Theorem evlsval2 22042

Description: Characterizing properties of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 12-Mar-2015.) (Revised by AV, 18-Sep-2021.)

Hypotheses

Ref	Expression
evlsval.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
evlsval.w	⊢ 𝑊 = (𝐼 mPoly 𝑈)
evlsval.v	⊢ 𝑉 = (𝐼 mVar 𝑈)
evlsval.u	⊢ 𝑈 = (𝑆 ↾s 𝑅)
evlsval.t	⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑m 𝐼))
evlsval.b	⊢ 𝐵 = (Base‘𝑆)
evlsval.a	⊢ 𝐴 = (algSc‘𝑊)
evlsval.x	⊢ 𝑋 = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))
evlsval.y	⊢ 𝑌 = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)))

Assertion

Ref	Expression
evlsval2	⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌)))

Distinct variable groups:   𝑔,𝐼,𝑥   𝑥,𝑅   𝑆,𝑔,𝑥   𝐵,𝑔,𝑥   𝑅,𝑔   𝑥,𝑇   𝑔,𝑍,𝑥

Allowed substitution hints:   𝐴(𝑥,𝑔)   𝑄(𝑥,𝑔)   𝑇(𝑔)   𝑈(𝑥,𝑔)   𝑉(𝑥,𝑔)   𝑊(𝑥,𝑔)   𝑋(𝑥,𝑔)   𝑌(𝑥,𝑔)

