Theorem mplval 21944

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 7393	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) ∈ V)
3		id 22	. . . . . . . 8 ⊢ (𝑠 = (𝑖 mPwSer 𝑟) → 𝑠 = (𝑖 mPwSer 𝑟))
4		oveq12 7367	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
5	3, 4	sylan9eqr 2793	. . . . . . 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 2789	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → 𝑠 = 𝑆)
8	7	fveq2d 6838	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (Base‘𝑠) = (Base‘𝑆))
9		mplval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑆)
10	8, 9	eqtr4di 2789	. . . . . . . 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 6838	. . . . . . . . . 10 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g‘𝑟) = (0g‘𝑅))
13		mplval.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑅)
14	12, 13	eqtr4di 2789	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (0g‘𝑟) = 0 )
15	14	breq2d 5110	. . . . . . . 8 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → (𝑓 finSupp (0g‘𝑟) ↔ 𝑓 finSupp 0 ))
16	10, 15	rabeqbidv 3417	. . . . . . 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 2789	. . . . . 6 ⊢ (((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) ∧ 𝑠 = (𝑖 mPwSer 𝑟)) → {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)} = 𝑈)
19	7, 18	oveq12d 7376	. . . . 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 3885	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ⦋(𝑖 mPwSer 𝑟) / 𝑠⦌(𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)}) = (𝑆 ↾s 𝑈))
21		df-mpl 21867	. . . 4 ⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑠⦌(𝑠 ↾s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g‘𝑟)}))
22		ovex 7391	. . . 4 ⊢ (𝑆 ↾s 𝑈) ∈ V
23	20, 21, 22	ovmpoa 7513	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (𝑆 ↾s 𝑈))
24		reldmmpl 21943	. . . . . 6 ⊢ Rel dom mPoly
25	24	ovprc 7396	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = ∅)
26		ress0 17170	. . . . 5 ⊢ (∅ ↾s 𝑈) = ∅
27	25, 26	eqtr4di 2789	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPoly 𝑅) = (∅ ↾s 𝑈))
28		reldmpsr 21870	. . . . . . 7 ⊢ Rel dom mPwSer
29	28	ovprc 7396	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
30	6, 29	eqtrid 2783	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
31	30	oveq1d 7373	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑆 ↾s 𝑈) = (∅ ↾s 𝑈))
32	27, 31	eqtr4d 2774	. . 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 2759	1 ⊢ 𝑃 = (𝑆 ↾s 𝑈)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440  ⦋csb 3849  ∅c0 4285   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358   finSupp cfsupp 9264  Basecbs 17136   ↾_s cress 17157  0_gc0g 17359   mPwSer cmps 21860   mPoly cmpl 21862

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-1cn 11084  ax-addcl 11086

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-nn 12146  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-psr 21865  df-mpl 21867

This theorem is referenced by:  mplbas  21945  mplval2  21951