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Theorem ig1pval 25347
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 3447 . . . . 5 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6766 . . . . . . . . . 10 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
4 ig1pval.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
53, 4eqtr4di 2796 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
65fveq2d 6770 . . . . . . . 8 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
7 ig1pval.u . . . . . . . 8 𝑈 = (LIdeal‘𝑃)
86, 7eqtr4di 2796 . . . . . . 7 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
95fveq2d 6770 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = (0g𝑃))
10 ig1pval.z . . . . . . . . . . 11 0 = (0g𝑃)
119, 10eqtr4di 2796 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = 0 )
1211sneqd 4573 . . . . . . . . 9 (𝑟 = 𝑅 → {(0g‘(Poly1𝑟))} = { 0 })
1312eqeq2d 2749 . . . . . . . 8 (𝑟 = 𝑅 → (𝑖 = {(0g‘(Poly1𝑟))} ↔ 𝑖 = { 0 }))
14 fveq2 6766 . . . . . . . . . . 11 (𝑟 = 𝑅 → (Monic1p𝑟) = (Monic1p𝑅))
15 ig1pval.m . . . . . . . . . . 11 𝑀 = (Monic1p𝑅)
1614, 15eqtr4di 2796 . . . . . . . . . 10 (𝑟 = 𝑅 → (Monic1p𝑟) = 𝑀)
1716ineq2d 4146 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑖 ∩ (Monic1p𝑟)) = (𝑖𝑀))
18 fveq2 6766 . . . . . . . . . . . 12 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
19 ig1pval.d . . . . . . . . . . . 12 𝐷 = ( deg1𝑅)
2018, 19eqtr4di 2796 . . . . . . . . . . 11 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
2120fveq1d 6768 . . . . . . . . . 10 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑔) = (𝐷𝑔))
2212difeq2d 4056 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (𝑖 ∖ {(0g‘(Poly1𝑟))}) = (𝑖 ∖ { 0 }))
2320, 22imaeq12d 5963 . . . . . . . . . . 11 (𝑟 = 𝑅 → (( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})) = (𝐷 “ (𝑖 ∖ { 0 })))
2423infeq1d 9223 . . . . . . . . . 10 (𝑟 = 𝑅 → inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < ) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))
2521, 24eqeq12d 2754 . . . . . . . . 9 (𝑟 = 𝑅 → ((( deg1𝑟)‘𝑔) = inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < ) ↔ (𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))
2617, 25riotaeqbidv 7227 . . . . . . . 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 4487 . . . . . . 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 5164 . . . . . 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 25309 . . . . . 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 7096 . . . . 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, 31eqtrid 2790 . . 3 (𝑅𝑉𝐺 = (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
3332fveq1d 6768 . 2 (𝑅𝑉 → (𝐺𝐼) = ((𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))‘𝐼))
34 eqeq1 2742 . . . 4 (𝑖 = 𝐼 → (𝑖 = { 0 } ↔ 𝐼 = { 0 }))
35 ineq1 4139 . . . . 5 (𝑖 = 𝐼 → (𝑖𝑀) = (𝐼𝑀))
36 difeq1 4049 . . . . . . . 8 (𝑖 = 𝐼 → (𝑖 ∖ { 0 }) = (𝐼 ∖ { 0 }))
3736imaeq2d 5962 . . . . . . 7 (𝑖 = 𝐼 → (𝐷 “ (𝑖 ∖ { 0 })) = (𝐷 “ (𝐼 ∖ { 0 })))
3837infeq1d 9223 . . . . . 6 (𝑖 = 𝐼 → inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))
3938eqeq2d 2749 . . . . 5 (𝑖 = 𝐼 → ((𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) ↔ (𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
4035, 39riotaeqbidv 7227 . . . 4 (𝑖 = 𝐼 → (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )) = (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
4134, 40ifbieq2d 4485 . . 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 2738 . . 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 6780 . . . 4 0 ∈ V
44 riotaex 7228 . . . 4 (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )) ∈ V
4543, 44ifex 4509 . . 3 if(𝐼 = { 0 }, 0 , (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))) ∈ V
4641, 42, 45fvmpt 6867 . 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 2798 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 396   = wceq 1539  wcel 2106  Vcvv 3429  cdif 3883  cin 3885  ifcif 4459  {csn 4561  cmpt 5156  cima 5587  cfv 6426  crio 7223  infcinf 9187  cr 10880   < clt 11019  0gc0g 17160  LIdealclidl 20442  Poly1cpl1 21358   deg1 cdg1 25226  Monic1pcmn1 25300  idlGen1pcig1p 25304
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5208  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-riota 7224  df-sup 9188  df-inf 9189  df-ig1p 25309
This theorem is referenced by:  ig1pval2  25348  ig1pval3  25349
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