Theorem mplval 21927

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 = (0g‘𝑅)
mplval.u	⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }

Assertion

Ref	Expression
mplval	⊢ 𝑃 = (𝑆 ↾s 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.

Step	Hyp	Ref	Expression
1		mplval.p	. 2 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
2		ovexd 7387	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) ∈ V)
3		id 22	. . . . . . . 8 ⊢ (𝑠 = (𝑖 mPwSer 𝑟) → 𝑠 = (𝑖 mPwSer 𝑟))
4		oveq12 7361	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
5	3, 4	sylan9eqr 2790	. . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = (𝐼 mPwSer 𝑅))
6		mplval.s	. . . . . . 7 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
7	5, 6	eqtr4di 2786	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = 𝑆)
8	7	fveq2d 6832	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = (Base‘𝑆))
9		mplval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑆)
10	8, 9	eqtr4di 2786	. . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = 𝐵)
11		simplr 768	. . . . . . . . . . 11 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑟 = 𝑅)
12	11	fveq2d 6832	. . . . . . . . . 10 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g‘𝑟) = (0g‘𝑅))
13		mplval.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑅)
14	12, 13	eqtr4di 2786	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g‘𝑟) = 0 )
15	14	breq2d 5105	. . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑓 finSupp (0g‘𝑟) ↔ 𝑓 finSupp 0 ))
16	10, 15	rabeqbidv 3414	. . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)} = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 })
17		mplval.u	. . . . . . 7 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }
18	16, 17	eqtr4di 2786	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)} = 𝑈)
19	7, 18	oveq12d 7370	. . . . 5 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)}) = (𝑆 ↾s 𝑈))
20	2, 19	csbied 3882	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ⦋(𝑖 mPwSer 𝑟) / 𝑠⦌(𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)}) = (𝑆 ↾s 𝑈))
21		df-mpl 21850	. . . 4 ⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑠⦌(𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)}))
22		ovex 7385	. . . 4 ⊢ (𝑆 ↾s 𝑈) ∈ V
23	20, 21, 22	ovmpoa 7507	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈))
24		reldmmpl 21926	. . . . . 6 ⊢ Rel dom mPoly
25	24	ovprc 7390	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = ∅)
26		ress0 17156	. . . . 5 ⊢ (∅ ↾s 𝑈) = ∅
27	25, 26	eqtr4di 2786	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (∅ ↾s 𝑈))
28		reldmpsr 21853	. . . . . . 7 ⊢ Rel dom mPwSer
29	28	ovprc 7390	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
30	6, 29	eqtrid 2780	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
31	30	oveq1d 7367	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑆 ↾s 𝑈) = (∅ ↾s 𝑈))
32	27, 31	eqtr4d 2771	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈))
33	23, 32	pm2.61i 182	. 2 ⊢ (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈)
34	1, 33	eqtri 2756	1 ⊢ 𝑃 = (𝑆 ↾s 𝑈)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3396  Vcvv 3437  ⦋csb 3846  ∅c0 4282   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352   finSupp cfsupp 9252  Basecbs 17122   ↾_s cress 17143  0_gc0g 17345   mPwSer cmps 21843   mPoly cmpl 21845

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-1cn 11071  ax-addcl 11073

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-nn 12133  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-psr 21848  df-mpl 21850

This theorem is referenced by:  mplbas  21928  mplval2  21934