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Theorem ig1pval 23926
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 3210 . . . . 5 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6189 . . . . . . . . . 10 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
4 ig1pval.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
53, 4syl6eqr 2673 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
65fveq2d 6193 . . . . . . . 8 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
7 ig1pval.u . . . . . . . 8 𝑈 = (LIdeal‘𝑃)
86, 7syl6eqr 2673 . . . . . . 7 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
95fveq2d 6193 . . . . . . . . . . 11 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = (0g𝑃))
10 ig1pval.z . . . . . . . . . . 11 0 = (0g𝑃)
119, 10syl6eqr 2673 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = 0 )
1211sneqd 4187 . . . . . . . . 9 (𝑟 = 𝑅 → {(0g‘(Poly1𝑟))} = { 0 })
1312eqeq2d 2631 . . . . . . . 8 (𝑟 = 𝑅 → (𝑖 = {(0g‘(Poly1𝑟))} ↔ 𝑖 = { 0 }))
14 fveq2 6189 . . . . . . . . . . 11 (𝑟 = 𝑅 → (Monic1p𝑟) = (Monic1p𝑅))
15 ig1pval.m . . . . . . . . . . 11 𝑀 = (Monic1p𝑅)
1614, 15syl6eqr 2673 . . . . . . . . . 10 (𝑟 = 𝑅 → (Monic1p𝑟) = 𝑀)
1716ineq2d 3812 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑖 ∩ (Monic1p𝑟)) = (𝑖𝑀))
18 fveq2 6189 . . . . . . . . . . . 12 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
19 ig1pval.d . . . . . . . . . . . 12 𝐷 = ( deg1𝑅)
2018, 19syl6eqr 2673 . . . . . . . . . . 11 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
2120fveq1d 6191 . . . . . . . . . 10 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑔) = (𝐷𝑔))
2212difeq2d 3726 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (𝑖 ∖ {(0g‘(Poly1𝑟))}) = (𝑖 ∖ { 0 }))
2320, 22imaeq12d 5465 . . . . . . . . . . 11 (𝑟 = 𝑅 → (( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})) = (𝐷 “ (𝑖 ∖ { 0 })))
2423infeq1d 8380 . . . . . . . . . 10 (𝑟 = 𝑅 → inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < ) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))
2521, 24eqeq12d 2636 . . . . . . . . 9 (𝑟 = 𝑅 → ((( deg1𝑟)‘𝑔) = inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < ) ↔ (𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))
2617, 25riotaeqbidv 6611 . . . . . . . 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 4111 . . . . . . 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 4731 . . . . . 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 23888 . . . . . 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 6199 . . . . . . . 8 (LIdeal‘𝑃) ∈ V
317, 30eqeltri 2696 . . . . . . 7 𝑈 ∈ V
3231mptex 6483 . . . . . 6 (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))) ∈ V
3328, 29, 32fvmpt 6280 . . . . 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 2667 . . 3 (𝑅𝑉𝐺 = (𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )))))
3635fveq1d 6191 . 2 (𝑅𝑉 → (𝐺𝐼) = ((𝑖𝑈 ↦ if(𝑖 = { 0 }, 0 , (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ))))‘𝐼))
37 eqeq1 2625 . . . 4 (𝑖 = 𝐼 → (𝑖 = { 0 } ↔ 𝐼 = { 0 }))
38 ineq1 3805 . . . . 5 (𝑖 = 𝐼 → (𝑖𝑀) = (𝐼𝑀))
39 difeq1 3719 . . . . . . . 8 (𝑖 = 𝐼 → (𝑖 ∖ { 0 }) = (𝐼 ∖ { 0 }))
4039imaeq2d 5464 . . . . . . 7 (𝑖 = 𝐼 → (𝐷 “ (𝑖 ∖ { 0 })) = (𝐷 “ (𝐼 ∖ { 0 })))
4140infeq1d 8380 . . . . . 6 (𝑖 = 𝐼 → inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))
4241eqeq2d 2631 . . . . 5 (𝑖 = 𝐼 → ((𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < ) ↔ (𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
4338, 42riotaeqbidv 6611 . . . 4 (𝑖 = 𝐼 → (𝑔 ∈ (𝑖𝑀)(𝐷𝑔) = inf((𝐷 “ (𝑖 ∖ { 0 })), ℝ, < )) = (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
4437, 43ifbieq2d 4109 . . 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 2621 . . 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 6199 . . . . 5 (0g𝑃) ∈ V
4710, 46eqeltri 2696 . . . 4 0 ∈ V
48 riotaex 6612 . . . 4 (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )) ∈ V
4947, 48ifex 4154 . . 3 if(𝐼 = { 0 }, 0 , (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))) ∈ V
5044, 45, 49fvmpt 6280 . 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 2675 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 384   = wceq 1482  wcel 1989  Vcvv 3198  cdif 3569  cin 3571  ifcif 4084  {csn 4175  cmpt 4727  cima 5115  cfv 5886  crio 6607  infcinf 8344  cr 9932   < clt 10071  0gc0g 16094  LIdealclidl 19164  Poly1cpl1 19541   deg1 cdg1 23808  Monic1pcmn1 23879  idlGen1pcig1p 23883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-sup 8345  df-inf 8346  df-ig1p 23888
This theorem is referenced by:  ig1pval2  23927  ig1pval3  23928
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