Theorem mplvalcoe 14707

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 2814	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3	2	adantr 276	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐼 ∈ V)
4		elex 2814	. . . 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 14684	. . . . . . 7 ⊢ mPwSer Fn (V × V)
8	7	a1i 9	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → mPwSer Fn (V × V))
9		fnovex 6051	. . . . . 6 ⊢ (( mPwSer Fn (V × V) ∧ 𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) ∈ V)
10	8, 3, 5, 9	syl3anc 1273	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐼 mPwSer 𝑅) ∈ V)
11	6, 10	eqeltrid 2318	. . . 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 13143	. . . . . . 7 ⊢ Base Fn V
15		funfvex 5656	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
16	15	funfni 5432	. . . . . . 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 2318	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐵 ∈ V)
19	12, 18	rabexd 4235	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑈 ∈ V)
20		ressex 13150	. . . 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 2805	. . . . . . 7 ⊢ 𝑖 ∈ V
23		vex 2805	. . . . . . 7 ⊢ 𝑟 ∈ V
24		fnovex 6051	. . . . . . 7 ⊢ (( mPwSer Fn (V × V) ∧ 𝑖 ∈ V ∧ 𝑟 ∈ V) → (𝑖 mPwSer 𝑟) ∈ V)
25	7, 22, 23, 24	mp3an 1373	. . . . . 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 6027	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
29	27, 28	sylan9eqr 2286	. . . . . . 7 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = (𝐼 mPwSer 𝑅))
30	29, 6	eqtr4di 2282	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = 𝑆)
31	30	fveq2d 5643	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = (Base‘𝑆))
32	31, 13	eqtr4di 2282	. . . . . . . 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 6034	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (ℕ0 ↑𝑚 𝑖) = (ℕ0 ↑𝑚 𝐼))
35	33	raleqdv 2736	. . . . . . . . . . 11 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) ↔ ∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘)))
36		simplr 529	. . . . . . . . . . . . . 14 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑟 = 𝑅)
37	36	fveq2d 5643	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g‘𝑟) = (0g‘𝑅))
38		mplval.z	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝑅)
39	37, 38	eqtr4di 2282	. . . . . . . . . . . 12 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g‘𝑟) = 0 )
40	39	eqeq2d 2243	. . . . . . . . . . 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 2746	. . . . . . . . 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 2747	. . . . . . . 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 2797	. . . . . . 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 2282	. . . . . 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 6036	. . . . 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 3174	. . . 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 14681	. . . 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 6151	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V ∧ (𝑆 ↾s 𝑈) ∈ V) → (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈))
50	3, 5, 21, 49	syl3anc 1273	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈))
51	1, 50	eqtrid 2276	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑃 = (𝑆 ↾s 𝑈))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  {crab 2514  Vcvv 2802  ⦋csb 3127   class class class wbr 4088   × cxp 4723   Fn wfn 5321  ‘cfv 5326  (class class class)co 6018   ↑_𝑚 cmap 6817   < clt 8214  ℕ₀cn0 9402  Basecbs 13084   ↾_s cress 13085  0_gc0g 13341   mPwSer cmps 14678   mPoly cmpl 14679

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-i2m1 8137

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235  df-1st 6303  df-2nd 6304  df-map 6819  df-ixp 6868  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-ndx 13087  df-slot 13088  df-base 13090  df-sets 13091  df-iress 13092  df-plusg 13175  df-mulr 13176  df-sca 13178  df-vsca 13179  df-tset 13181  df-rest 13326  df-topn 13327  df-topgen 13345  df-pt 13346  df-psr 14680  df-mplcoe 14681

This theorem is referenced by:  mplbascoe  14708  mplval2g  14712