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Theorem mplval 19249
 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 ovex 6577 . . . . . 6 (𝑖 mPwSer 𝑟) ∈ V
32a1i 11 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) ∈ V)
4 id 22 . . . . . . . 8 (𝑠 = (𝑖 mPwSer 𝑟) → 𝑠 = (𝑖 mPwSer 𝑟))
5 oveq12 6558 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
64, 5sylan9eqr 2666 . . . . . . 7 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = (𝐼 mPwSer 𝑅))
7 mplval.s . . . . . . 7 𝑆 = (𝐼 mPwSer 𝑅)
86, 7syl6eqr 2662 . . . . . 6 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = 𝑆)
98fveq2d 6107 . . . . . . . . 9 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = (Base‘𝑆))
10 mplval.b . . . . . . . . 9 𝐵 = (Base‘𝑆)
119, 10syl6eqr 2662 . . . . . . . 8 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = 𝐵)
12 simplr 788 . . . . . . . . . . 11 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑟 = 𝑅)
1312fveq2d 6107 . . . . . . . . . 10 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g𝑟) = (0g𝑅))
14 mplval.z . . . . . . . . . 10 0 = (0g𝑅)
1513, 14syl6eqr 2662 . . . . . . . . 9 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g𝑟) = 0 )
1615breq2d 4595 . . . . . . . 8 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑓 finSupp (0g𝑟) ↔ 𝑓 finSupp 0 ))
1711, 16rabeqbidv 3168 . . . . . . 7 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)} = {𝑓𝐵𝑓 finSupp 0 })
18 mplval.u . . . . . . 7 𝑈 = {𝑓𝐵𝑓 finSupp 0 }
1917, 18syl6eqr 2662 . . . . . 6 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)} = 𝑈)
208, 19oveq12d 6567 . . . . 5 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}) = (𝑆s 𝑈))
213, 20csbied 3526 . . . 4 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) / 𝑠(𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}) = (𝑆s 𝑈))
22 df-mpl 19179 . . . 4 mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑖 mPwSer 𝑟) / 𝑠(𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}))
23 ovex 6577 . . . 4 (𝑆s 𝑈) ∈ V
2421, 22, 23ovmpt2a 6689 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆s 𝑈))
25 reldmmpl 19248 . . . . . 6 Rel dom mPoly
2625ovprc 6581 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = ∅)
27 ress0 15761 . . . . 5 (∅ ↾s 𝑈) = ∅
2826, 27syl6eqr 2662 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (∅ ↾s 𝑈))
29 reldmpsr 19182 . . . . . . 7 Rel dom mPwSer
3029ovprc 6581 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
317, 30syl5eq 2656 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
3231oveq1d 6564 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑆s 𝑈) = (∅ ↾s 𝑈))
3328, 32eqtr4d 2647 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆s 𝑈))
3424, 33pm2.61i 175 . 2 (𝐼 mPoly 𝑅) = (𝑆s 𝑈)
351, 34eqtri 2632 1 𝑃 = (𝑆s 𝑈)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ⦋csb 3499  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   finSupp cfsupp 8158  Basecbs 15695   ↾s cress 15696  0gc0g 15923   mPwSer cmps 19172   mPoly cmpl 19174 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-slot 15699  df-base 15700  df-ress 15702  df-psr 19177  df-mpl 19179 This theorem is referenced by:  mplbas  19250  mplval2  19252
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