Theorem mplval 22020

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 7427	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) ∈ V)
3		id 22	. . . . . . . 8 ⊢ (𝑠 = (𝑖 mPwSer 𝑟) → 𝑠 = (𝑖 mPwSer 𝑟))
4		oveq12 7401	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
5	3, 4	sylan9eqr 2818	. . . . . . 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 2814	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = 𝑆)
8	7	fveq2d 6867	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = (Base‘𝑆))
9		mplval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑆)
10	8, 9	eqtr4di 2814	. . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = 𝐵)
11		simplr 778	. . . . . . . . . . 11 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑟 = 𝑅)
12	11	fveq2d 6867	. . . . . . . . . 10 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g‘𝑟) = (0g‘𝑅))
13		mplval.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑅)
14	12, 13	eqtr4di 2814	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g‘𝑟) = 0 )
15	14	breq2d 5111	. . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑓 finSupp (0g‘𝑟) ↔ 𝑓 finSupp 0 ))
16	10, 15	rabeqbidv 3431	. . . . . . 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 2814	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)} = 𝑈)
19	7, 18	oveq12d 7410	. . . . 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 3888	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ⦋(𝑖 mPwSer 𝑟) / 𝑠⦌(𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)}) = (𝑆 ↾s 𝑈))
21		df-mpl 21943	. . . 4 ⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑠⦌(𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)}))
22		ovex 7425	. . . 4 ⊢ (𝑆 ↾s 𝑈) ∈ V
23	20, 21, 22	ovmpoa 7547	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈))
24		reldmmpl 22019	. . . . . 6 ⊢ Rel dom mPoly
25	24	ovprc 7430	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = ∅)
26		ress0 17262	. . . . 5 ⊢ (∅ ↾s 𝑈) = ∅
27	25, 26	eqtr4di 2814	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (∅ ↾s 𝑈))
28		reldmpsr 21946	. . . . . . 7 ⊢ Rel dom mPwSer
29	28	ovprc 7430	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
30	6, 29	eqtrid 2808	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
31	30	oveq1d 7407	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑆 ↾s 𝑈) = (∅ ↾s 𝑈))
32	27, 31	eqtr4d 2799	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈))
33	23, 32	pm2.61i 183	. 2 ⊢ (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈)
34	1, 33	eqtri 2784	1 ⊢ 𝑃 = (𝑆 ↾s 𝑈)

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453  ⦋csb 3852  ∅c0 4285   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392   finSupp cfsupp 9304  Basecbs 17228   ↾_s cress 17249  0_gc0g 17451   mPwSer cmps 21936   mPoly cmpl 21938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-1cn 11128  ax-addcl 11130

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-nn 12208  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-psr 21941  df-mpl 21943

This theorem is referenced by:  mplbas  22021  mplval2  22027  mplnzr  33771