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Theorem mplval 22027
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 7466 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) ∈ V)
3 id 22 . . . . . . . 8 (𝑠 = (𝑖 mPwSer 𝑟) → 𝑠 = (𝑖 mPwSer 𝑟))
4 oveq12 7440 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
53, 4sylan9eqr 2797 . . . . . . 7 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = (𝐼 mPwSer 𝑅))
6 mplval.s . . . . . . 7 𝑆 = (𝐼 mPwSer 𝑅)
75, 6eqtr4di 2793 . . . . . 6 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = 𝑆)
87fveq2d 6911 . . . . . . . . 9 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = (Base‘𝑆))
9 mplval.b . . . . . . . . 9 𝐵 = (Base‘𝑆)
108, 9eqtr4di 2793 . . . . . . . 8 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = 𝐵)
11 simplr 769 . . . . . . . . . . 11 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑟 = 𝑅)
1211fveq2d 6911 . . . . . . . . . 10 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g𝑟) = (0g𝑅))
13 mplval.z . . . . . . . . . 10 0 = (0g𝑅)
1412, 13eqtr4di 2793 . . . . . . . . 9 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g𝑟) = 0 )
1514breq2d 5160 . . . . . . . 8 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑓 finSupp (0g𝑟) ↔ 𝑓 finSupp 0 ))
1610, 15rabeqbidv 3452 . . . . . . 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, 17eqtr4di 2793 . . . . . 6 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)} = 𝑈)
197, 18oveq12d 7449 . . . . 5 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}) = (𝑆s 𝑈))
202, 19csbied 3946 . . . 4 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) / 𝑠(𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}) = (𝑆s 𝑈))
21 df-mpl 21949 . . . 4 mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑖 mPwSer 𝑟) / 𝑠(𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}))
22 ovex 7464 . . . 4 (𝑆s 𝑈) ∈ V
2320, 21, 22ovmpoa 7588 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆s 𝑈))
24 reldmmpl 22026 . . . . . 6 Rel dom mPoly
2524ovprc 7469 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = ∅)
26 ress0 17289 . . . . 5 (∅ ↾s 𝑈) = ∅
2725, 26eqtr4di 2793 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (∅ ↾s 𝑈))
28 reldmpsr 21952 . . . . . . 7 Rel dom mPwSer
2928ovprc 7469 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
306, 29eqtrid 2787 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
3130oveq1d 7446 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑆s 𝑈) = (∅ ↾s 𝑈))
3227, 31eqtr4d 2778 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆s 𝑈))
3323, 32pm2.61i 182 . 2 (𝐼 mPoly 𝑅) = (𝑆s 𝑈)
341, 33eqtri 2763 1 𝑃 = (𝑆s 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  csb 3908  c0 4339   class class class wbr 5148  cfv 6563  (class class class)co 7431   finSupp cfsupp 9399  Basecbs 17245  s cress 17274  0gc0g 17486   mPwSer cmps 21942   mPoly cmpl 21944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-psr 21947  df-mpl 21949
This theorem is referenced by:  mplbas  22028  mplval2  22034
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