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Theorem ig1pval 24758
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 3511 . . . . 5 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6663 . . . . . . . . . 10 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
4 ig1pval.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
53, 4syl6eqr 2872 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
65fveq2d 6667 . . . . . . . 8 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
7 ig1pval.u . . . . . . . 8 𝑈 = (LIdeal‘𝑃)
86, 7syl6eqr 2872 . . . . . . 7 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
95fveq2d 6667 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = (0g𝑃))
10 ig1pval.z . . . . . . . . . . 11 0 = (0g𝑃)
119, 10syl6eqr 2872 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = 0 )
1211sneqd 4571 . . . . . . . . 9 (𝑟 = 𝑅 → {(0g‘(Poly1𝑟))} = { 0 })
1312eqeq2d 2830 . . . . . . . 8 (𝑟 = 𝑅 → (𝑖 = {(0g‘(Poly1𝑟))} ↔ 𝑖 = { 0 }))
14 fveq2 6663 . . . . . . . . . . 11 (𝑟 = 𝑅 → (Monic1p𝑟) = (Monic1p𝑅))
15 ig1pval.m . . . . . . . . . . 11 𝑀 = (Monic1p𝑅)
1614, 15syl6eqr 2872 . . . . . . . . . 10 (𝑟 = 𝑅 → (Monic1p𝑟) = 𝑀)
1716ineq2d 4187 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑖 ∩ (Monic1p𝑟)) = (𝑖𝑀))
18 fveq2 6663 . . . . . . . . . . . 12 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
19 ig1pval.d . . . . . . . . . . . 12 𝐷 = ( deg1𝑅)
2018, 19syl6eqr 2872 . . . . . . . . . . 11 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
2120fveq1d 6665 . . . . . . . . . 10 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑔) = (𝐷𝑔))
2212difeq2d 4097 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (𝑖 ∖ {(0g‘(Poly1𝑟))}) = (𝑖 ∖ { 0 }))
2320, 22imaeq12d 5923 . . . . . . . . . . 11 (𝑟 = 𝑅 → (( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})) = (𝐷 “ (𝑖 ∖ { 0 })))
2423infeq1d 8933 . . . . . . . . . 10 (𝑟 = 𝑅 → inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < ) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))
2521, 24eqeq12d 2835 . . . . . . . . 9 (𝑟 = 𝑅 → ((( deg1𝑟)‘𝑔) = inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < ) ↔ (𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))
2617, 25riotaeqbidv 7109 . . . . . . . 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 4492 . . . . . . 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 5142 . . . . . 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 24720 . . . . . 6 idlGen1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ if(𝑖 = {(0g‘(Poly1𝑟))}, (0g‘(Poly1𝑟)), (𝑔 ∈ (𝑖 ∩ (Monic1p𝑟))(( deg1𝑟)‘𝑔) = inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < )))))
3028, 29, 7mptfvmpt 6982 . . . . 5 (𝑅 ∈ V → (idlGen1p𝑅) = (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
312, 30syl 17 . . . 4 (𝑅𝑉 → (idlGen1p𝑅) = (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
321, 31syl5eq 2866 . . 3 (𝑅𝑉𝐺 = (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
3332fveq1d 6665 . 2 (𝑅𝑉 → (𝐺𝐼) = ((𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))‘𝐼))
34 eqeq1 2823 . . . 4 (𝑖 = 𝐼 → (𝑖 = { 0 } ↔ 𝐼 = { 0 }))
35 ineq1 4179 . . . . 5 (𝑖 = 𝐼 → (𝑖𝑀) = (𝐼𝑀))
36 difeq1 4090 . . . . . . . 8 (𝑖 = 𝐼 → (𝑖 ∖ { 0 }) = (𝐼 ∖ { 0 }))
3736imaeq2d 5922 . . . . . . 7 (𝑖 = 𝐼 → (𝐷 “ (𝑖 ∖ { 0 })) = (𝐷 “ (𝐼 ∖ { 0 })))
3837infeq1d 8933 . . . . . 6 (𝑖 = 𝐼 → inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))
3938eqeq2d 2830 . . . . 5 (𝑖 = 𝐼 → ((𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) ↔ (𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
4035, 39riotaeqbidv 7109 . . . 4 (𝑖 = 𝐼 → (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )) = (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
4134, 40ifbieq2d 4490 . . 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, < ))))
42 eqid 2819 . . 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, < ))))
4310fvexi 6677 . . . 4 0 ∈ V
44 riotaex 7110 . . . 4 (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )) ∈ V
4543, 44ifex 4513 . . 3 if(𝐼 = { 0 }, 0 , (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))) ∈ V
4641, 42, 45fvmpt 6761 . 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, < ))))
4733, 46sylan9eq 2874 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 398   = wceq 1530  wcel 2107  Vcvv 3493  cdif 3931  cin 3933  ifcif 4465  {csn 4559  cmpt 5137  cima 5551  cfv 6348  crio 7105  infcinf 8897  cr 10528   < clt 10667  0gc0g 16705  LIdealclidl 19934  Poly1cpl1 20337   deg1 cdg1 24640  Monic1pcmn1 24711  idlGen1pcig1p 24715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-sup 8898  df-inf 8899  df-ig1p 24720
This theorem is referenced by:  ig1pval2  24759  ig1pval3  24760
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