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Theorem mplval 20136
Description: Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.)
Hypotheses
Ref Expression
mplval.p 𝑃 = (𝐼 mPoly 𝑅)
mplval.s 𝑆 = (𝐼 mPwSer 𝑅)
mplval.b 𝐵 = (Base‘𝑆)
mplval.z 0 = (0g𝑅)
mplval.u 𝑈 = {𝑓𝐵𝑓 finSupp 0 }
Assertion
Ref Expression
mplval 𝑃 = (𝑆s 𝑈)
Distinct variable groups:   𝐵,𝑓   𝑓,𝐼   𝑅,𝑓   0 ,𝑓
Allowed substitution hints:   𝑃(𝑓)   𝑆(𝑓)   𝑈(𝑓)

Proof of Theorem mplval
Dummy variables 𝑖 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mplval.p . 2 𝑃 = (𝐼 mPoly 𝑅)
2 ovexd 7180 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) ∈ V)
3 id 22 . . . . . . . 8 (𝑠 = (𝑖 mPwSer 𝑟) → 𝑠 = (𝑖 mPwSer 𝑟))
4 oveq12 7154 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
53, 4sylan9eqr 2875 . . . . . . 7 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = (𝐼 mPwSer 𝑅))
6 mplval.s . . . . . . 7 𝑆 = (𝐼 mPwSer 𝑅)
75, 6syl6eqr 2871 . . . . . 6 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = 𝑆)
87fveq2d 6667 . . . . . . . . 9 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = (Base‘𝑆))
9 mplval.b . . . . . . . . 9 𝐵 = (Base‘𝑆)
108, 9syl6eqr 2871 . . . . . . . 8 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = 𝐵)
11 simplr 765 . . . . . . . . . . 11 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑟 = 𝑅)
1211fveq2d 6667 . . . . . . . . . 10 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g𝑟) = (0g𝑅))
13 mplval.z . . . . . . . . . 10 0 = (0g𝑅)
1412, 13syl6eqr 2871 . . . . . . . . 9 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g𝑟) = 0 )
1514breq2d 5069 . . . . . . . 8 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑓 finSupp (0g𝑟) ↔ 𝑓 finSupp 0 ))
1610, 15rabeqbidv 3483 . . . . . . 7 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)} = {𝑓𝐵𝑓 finSupp 0 })
17 mplval.u . . . . . . 7 𝑈 = {𝑓𝐵𝑓 finSupp 0 }
1816, 17syl6eqr 2871 . . . . . 6 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)} = 𝑈)
197, 18oveq12d 7163 . . . . 5 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}) = (𝑆s 𝑈))
202, 19csbied 3916 . . . 4 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) / 𝑠(𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}) = (𝑆s 𝑈))
21 df-mpl 20066 . . . 4 mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑖 mPwSer 𝑟) / 𝑠(𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}))
22 ovex 7178 . . . 4 (𝑆s 𝑈) ∈ V
2320, 21, 22ovmpoa 7294 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆s 𝑈))
24 reldmmpl 20135 . . . . . 6 Rel dom mPoly
2524ovprc 7183 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = ∅)
26 ress0 16546 . . . . 5 (∅ ↾s 𝑈) = ∅
2725, 26syl6eqr 2871 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (∅ ↾s 𝑈))
28 reldmpsr 20069 . . . . . . 7 Rel dom mPwSer
2928ovprc 7183 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
306, 29syl5eq 2865 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
3130oveq1d 7160 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑆s 𝑈) = (∅ ↾s 𝑈))
3227, 31eqtr4d 2856 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆s 𝑈))
3323, 32pm2.61i 183 . 2 (𝐼 mPoly 𝑅) = (𝑆s 𝑈)
341, 33eqtri 2841 1 𝑃 = (𝑆s 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  csb 3880  c0 4288   class class class wbr 5057  cfv 6348  (class class class)co 7145   finSupp cfsupp 8821  Basecbs 16471  s cress 16472  0gc0g 16701   mPwSer cmps 20059   mPoly cmpl 20061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  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-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-slot 16475  df-base 16477  df-ress 16479  df-psr 20064  df-mpl 20066
This theorem is referenced by:  mplbas  20137  mplval2  20139  mhpinvcl  20267
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