Theorem mplvalcoe 14619

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 2791	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3	2	adantr 276	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐼 ∈ V)
4		elex 2791	. . . 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 14596	. . . . . . 7 ⊢ mPwSer Fn (V × V)
8	7	a1i 9	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → mPwSer Fn (V × V))
9		fnovex 6007	. . . . . 6 ⊢ (( mPwSer Fn (V × V) ∧ 𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) ∈ V)
10	8, 3, 5, 9	syl3anc 1252	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐼 mPwSer 𝑅) ∈ V)
11	6, 10	eqeltrid 2296	. . . 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 13057	. . . . . . 7 ⊢ Base Fn V
15		funfvex 5620	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
16	15	funfni 5399	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
17	14, 11, 16	sylancr 414	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (Base‘𝑆) ∈ V)
18	13, 17	eqeltrid 2296	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐵 ∈ V)
19	12, 18	rabexd 4208	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑈 ∈ V)
20		ressex 13064	. . . 4 ⊢ ((𝑆 ∈ V ∧ 𝑈 ∈ V) → (𝑆 ↾s 𝑈) ∈ V)
21	11, 19, 20	syl2anc 411	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑆 ↾s 𝑈) ∈ V)
22		vex 2782	. . . . . . 7 ⊢ 𝑖 ∈ V
23		vex 2782	. . . . . . 7 ⊢ 𝑟 ∈ V
24		fnovex 6007	. . . . . . 7 ⊢ (( mPwSer Fn (V × V) ∧ 𝑖 ∈ V ∧ 𝑟 ∈ V) → (𝑖 mPwSer 𝑟) ∈ V)
25	7, 22, 23, 24	mp3an 1352	. . . . . 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 5983	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
29	27, 28	sylan9eqr 2264	. . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = (𝐼 mPwSer 𝑅))
30	29, 6	eqtr4di 2260	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = 𝑆)
31	30	fveq2d 5607	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = (Base‘𝑆))
32	31, 13	eqtr4di 2260	. . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = 𝐵)
33		simpll 527	. . . . . . . . . 10 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑖 = 𝐼)
34	33	oveq2d 5990	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (ℕ0 ↑𝑚 𝑖) = (ℕ0 ↑𝑚 𝐼))
35	33	raleqdv 2714	. . . . . . . . . . 11 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) ↔ ∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘)))
36		simplr 528	. . . . . . . . . . . . . 14 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑟 = 𝑅)
37	36	fveq2d 5607	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g‘𝑟) = (0g‘𝑅))
38		mplval.z	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝑅)
39	37, 38	eqtr4di 2260	. . . . . . . . . . . 12 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g‘𝑟) = 0 )
40	39	eqeq2d 2221	. . . . . . . . . . 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 2724	. . . . . . . . 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 2725	. . . . . . . 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 2774	. . . . . . 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 2260	. . . . . 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 5992	. . . . 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 3151	. . . 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 14593	. . . 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 6105	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V ∧ (𝑆 ↾s 𝑈) ∈ V) → (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈))
50	3, 5, 21, 49	syl3anc 1252	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈))
51	1, 50	eqtrid 2254	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑃 = (𝑆 ↾s 𝑈))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489  {crab 2492  Vcvv 2779  ⦋csb 3104   class class class wbr 4062   × cxp 4694   Fn wfn 5289  ‘cfv 5294  (class class class)co 5974   ↑_𝑚 cmap 6765   < clt 8149  ℕ₀cn0 9337  Basecbs 12998   ↾_s cress 12999  0_gc0g 13255   mPwSer cmps 14590   mPoly cmpl 14591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-i2m1 8072

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-tp 3654  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-of 6188  df-1st 6256  df-2nd 6257  df-map 6767  df-ixp 6816  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-ndx 13001  df-slot 13002  df-base 13004  df-sets 13005  df-iress 13006  df-plusg 13089  df-mulr 13090  df-sca 13092  df-vsca 13093  df-tset 13095  df-rest 13240  df-topn 13241  df-topgen 13259  df-pt 13260  df-psr 14592  df-mplcoe 14593

This theorem is referenced by:  mplbascoe  14620  mplval2g  14624