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Theorem mplval 19935
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 7009 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) ∈ V)
3 id 22 . . . . . . . 8 (𝑠 = (𝑖 mPwSer 𝑟) → 𝑠 = (𝑖 mPwSer 𝑟))
4 oveq12 6984 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
53, 4sylan9eqr 2831 . . . . . . 7 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = (𝐼 mPwSer 𝑅))
6 mplval.s . . . . . . 7 𝑆 = (𝐼 mPwSer 𝑅)
75, 6syl6eqr 2827 . . . . . 6 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = 𝑆)
87fveq2d 6501 . . . . . . . . 9 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = (Base‘𝑆))
9 mplval.b . . . . . . . . 9 𝐵 = (Base‘𝑆)
108, 9syl6eqr 2827 . . . . . . . 8 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = 𝐵)
11 simplr 757 . . . . . . . . . . 11 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑟 = 𝑅)
1211fveq2d 6501 . . . . . . . . . 10 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g𝑟) = (0g𝑅))
13 mplval.z . . . . . . . . . 10 0 = (0g𝑅)
1412, 13syl6eqr 2827 . . . . . . . . 9 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g𝑟) = 0 )
1514breq2d 4938 . . . . . . . 8 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑓 finSupp (0g𝑟) ↔ 𝑓 finSupp 0 ))
1610, 15rabeqbidv 3403 . . . . . . 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 2827 . . . . . 6 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)} = 𝑈)
197, 18oveq12d 6993 . . . . 5 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}) = (𝑆s 𝑈))
202, 19csbied 3810 . . . 4 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) / 𝑠(𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}) = (𝑆s 𝑈))
21 df-mpl 19865 . . . 4 mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑖 mPwSer 𝑟) / 𝑠(𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}))
22 ovex 7007 . . . 4 (𝑆s 𝑈) ∈ V
2320, 21, 22ovmpoa 7120 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆s 𝑈))
24 reldmmpl 19934 . . . . . 6 Rel dom mPoly
2524ovprc 7012 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = ∅)
26 ress0 16413 . . . . 5 (∅ ↾s 𝑈) = ∅
2725, 26syl6eqr 2827 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (∅ ↾s 𝑈))
28 reldmpsr 19868 . . . . . . 7 Rel dom mPwSer
2928ovprc 7012 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
306, 29syl5eq 2821 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
3130oveq1d 6990 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑆s 𝑈) = (∅ ↾s 𝑈))
3227, 31eqtr4d 2812 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆s 𝑈))
3323, 32pm2.61i 177 . 2 (𝐼 mPoly 𝑅) = (𝑆s 𝑈)
341, 33eqtri 2797 1 𝑃 = (𝑆s 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 387   = wceq 1508  wcel 2051  {crab 3087  Vcvv 3410  csb 3781  c0 4173   class class class wbr 4926  cfv 6186  (class class class)co 6975   finSupp cfsupp 8627  Basecbs 16338  s cress 16339  0gc0g 16568   mPwSer cmps 19858   mPoly cmpl 19860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-iota 6150  df-fun 6188  df-fv 6194  df-ov 6978  df-oprab 6979  df-mpo 6980  df-slot 16342  df-base 16344  df-ress 16346  df-psr 19863  df-mpl 19865
This theorem is referenced by:  mplbas  19936  mplval2  19938  mhpinvcl  20058
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