Theorem evlsval2 20299

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 20298	. . 3 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (℩𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌)))
11		eqid 2821	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
12		elex 3512	. . . . . 6 ⊢ (𝐼 ∈ 𝑍 → 𝐼 ∈ V)
13	12	3ad2ant1 1129	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝐼 ∈ V)
14	4	subrgcrng 19538	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
15	14	3adant1 1126	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
16		simp2 1133	. . . . . 6 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ CRing)
17		ovex 7188	. . . . . 6 ⊢ (𝐵 ↑m 𝐼) ∈ V
18	5	pwscrng 19366	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ (𝐵 ↑m 𝐼) ∈ V) → 𝑇 ∈ CRing)
19	16, 17, 18	sylancl 588	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑇 ∈ CRing)
20	6	subrgss 19535	. . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
21	20	3ad2ant3 1131	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ⊆ 𝐵)
22	21	resmptd 5907	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})))
23	22, 8	syl6eqr 2874	. . . . . 6 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ↾ 𝑅) = 𝑋)
24		crngring 19307	. . . . . . . . 9 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
25	24	3ad2ant2 1130	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ Ring)
26		eqid 2821	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) = (𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))
27	5, 6, 26	pwsdiagrhm 19568	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ (𝐵 ↑m 𝐼) ∈ V) → (𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
28	25, 17, 27	sylancl 588	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
29		simp3 1134	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ (SubRing‘𝑆))
30	4	resrhm 19563	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇) ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ↾ 𝑅) ∈ (𝑈 RingHom 𝑇))
31	28, 29, 30	syl2anc 586	. . . . . 6 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ↾ 𝑅) ∈ (𝑈 RingHom 𝑇))
32	23, 31	eqeltrrd 2914	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑋 ∈ (𝑈 RingHom 𝑇))
33	6	fvexi 6683	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
34		simpl1 1187	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑍)
35		elmapg 8418	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ V ∧ 𝐼 ∈ 𝑍) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↔ 𝑔:𝐼⟶𝐵))
36	33, 34, 35	sylancr 589	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↔ 𝑔:𝐼⟶𝐵))
37	36	biimpa 479	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑔 ∈ (𝐵 ↑m 𝐼)) → 𝑔:𝐼⟶𝐵)
38		simplr 767	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑔 ∈ (𝐵 ↑m 𝐼)) → 𝑥 ∈ 𝐼)
39	37, 38	ffvelrnd 6851	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑔 ∈ (𝐵 ↑m 𝐼)) → (𝑔‘𝑥) ∈ 𝐵)
40	39	fmpttd 6878	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)):(𝐵 ↑m 𝐼)⟶𝐵)
41		simpl2 1188	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ CRing)
42	5, 6, 11	pwselbasb 16760	. . . . . . . 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))
43	41, 17, 42	sylancl 588	. . . . . . 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))
44	40, 43	mpbird 259	. . . . . 6 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)) ∈ (Base‘𝑇))
45	44, 9	fmptd 6877	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑌:𝐼⟶(Base‘𝑇))
46	2, 11, 7, 3, 13, 15, 19, 32, 45	evlseu 20295	. . . 4 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ∃!𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌))
47		riotacl2 7129	. . . 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)})
48	46, 47	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)})
49	10, 48	eqeltrd 2913	. 2 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌)})
50		coeq1 5727	. . . . 5 ⊢ (𝑚 = 𝑄 → (𝑚 ∘ 𝐴) = (𝑄 ∘ 𝐴))
51	50	eqeq1d 2823	. . . 4 ⊢ (𝑚 = 𝑄 → ((𝑚 ∘ 𝐴) = 𝑋 ↔ (𝑄 ∘ 𝐴) = 𝑋))
52		coeq1 5727	. . . . 5 ⊢ (𝑚 = 𝑄 → (𝑚 ∘ 𝑉) = (𝑄 ∘ 𝑉))
53	52	eqeq1d 2823	. . . 4 ⊢ (𝑚 = 𝑄 → ((𝑚 ∘ 𝑉) = 𝑌 ↔ (𝑄 ∘ 𝑉) = 𝑌))
54	51, 53	anbi12d 632	. . 3 ⊢ (𝑚 = 𝑄 → (((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌) ↔ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌)))
55	54	elrab 3679	. 2 ⊢ (𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌)} ↔ (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌)))
56	49, 55	sylib 220	1 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∃!wreu 3140  {crab 3142  Vcvv 3494   ⊆ wss 3935  {csn 4566   ↦ cmpt 5145   × cxp 5552   ↾ cres 5556   ∘ ccom 5558  ⟶wf 6350  ‘cfv 6354  ℩crio 7112  (class class class)co 7155   ↑_m cmap 8405  Basecbs 16482   ↾_s cress 16483   ↑_s cpws 16719  Ringcrg 19296  CRingccrg 19297   RingHom crh 19463  SubRingcsubrg 19530  algSccascl 20083   mVar cmvr 20131   mPoly cmpl 20132   evalSub ces 20283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-iin 4921  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7408  df-ofr 7409  df-om 7580  df-1st 7688  df-2nd 7689  df-supp 7830  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-2o 8102  df-oadd 8105  df-er 8288  df-map 8407  df-pm 8408  df-ixp 8461  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-fsupp 8833  df-sup 8905  df-oi 8973  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-z 11981  df-dec 12098  df-uz 12243  df-fz 12892  df-fzo 13033  df-seq 13369  df-hash 13690  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-sets 16489  df-ress 16490  df-plusg 16577  df-mulr 16578  df-sca 16580  df-vsca 16581  df-ip 16582  df-tset 16583  df-ple 16584  df-ds 16586  df-hom 16588  df-cco 16589  df-0g 16714  df-gsum 16715  df-prds 16720  df-pws 16722  df-mre 16856  df-mrc 16857  df-acs 16859  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-mhm 17955  df-submnd 17956  df-grp 18105  df-minusg 18106  df-sbg 18107  df-mulg 18224  df-subg 18275  df-ghm 18355  df-cntz 18446  df-cmn 18907  df-abl 18908  df-mgp 19239  df-ur 19251  df-srg 19255  df-ring 19298  df-cring 19299  df-rnghom 19466  df-subrg 19532  df-lmod 19635  df-lss 19703  df-lsp 19743  df-assa 20084  df-asp 20085  df-ascl 20086  df-psr 20135  df-mvr 20136  df-mpl 20137  df-evls 20285

This theorem is referenced by:  evlsrhm  20300  evlssca  20301  evlsvar  20302