Proof of Theorem evlsval2

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlsval.q	. . . 4 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
2		evlsval.w	. . . 4 ⊢ 𝑊 = (𝐼 mPoly 𝑈)
3		evlsval.v	. . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑈)
4		evlsval.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑅)
5		evlsval.t	. . . 4 ⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑m 𝐼))
6		evlsval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
7		evlsval.a	. . . 4 ⊢ 𝐴 = (algSc‘𝑊)
8		evlsval.x	. . . 4 ⊢ 𝑋 = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))
9		evlsval.y	. . . 4 ⊢ 𝑌 = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	evlsval 22041	. . 3 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (℩𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌)))
11		eqid 2736	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
12		simp1 1136	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝐼 ∈ 𝑍)
13	4	subrgcrng 20508	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
14	13	3adant1 1130	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
15		simp2 1137	. . . . . 6 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ CRing)
16		ovex 7391	. . . . . 6 ⊢ (𝐵 ↑m 𝐼) ∈ V
17	5	pwscrng 20261	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ (𝐵 ↑m 𝐼) ∈ V) → 𝑇 ∈ CRing)
18	15, 16, 17	sylancl 586	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑇 ∈ CRing)
19	6	subrgss 20505	. . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
20	19	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ⊆ 𝐵)
21	20	resmptd 5999	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})))
22	21, 8	eqtr4di 2789	. . . . . 6 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ↾ 𝑅) = 𝑋)
23		crngring 20180	. . . . . . . . 9 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
24	23	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ Ring)
25		eqid 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) = (𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))
26	5, 6, 25	pwsdiagrhm 20540	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ (𝐵 ↑m 𝐼) ∈ V) → (𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
27	24, 16, 26	sylancl 586	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
28		simp3 1138	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ (SubRing‘𝑆))
29	4	resrhm 20534	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇) ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ↾ 𝑅) ∈ (𝑈 RingHom 𝑇))
30	27, 28, 29	syl2anc 584	. . . . . 6 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ↾ 𝑅) ∈ (𝑈 RingHom 𝑇))
31	22, 30	eqeltrrd 2837	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑋 ∈ (𝑈 RingHom 𝑇))
32	6	fvexi 6848	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
33		simpl1 1192	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑍)
34		elmapg 8776	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ V ∧ 𝐼 ∈ 𝑍) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↔ 𝑔:𝐼⟶𝐵))
35	32, 33, 34	sylancr 587	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↔ 𝑔:𝐼⟶𝐵))
36	35	biimpa 476	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑔 ∈ (𝐵 ↑m 𝐼)) → 𝑔:𝐼⟶𝐵)
37		simplr 768	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑔 ∈ (𝐵 ↑m 𝐼)) → 𝑥 ∈ 𝐼)
38	36, 37	ffvelcdmd 7030	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑔 ∈ (𝐵 ↑m 𝐼)) → (𝑔‘𝑥) ∈ 𝐵)
39	38	fmpttd 7060	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)):(𝐵 ↑m 𝐼)⟶𝐵)
40		simpl2 1193	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ CRing)
41	5, 6, 11	pwselbasb 17408	. . . . . . . 8 ⊢ ((𝑆 ∈ CRing ∧ (𝐵 ↑m 𝐼) ∈ V) → ((𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)) ∈ (Base‘𝑇) ↔ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)):(𝐵 ↑m 𝐼)⟶𝐵))
42	40, 16, 41	sylancl 586	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → ((𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)) ∈ (Base‘𝑇) ↔ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)):(𝐵 ↑m 𝐼)⟶𝐵))
43	39, 42	mpbird 257	. . . . . 6 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)) ∈ (Base‘𝑇))
44	43, 9	fmptd 7059	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑌:𝐼⟶(Base‘𝑇))
45	2, 11, 7, 3, 12, 14, 18, 31, 44	evlseu 22038	. . . 4 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ∃!𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌))
46		riotacl2 7331	. . . 4 ⊢ (∃!𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌) → (℩𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌)) ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌)})
47	45, 46	syl 17	. . 3 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (℩𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌)) ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌)})
48	10, 47	eqeltrd 2836	. 2 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌)})
49		coeq1 5806	. . . . 5 ⊢ (𝑚 = 𝑄 → (𝑚 ∘ 𝐴) = (𝑄 ∘ 𝐴))
50	49	eqeq1d 2738	. . . 4 ⊢ (𝑚 = 𝑄 → ((𝑚 ∘ 𝐴) = 𝑋 ↔ (𝑄 ∘ 𝐴) = 𝑋))
51		coeq1 5806	. . . . 5 ⊢ (𝑚 = 𝑄 → (𝑚 ∘ 𝑉) = (𝑄 ∘ 𝑉))
52	51	eqeq1d 2738	. . . 4 ⊢ (𝑚 = 𝑄 → ((𝑚 ∘ 𝑉) = 𝑌 ↔ (𝑄 ∘ 𝑉) = 𝑌))
53	50, 52	anbi12d 632	. . 3 ⊢ (𝑚 = 𝑄 → (((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌) ↔ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌)))
54	53	elrab 3646	. 2 ⊢ (𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌)} ↔ (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌)))
55	48, 54	sylib 218	1 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∃!wreu 3348  {crab 3399  Vcvv 3440   ⊆ wss 3901  {csn 4580   ↦ cmpt 5179   × cxp 5622   ↾ cres 5626   ∘ ccom 5628  ⟶wf 6488  ‘cfv 6492  ℩crio 7314  (class class class)co 7358   ↑_m cmap 8763  Basecbs 17136   ↾_s cress 17157   ↑_s cpws 17366  Ringcrg 20168  CRingccrg 20169   RingHom crh 20405  SubRingcsubrg 20502  algSccascl 21807   mVar cmvr 21861   mPoly cmpl 21862   evalSub ces 22027

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-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

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-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  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-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  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-se 5578  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-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-hom 17201  df-cco 17202  df-0g 17361  df-gsum 17362  df-prds 17367  df-pws 17369  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-ghm 19142  df-cntz 19246  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-srg 20122  df-ring 20170  df-cring 20171  df-rhm 20408  df-subrng 20479  df-subrg 20503  df-lmod 20813  df-lss 20883  df-lsp 20923  df-assa 21808  df-asp 21809  df-ascl 21810  df-psr 21865  df-mvr 21866  df-mpl 21867  df-evls 22029

This theorem is referenced by:  evlsrhm  22043  evlssca  22049  evlsvar  22050