Theorem evls1rhm 19735

Description: Polynomial evaluation is a homomorphism (into the product ring). (Contributed by AV, 11-Sep-2019.)

Hypotheses

Ref	Expression
evls1rhm.q	⊢ 𝑄 = (𝑆 evalSub1 𝑅)
evls1rhm.b	⊢ 𝐵 = (Base‘𝑆)
evls1rhm.t	⊢ 𝑇 = (𝑆 ↑s 𝐵)
evls1rhm.u	⊢ 𝑈 = (𝑆 ↾s 𝑅)
evls1rhm.w	⊢ 𝑊 = (Poly1‘𝑈)

Assertion

Ref	Expression
evls1rhm	⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))

Proof of Theorem evls1rhm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evls1rhm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
2	1	subrgss 18829	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
3	2	adantl 481	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ⊆ 𝐵)
4		elpwg 4199	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵))
5	4	adantl 481	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵))
6	3, 5	mpbird 247	. . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ 𝒫 𝐵)
7		evls1rhm.q	. . . 4 ⊢ 𝑄 = (𝑆 evalSub1 𝑅)
8		eqid 2651	. . . 4 ⊢ (1𝑜 evalSub 𝑆) = (1𝑜 evalSub 𝑆)
9	7, 8, 1	evls1fval 19732	. . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅)))
10	6, 9	syldan 486	. 2 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅)))
11		evls1rhm.t	. . . . 5 ⊢ 𝑇 = (𝑆 ↑s 𝐵)
12		eqid 2651	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) = (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))
13	1, 11, 12	evls1rhmlem 19734	. . . 4 ⊢ (𝑆 ∈ CRing → (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∈ ((𝑆 ↑s (𝐵 ↑𝑚 1𝑜)) RingHom 𝑇))
14	13	adantr 480	. . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∈ ((𝑆 ↑s (𝐵 ↑𝑚 1𝑜)) RingHom 𝑇))
15		1on 7612	. . . . 5 ⊢ 1𝑜 ∈ On
16		eqid 2651	. . . . . 6 ⊢ ((1𝑜 evalSub 𝑆)‘𝑅) = ((1𝑜 evalSub 𝑆)‘𝑅)
17		eqid 2651	. . . . . 6 ⊢ (1𝑜 mPoly 𝑈) = (1𝑜 mPoly 𝑈)
18		evls1rhm.u	. . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅)
19		eqid 2651	. . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑𝑚 1𝑜)) = (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))
20	16, 17, 18, 19, 1	evlsrhm 19569	. . . . 5 ⊢ ((1𝑜 ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ ((1𝑜 mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))
21	15, 20	mp3an1 1451	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ ((1𝑜 mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))
22		eqidd 2652	. . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘𝑊))
23		eqidd 2652	. . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))) = (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))
24		evls1rhm.w	. . . . . . 7 ⊢ 𝑊 = (Poly1‘𝑈)
25		eqid 2651	. . . . . . 7 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈)
26		eqid 2651	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
27	24, 25, 26	ply1bas 19613	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘(1𝑜 mPoly 𝑈))
28	27	a1i 11	. . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘(1𝑜 mPoly 𝑈)))
29		eqid 2651	. . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊)
30	24, 17, 29	ply1plusg 19643	. . . . . . 7 ⊢ (+g‘𝑊) = (+g‘(1𝑜 mPoly 𝑈))
31	30	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (+g‘𝑊) = (+g‘(1𝑜 mPoly 𝑈)))
32	31	oveqdr 6714	. . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘(1𝑜 mPoly 𝑈))𝑦))
33		eqidd 2652	. . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))) ∧ 𝑦 ∈ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))) → (𝑥(+g‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))𝑦) = (𝑥(+g‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))𝑦))
34		eqid 2651	. . . . . . . 8 ⊢ (.r‘𝑊) = (.r‘𝑊)
35	24, 17, 34	ply1mulr 19645	. . . . . . 7 ⊢ (.r‘𝑊) = (.r‘(1𝑜 mPoly 𝑈))
36	35	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (.r‘𝑊) = (.r‘(1𝑜 mPoly 𝑈)))
37	36	oveqdr 6714	. . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)𝑦) = (𝑥(.r‘(1𝑜 mPoly 𝑈))𝑦))
38		eqidd 2652	. . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))) ∧ 𝑦 ∈ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))) → (𝑥(.r‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))𝑦) = (𝑥(.r‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))𝑦))
39	22, 23, 28, 23, 32, 33, 37, 38	rhmpropd 18863	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑊 RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))) = ((1𝑜 mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))
40	21, 39	eleqtrrd 2733	. . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))
41		rhmco 18785	. . 3 ⊢ (((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∈ ((𝑆 ↑s (𝐵 ↑𝑚 1𝑜)) RingHom 𝑇) ∧ ((1𝑜 evalSub 𝑆)‘𝑅) ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))) → ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅)) ∈ (𝑊 RingHom 𝑇))
42	14, 40, 41	syl2anc 694	. 2 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅)) ∈ (𝑊 RingHom 𝑇))
43	10, 42	eqeltrd 2730	1 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ⊆ wss 3607  𝒫 cpw 4191  {csn 4210   ↦ cmpt 4762   × cxp 5141   ∘ ccom 5147  Oncon0 5761  ‘cfv 5926  (class class class)co 6690  1_𝑜c1o 7598   ↑_𝑚 cmap 7899  Basecbs 15904   ↾_s cress 15905  +_gcplusg 15988  ._rcmulr 15989   ↑_s cpws 16154  CRingccrg 18594   RingHom crh 18760  SubRingcsubrg 18824   mPoly cmpl 19401   evalSub ces 19552  PwSer₁cps1 19593  Poly₁cpl1 19595   evalSub₁ ces1 19726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-hom 16013  df-cco 16014  df-0g 16149  df-gsum 16150  df-prds 16155  df-pws 16157  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-srg 18552  df-ring 18595  df-cring 18596  df-rnghom 18763  df-subrg 18826  df-lmod 18913  df-lss 18981  df-lsp 19020  df-assa 19360  df-asp 19361  df-ascl 19362  df-psr 19404  df-mvr 19405  df-mpl 19406  df-opsr 19408  df-evls 19554  df-psr1 19598  df-ply1 19600  df-evls1 19728

This theorem is referenced by:  evls1gsumadd  19737  evls1gsummul  19738  evls1varpw  19739