Theorem evlsval2 21497

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 21496	. . 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 20226	. . . . . 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 7390	. . . . . 6 ⊢ (𝐵 ↑m 𝐼) ∈ V
17	5	pwscrng 20041	. . . . . 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 20223	. . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
20	19	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ⊆ 𝐵)
21	20	resmptd 5994	. . . . . . 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 2794	. . . . . 6 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ↾ 𝑅) = 𝑋)
23		crngring 19976	. . . . . . . . 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 20256	. . . . . . . 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 20251	. . . . . . 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 2839	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑋 ∈ (𝑈 RingHom 𝑇))
32	6	fvexi 6856	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
33		simpl1 1191	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑍)
34		elmapg 8778	. . . . . . . . . . 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 477	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑔 ∈ (𝐵 ↑m 𝐼)) → 𝑔:𝐼⟶𝐵)
37		simplr 767	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑔 ∈ (𝐵 ↑m 𝐼)) → 𝑥 ∈ 𝐼)
38	36, 37	ffvelcdmd 7036	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑔 ∈ (𝐵 ↑m 𝐼)) → (𝑔‘𝑥) ∈ 𝐵)
39	38	fmpttd 7063	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)):(𝐵 ↑m 𝐼)⟶𝐵)
40		simpl2 1192	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ CRing)
41	5, 6, 11	pwselbasb 17370	. . . . . . . 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 256	. . . . . 6 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)) ∈ (Base‘𝑇))
44	43, 9	fmptd 7062	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑌:𝐼⟶(Base‘𝑇))
45	2, 11, 7, 3, 12, 14, 18, 31, 44	evlseu 21493	. . . 4 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ∃!𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌))
46		riotacl2 7330	. . . 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 2838	. 2 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌)})
49		coeq1 5813	. . . . 5 ⊢ (𝑚 = 𝑄 → (𝑚 ∘ 𝐴) = (𝑄 ∘ 𝐴))
50	49	eqeq1d 2738	. . . 4 ⊢ (𝑚 = 𝑄 → ((𝑚 ∘ 𝐴) = 𝑋 ↔ (𝑄 ∘ 𝐴) = 𝑋))
51		coeq1 5813	. . . . 5 ⊢ (𝑚 = 𝑄 → (𝑚 ∘ 𝑉) = (𝑄 ∘ 𝑉))
52	51	eqeq1d 2738	. . . 4 ⊢ (𝑚 = 𝑄 → ((𝑚 ∘ 𝑉) = 𝑌 ↔ (𝑄 ∘ 𝑉) = 𝑌))
53	50, 52	anbi12d 631	. . 3 ⊢ (𝑚 = 𝑄 → (((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌) ↔ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌)))
54	53	elrab 3645	. 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 217	1 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃!wreu 3351  {crab 3407  Vcvv 3445   ⊆ wss 3910  {csn 4586   ↦ cmpt 5188   × cxp 5631   ↾ cres 5635   ∘ ccom 5637  ⟶wf 6492  ‘cfv 6496  ℩crio 7312  (class class class)co 7357   ↑_m cmap 8765  Basecbs 17083   ↾_s cress 17112   ↑_s cpws 17328  Ringcrg 19964  CRingccrg 19965   RingHom crh 20143  SubRingcsubrg 20218  algSccascl 21258   mVar cmvr 21307   mPoly cmpl 21308   evalSub ces 21480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-srg 19918  df-ring 19966  df-cring 19967  df-rnghom 20146  df-subrg 20220  df-lmod 20324  df-lss 20393  df-lsp 20433  df-assa 21259  df-asp 21260  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-evls 21482

This theorem is referenced by:  evlsrhm  21498  evlssca  21499  evlsvar  21500