 6229 . . . . . . . . . . . 12 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
19 ig1pval.d . . . . . . . . . . . 12 𝐷 = ( deg1𝑅)
2018, 19syl6eqr 2703 . . . . . . . . . . 11 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
2120fveq1d 6231 . . . . . . . . . 10 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑔) = (𝐷𝑔))
2212difeq2d 3761 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (𝑖 ∖ {(0g‘(Poly1𝑟))}) = (𝑖 ∖ { 0 }))
2320, 22imaeq12d 5502 . . . . . . . . . . 11 (𝑟 = 𝑅 → (( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})) = (𝐷 “ (𝑖 ∖ { 0 })))
2423infeq1d 8424 . . . . . . . . . 10 (𝑟 = 𝑅 → inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < ) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))
2521, 24eqeq12d 2666 . . . . . . . . 9 (𝑟 = 𝑅 → ((( deg1𝑟)‘𝑔) = inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < ) ↔ (𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))
2617, 25riotaeqbidv 6654 . . . . . . . 8 (𝑟 = 𝑅 → (𝑔 ∈ (𝑖 ∩ (Monic1p𝑟))(( deg1𝑟)‘𝑔) = inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < )) = (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))
2713, 11, 26ifbieq12d 4146 . . . . . . 7 (𝑟 = 𝑅 → if(𝑖 = {(0g‘(Poly1𝑟))}, (0g‘(Poly1𝑟)), (𝑔 ∈ (𝑖 ∩ (Monic1p𝑟))(( deg1𝑟)‘𝑔) = inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < ))) = if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))
288, 27mpteq12dv 4766 . . . . . 6 (𝑟 = 𝑅 → (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ if(𝑖 = {(0g‘(Poly1𝑟))}, (0g‘(Poly1𝑟)), (𝑔 ∈ (𝑖 ∩ (Monic1p𝑟))(( deg1𝑟)‘𝑔) = inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < )))) = (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
29 df-ig1p 23939 . . . . . 6 idlGen1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ if(𝑖 = {(0g‘(Poly1𝑟))}, (0g‘(Poly1𝑟)), (𝑔 ∈ (𝑖 ∩ (Monic1p𝑟))(( deg1𝑟)‘𝑔) = inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < )))))
30 fvex 6239 . . . . . . . 8 (LIdeal‘𝑃) ∈ V
317, 30eqeltri 2726 . . . . . . 7 𝑈 ∈ V
3231mptex 6527 . . . . . 6 (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))) ∈ V
3328, 29, 32fvmpt 6321 . . . . 5 (𝑅 ∈ V → (idlGen1p𝑅) = (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
342, 33syl 17 . . . 4 (𝑅𝑉 → (idlGen1p𝑅) = (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
351, 34syl5eq 2697 . . 3 (𝑅𝑉𝐺 = (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
3635fveq1d 6231 . 2 (𝑅𝑉 → (𝐺𝐼) = ((𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))‘𝐼))
37 eqeq1 2655 . . . 4 (𝑖 = 𝐼 → (𝑖 = { 0 } ↔ 𝐼 = { 0 }))
38 ineq1 3840 . . . . 5 (𝑖 = 𝐼 → (𝑖𝑀) = (𝐼𝑀))
39 difeq1 3754 . . . . . . . 8 (𝑖 = 𝐼 → (𝑖 ∖ { 0 }) = (𝐼 ∖ { 0 }))
4039imaeq2d 5501 . . . . . . 7 (𝑖 = 𝐼 → (𝐷 “ (𝑖 ∖ { 0 })) = (𝐷 “ (𝐼 ∖ { 0 })))
4140infeq1d 8424 . . . . . 6 (𝑖 = 𝐼 → inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))
4241eqeq2d 2661 . . . . 5 (𝑖 = 𝐼 → ((𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) ↔ (𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
4338, 42riotaeqbidv 6654 . . . 4 (𝑖 = 𝐼 → (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )) = (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
4437, 43ifbieq2d 4144 . . 3 (𝑖 = 𝐼 → if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))) = if(𝐼 = { 0 }, 0 , (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))))
45 eqid 2651 . . 3 (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))) = (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))
46 fvex 6239 . . . . 5 (0g𝑃) ∈ V
4710, 46eqeltri 2726 . . . 4 0 ∈ V
48 riotaex 6655 . . . 4 (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )) ∈ V
4947, 48ifex 4189 . . 3 if(𝐼 = { 0 }, 0 , (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))) ∈ V
5044, 45, 49fvmpt 6321 . 2 (𝐼𝑈 → ((𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))‘𝐼) = if(𝐼 = { 0 }, 0 , (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))))
5136, 50sylan9eq 2705 1 ((𝑅𝑉𝐼𝑈) → (𝐺𝐼) = if(𝐼 = { 0 }, 0 , (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  cin 3606  ifcif 4119  {csn 4210  cmpt 4762  cima 5146  cfv 5926  crio 6650  infcinf 8388  cr 9973   < clt 10112  0gc0g 16147  LIdealclidl 19218  Poly1cpl1 19595   deg1 cdg1 23859  Monic1pcmn1 23930  idlGen1pcig1p 23934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-sup 8389  df-inf 8390  df-ig1p 23939
This theorem is referenced by:  ig1pval2  23978  ig1pval3  23979
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