Theorem evls1rhm 19687

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 18781	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
3	2	adantl 482	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ⊆ 𝐵)
4		elpwg 4166	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵))
5	4	adantl 482	. . . 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 2622	. . . 4 ⊢ (1𝑜 evalSub 𝑆) = (1𝑜 evalSub 𝑆)
9	7, 8, 1	evls1fval 19684	. . 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 487	. 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 2622	. . . . 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 19686	. . . 4 ⊢ (𝑆 ∈ CRing → (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∈ ((𝑆 ↑s (𝐵 ↑𝑚 1𝑜)) RingHom 𝑇))
14	13	adantr 481	. . 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 7567	. . . . 5 ⊢ 1𝑜 ∈ On
16		eqid 2622	. . . . . 6 ⊢ ((1𝑜 evalSub 𝑆)‘𝑅) = ((1𝑜 evalSub 𝑆)‘𝑅)
17		eqid 2622	. . . . . 6 ⊢ (1𝑜 mPoly 𝑈) = (1𝑜 mPoly 𝑈)
18		evls1rhm.u	. . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅)
19		eqid 2622	. . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑𝑚 1𝑜)) = (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))
20	16, 17, 18, 19, 1	evlsrhm 19521	. . . . 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 1411	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ ((1𝑜 mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))
22		eqidd 2623	. . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘𝑊))
23		eqidd 2623	. . . . 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 2622	. . . . . . 7 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈)
26		eqid 2622	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
27	24, 25, 26	ply1bas 19565	. . . . . 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 2622	. . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊)
30	24, 17, 29	ply1plusg 19595	. . . . . . 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 6674	. . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘(1𝑜 mPoly 𝑈))𝑦))
33		eqidd 2623	. . . . 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 2622	. . . . . . . 8 ⊢ (.r‘𝑊) = (.r‘𝑊)
35	24, 17, 34	ply1mulr 19597	. . . . . . 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 6674	. . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)𝑦) = (𝑥(.r‘(1𝑜 mPoly 𝑈))𝑦))
38		eqidd 2623	. . . . 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 18815	. . . 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 2704	. . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))
41		rhmco 18737	. . 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 693	. 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 2701	1 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177   ↦ cmpt 4729   × cxp 5112   ∘ ccom 5118  Oncon0 5723  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553   ↑_𝑚 cmap 7857  Basecbs 15857   ↾_s cress 15858  +_gcplusg 15941  ._rcmulr 15942   ↑_s cpws 16107  CRingccrg 18548   RingHom crh 18712  SubRingcsubrg 18776   mPoly cmpl 19353   evalSub ces 19504  PwSer₁cps1 19545  Poly₁cpl1 19547   evalSub₁ ces1 19678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-evls 19506  df-psr1 19550  df-ply1 19552  df-evls1 19680

This theorem is referenced by:  evls1gsumadd  19689  evls1gsummul  19690  evls1varpw  19691