Theorem mplvalcoe 15083

Description: Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)

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)
mplvalcoe.u	⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = 0 )}

Assertion

Ref	Expression
mplvalcoe	⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑃 = (𝑆 ↾s 𝑈))

Distinct variable groups:   𝐵,𝑓   𝑓,𝑎,𝑏,𝑘,𝐼   𝑅,𝑓,𝑎,𝑏,𝑘   0 ,𝑓

Allowed substitution hints:   𝐵(𝑘, 𝑎, 𝑏)   𝑃(𝑓, 𝑘, 𝑎, 𝑏)   𝑆(𝑓, 𝑘, 𝑎, 𝑏)   𝑈(𝑓, 𝑘, 𝑎, 𝑏)   𝑉(𝑓, 𝑘, 𝑎, 𝑏)   𝑊(𝑓, 𝑘, 𝑎, 𝑏)   0 (𝑘, 𝑎, 𝑏)

Proof of Theorem mplvalcoe

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		elex 2833	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3	2	adantr 276	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐼 ∈ V)
4		elex 2833	. . . 4 ⊢ (𝑅 ∈ 𝑊 → 𝑅 ∈ V)
5	4	adantl 277	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑅 ∈ V)
6		mplval.s	. . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
7		fnpsr 15053	. . . . . . 7 ⊢ mPwSer Fn (V × V)
8	7	a1i 9	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → mPwSer Fn (V × V))
9		fnovex 6118	. . . . . 6 ⊢ (( mPwSer Fn (V × V) ∧ 𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) ∈ V)
10	8, 3, 5, 9	syl3anc 1278	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐼 mPwSer 𝑅) ∈ V)
11	6, 10	eqeltrid 2325	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑆 ∈ V)
12		mplvalcoe.u	. . . . 5 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = 0 )}
13		mplval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
14		basfn 13413	. . . . . . 7 ⊢ Base Fn V
15		funfvex 5712	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
16	15	funfni 5483	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
17	14, 11, 16	sylancr 418	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (Base‘𝑆) ∈ V)
18	13, 17	eqeltrid 2325	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐵 ∈ V)
19	12, 18	rabexd 4281	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑈 ∈ V)
20		ressex 13421	. . . 4 ⊢ ((𝑆 ∈ V ∧ 𝑈 ∈ V) → (𝑆 ↾s 𝑈) ∈ V)
21	11, 19, 20	syl2anc 415	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑆 ↾s 𝑈) ∈ V)
22		vex 2824	. . . . . . 7 ⊢ 𝑖 ∈ V
23		vex 2824	. . . . . . 7 ⊢ 𝑟 ∈ V
24		fnovex 6118	. . . . . . 7 ⊢ (( mPwSer Fn (V × V) ∧ 𝑖 ∈ V ∧ 𝑟 ∈ V) → (𝑖 mPwSer 𝑟) ∈ V)
25	7, 22, 23, 24	mp3an 1378	. . . . . 6 ⊢ (𝑖 mPwSer 𝑟) ∈ V
26	25	a1i 9	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) ∈ V)
27		id 19	. . . . . . . 8 ⊢ (𝑠 = (𝑖 mPwSer 𝑟) → 𝑠 = (𝑖 mPwSer 𝑟))
28		oveq12 6094	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
29	27, 28	sylan9eqr 2293	. . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = (𝐼 mPwSer 𝑅))
30	29, 6	eqtr4di 2289	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = 𝑆)
31	30	fveq2d 5699	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = (Base‘𝑆))
32	31, 13	eqtr4di 2289	. . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = 𝐵)
33		simpll 531	. . . . . . . . . 10 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑖 = 𝐼)
34	33	oveq2d 6101	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (ℕ0 ↑𝑚 𝑖) = (ℕ0 ↑𝑚 𝐼))
35	33	raleqdv 2755	. . . . . . . . . . 11 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) ↔ ∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘)))
36		simplr 533	. . . . . . . . . . . . . 14 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑟 = 𝑅)
37	36	fveq2d 5699	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g‘𝑟) = (0g‘𝑅))
38		mplval.z	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝑅)
39	37, 38	eqtr4di 2289	. . . . . . . . . . . 12 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g‘𝑟) = 0 )
40	39	eqeq2d 2250	. . . . . . . . . . 11 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → ((𝑓‘𝑏) = (0g‘𝑟) ↔ (𝑓‘𝑏) = 0 ))
41	35, 40	imbi12d 234	. . . . . . . . . 10 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → ((∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟)) ↔ (∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = 0 )))
42	34, 41	raleqbidv 2765	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟)) ↔ ∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = 0 )))
43	34, 42	rexeqbidv 2766	. . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟)) ↔ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = 0 )))
44	32, 43	rabeqbidv 2816	. . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))} = {𝑓 ∈ 𝐵 ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = 0 )})
45	44, 12	eqtr4di 2289	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))} = 𝑈)
46	30, 45	oveq12d 6103	. . . . 5 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))}) = (𝑆 ↾s 𝑈))
47	26, 46	csbied 3194	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ⦋(𝑖 mPwSer 𝑟) / 𝑠⦌(𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))}) = (𝑆 ↾s 𝑈))
48		df-mplcoe 15050	. . . 4 ⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑠⦌(𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))}))
49	47, 48	ovmpoga 6218	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V ∧ (𝑆 ↾s 𝑈) ∈ V) → (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈))
50	3, 5, 21, 49	syl3anc 1278	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈))
51	1, 50	eqtrid 2283	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑃 = (𝑆 ↾s 𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532  Vcvv 2821  ⦋csb 3147   class class class wbr 4130   × cxp 4772   Fn wfn 5372  ‘cfv 5377  (class class class)co 6085   ↑_𝑚 cmap 6922   < clt 8360  ℕ₀cn0 9565  Basecbs 13354   ↾_s cress 13355  0_gc0g 13612   mPwSer cmps 15047   mPoly cmpl 15048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-i2m1 8284

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-map 6924  df-ixp 6981  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-iress 13362  df-plusg 13446  df-mulr 13447  df-sca 13449  df-vsca 13450  df-tset 13452  df-rest 13597  df-topn 13598  df-topgen 13616  df-pt 13617  df-psr 15049  df-mplcoe 15050

This theorem is used by:  mplbascoe  15084  mplval2g  15088