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Theorem mplval 21768
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 7447 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) ∈ V)
3 id 22 . . . . . . . 8 (𝑠 = (𝑖 mPwSer 𝑟) → 𝑠 = (𝑖 mPwSer 𝑟))
4 oveq12 7421 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
53, 4sylan9eqr 2793 . . . . . . 7 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = (𝐼 mPwSer 𝑅))
6 mplval.s . . . . . . 7 𝑆 = (𝐼 mPwSer 𝑅)
75, 6eqtr4di 2789 . . . . . 6 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = 𝑆)
87fveq2d 6895 . . . . . . . . 9 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = (Base‘𝑆))
9 mplval.b . . . . . . . . 9 𝐵 = (Base‘𝑆)
108, 9eqtr4di 2789 . . . . . . . 8 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = 𝐵)
11 simplr 766 . . . . . . . . . . 11 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑟 = 𝑅)
1211fveq2d 6895 . . . . . . . . . 10 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g𝑟) = (0g𝑅))
13 mplval.z . . . . . . . . . 10 0 = (0g𝑅)
1412, 13eqtr4di 2789 . . . . . . . . 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 3448 . . . . . . 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 2789 . . . . . 6 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)} = 𝑈)
197, 18oveq12d 7430 . . . . 5 (((𝑖 = 𝐼𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}) = (𝑆s 𝑈))
202, 19csbied 3931 . . . 4 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) / 𝑠(𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}) = (𝑆s 𝑈))
21 df-mpl 21684 . . . 4 mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑖 mPwSer 𝑟) / 𝑠(𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}))
22 ovex 7445 . . . 4 (𝑆s 𝑈) ∈ V
2320, 21, 22ovmpoa 7566 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆s 𝑈))
24 reldmmpl 21767 . . . . . 6 Rel dom mPoly
2524ovprc 7450 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = ∅)
26 ress0 17193 . . . . 5 (∅ ↾s 𝑈) = ∅
2725, 26eqtr4di 2789 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (∅ ↾s 𝑈))
28 reldmpsr 21687 . . . . . . 7 Rel dom mPwSer
2928ovprc 7450 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
306, 29eqtrid 2783 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
3130oveq1d 7427 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑆s 𝑈) = (∅ ↾s 𝑈))
3227, 31eqtr4d 2774 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆s 𝑈))
3323, 32pm2.61i 182 . 2 (𝐼 mPoly 𝑅) = (𝑆s 𝑈)
341, 33eqtri 2759 1 𝑃 = (𝑆s 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wcel 2105  {crab 3431  Vcvv 3473  csb 3893  c0 4322   class class class wbr 5148  cfv 6543  (class class class)co 7412   finSupp cfsupp 9364  Basecbs 17149  s cress 17178  0gc0g 17390   mPwSer cmps 21677   mPoly cmpl 21679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-cnex 11169  ax-1cn 11171  ax-addcl 11173
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-nn 12218  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-psr 21682  df-mpl 21684
This theorem is referenced by:  mplbas  21769  mplval2  21775
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