Theorem evlsval2 22065

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 22064	. . 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 1137	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝐼 ∈ 𝑍)
13	4	subrgcrng 20552	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
14	13	3adant1 1131	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
15		simp2 1138	. . . . . 6 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ CRing)
16		ovex 7400	. . . . . 6 ⊢ (𝐵 ↑m 𝐼) ∈ V
17	5	pwscrng 20305	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ (𝐵 ↑m 𝐼) ∈ V) → 𝑇 ∈ CRing)
18	15, 16, 17	sylancl 587	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑇 ∈ CRing)
19	6	subrgss 20549	. . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
20	19	3ad2ant3 1136	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ⊆ 𝐵)
21	20	resmptd 6005	. . . . . . 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 20226	. . . . . . . . 9 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
24	23	3ad2ant2 1135	. . . . . . . 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 20584	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ (𝐵 ↑m 𝐼) ∈ V) → (𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
27	24, 16, 26	sylancl 587	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑥 ∈ 𝐵 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
28		simp3 1139	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ (SubRing‘𝑆))
29	4	resrhm 20578	. . . . . . 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 585	. . . . . 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 6854	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
33		simpl1 1193	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑍)
34		elmapg 8786	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ V ∧ 𝐼 ∈ 𝑍) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↔ 𝑔:𝐼⟶𝐵))
35	32, 33, 34	sylancr 588	. . . . . . . . . 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 769	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑔 ∈ (𝐵 ↑m 𝐼)) → 𝑥 ∈ 𝐼)
38	36, 37	ffvelcdmd 7037	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑔 ∈ (𝐵 ↑m 𝐼)) → (𝑔‘𝑥) ∈ 𝐵)
39	38	fmpttd 7067	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)):(𝐵 ↑m 𝐼)⟶𝐵)
40		simpl2 1194	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ CRing)
41	5, 6, 11	pwselbasb 17451	. . . . . . . 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 587	. . . . . . 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 7066	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑌:𝐼⟶(Base‘𝑇))
45	2, 11, 7, 3, 12, 14, 18, 31, 44	evlseu 22061	. . . 4 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ∃!𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌))
46		riotacl2 7340	. . . 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 5812	. . . . 5 ⊢ (𝑚 = 𝑄 → (𝑚 ∘ 𝐴) = (𝑄 ∘ 𝐴))
50	49	eqeq1d 2738	. . . 4 ⊢ (𝑚 = 𝑄 → ((𝑚 ∘ 𝐴) = 𝑋 ↔ (𝑄 ∘ 𝐴) = 𝑋))
51		coeq1 5812	. . . . 5 ⊢ (𝑚 = 𝑄 → (𝑚 ∘ 𝑉) = (𝑄 ∘ 𝑉))
52	51	eqeq1d 2738	. . . 4 ⊢ (𝑚 = 𝑄 → ((𝑚 ∘ 𝑉) = 𝑌 ↔ (𝑄 ∘ 𝑉) = 𝑌))
53	50, 52	anbi12d 633	. . 3 ⊢ (𝑚 = 𝑄 → (((𝑚 ∘ 𝐴) = 𝑋 ∧ (𝑚 ∘ 𝑉) = 𝑌) ↔ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌)))
54	53	elrab 3634	. 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 1087   = wceq 1542   ∈ wcel 2114  ∃!wreu 3340  {crab 3389  Vcvv 3429   ⊆ wss 3889  {csn 4567   ↦ cmpt 5166   × cxp 5629   ↾ cres 5633   ∘ ccom 5635  ⟶wf 6494  ‘cfv 6498  ℩crio 7323  (class class class)co 7367   ↑_m cmap 8773  Basecbs 17179   ↾_s cress 17200   ↑_s cpws 17409  Ringcrg 20214  CRingccrg 20215   RingHom crh 20449  SubRingcsubrg 20546  algSccascl 21832   mVar cmvr 21885   mPoly cmpl 21886   evalSub ces 22050

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-ofr 7632  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-srg 20168  df-ring 20216  df-cring 20217  df-rhm 20452  df-subrng 20523  df-subrg 20547  df-lmod 20857  df-lss 20927  df-lsp 20967  df-assa 21833  df-asp 21834  df-ascl 21835  df-psr 21889  df-mvr 21890  df-mpl 21891  df-evls 22052

This theorem is referenced by:  evlsrhm  22066  evlssca  22072  evlsvar  22073