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Theorem evls1rhm 20944
 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 = (𝑆s 𝐵)
evls1rhm.u 𝑈 = (𝑆s 𝑅)
evls1rhm.w 𝑊 = (Poly1𝑈)
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.
StepHypRef Expression
1 evls1rhm.b . . . . . 6 𝐵 = (Base‘𝑆)
21subrgss 19527 . . . . 5 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
32adantl 485 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅𝐵)
4 elpwg 4514 . . . . 5 (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵𝑅𝐵))
54adantl 485 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑅 ∈ 𝒫 𝐵𝑅𝐵))
63, 5mpbird 260 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ 𝒫 𝐵)
7 evls1rhm.q . . . 4 𝑄 = (𝑆 evalSub1 𝑅)
8 eqid 2822 . . . 4 (1o evalSub 𝑆) = (1o evalSub 𝑆)
97, 8, 1evls1fval 20941 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)))
106, 9syldan 594 . 2 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)))
11 evls1rhm.t . . . 4 𝑇 = (𝑆s 𝐵)
12 eqid 2822 . . . 4 (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
131, 11, 12evls1rhmlem 20943 . . 3 (𝑆 ∈ CRing → (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑆s (𝐵m 1o)) RingHom 𝑇))
14 1on 8096 . . . . 5 1o ∈ On
15 eqid 2822 . . . . . 6 ((1o evalSub 𝑆)‘𝑅) = ((1o evalSub 𝑆)‘𝑅)
16 eqid 2822 . . . . . 6 (1o mPoly 𝑈) = (1o mPoly 𝑈)
17 evls1rhm.u . . . . . 6 𝑈 = (𝑆s 𝑅)
18 eqid 2822 . . . . . 6 (𝑆s (𝐵m 1o)) = (𝑆s (𝐵m 1o))
1915, 16, 17, 18, 1evlsrhm 20758 . . . . 5 ((1o ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆s (𝐵m 1o))))
2014, 19mp3an1 1445 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆s (𝐵m 1o))))
21 eqidd 2823 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘𝑊))
22 eqidd 2823 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘(𝑆s (𝐵m 1o))) = (Base‘(𝑆s (𝐵m 1o))))
23 evls1rhm.w . . . . . . 7 𝑊 = (Poly1𝑈)
24 eqid 2822 . . . . . . 7 (PwSer1𝑈) = (PwSer1𝑈)
25 eqid 2822 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
2623, 24, 25ply1bas 20822 . . . . . 6 (Base‘𝑊) = (Base‘(1o mPoly 𝑈))
2726a1i 11 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘(1o mPoly 𝑈)))
28 eqid 2822 . . . . . . . 8 (+g𝑊) = (+g𝑊)
2923, 16, 28ply1plusg 20852 . . . . . . 7 (+g𝑊) = (+g‘(1o mPoly 𝑈))
3029a1i 11 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (+g𝑊) = (+g‘(1o mPoly 𝑈)))
3130oveqdr 7168 . . . . 5 (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g‘(1o mPoly 𝑈))𝑦))
32 eqidd 2823 . . . . 5 (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘(𝑆s (𝐵m 1o))) ∧ 𝑦 ∈ (Base‘(𝑆s (𝐵m 1o))))) → (𝑥(+g‘(𝑆s (𝐵m 1o)))𝑦) = (𝑥(+g‘(𝑆s (𝐵m 1o)))𝑦))
33 eqid 2822 . . . . . . . 8 (.r𝑊) = (.r𝑊)
3423, 16, 33ply1mulr 20854 . . . . . . 7 (.r𝑊) = (.r‘(1o mPoly 𝑈))
3534a1i 11 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (.r𝑊) = (.r‘(1o mPoly 𝑈)))
3635oveqdr 7168 . . . . 5 (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(.r𝑊)𝑦) = (𝑥(.r‘(1o mPoly 𝑈))𝑦))
37 eqidd 2823 . . . . 5 (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘(𝑆s (𝐵m 1o))) ∧ 𝑦 ∈ (Base‘(𝑆s (𝐵m 1o))))) → (𝑥(.r‘(𝑆s (𝐵m 1o)))𝑦) = (𝑥(.r‘(𝑆s (𝐵m 1o)))𝑦))
3821, 22, 27, 22, 31, 32, 36, 37rhmpropd 19562 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑊 RingHom (𝑆s (𝐵m 1o))) = ((1o mPoly 𝑈) RingHom (𝑆s (𝐵m 1o))))
3920, 38eleqtrrd 2917 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ (𝑊 RingHom (𝑆s (𝐵m 1o))))
40 rhmco 19483 . . 3 (((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑆s (𝐵m 1o)) RingHom 𝑇) ∧ ((1o evalSub 𝑆)‘𝑅) ∈ (𝑊 RingHom (𝑆s (𝐵m 1o)))) → ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)) ∈ (𝑊 RingHom 𝑇))
4113, 39, 40syl2an2r 684 . 2 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)) ∈ (𝑊 RingHom 𝑇))
4210, 41eqeltrd 2914 1 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114   ⊆ wss 3908  𝒫 cpw 4511  {csn 4539   ↦ cmpt 5122   × cxp 5530   ∘ ccom 5536  Oncon0 6169  ‘cfv 6334  (class class class)co 7140  1oc1o 8082   ↑m cmap 8393  Basecbs 16474   ↾s cress 16475  +gcplusg 16556  .rcmulr 16557   ↑s cpws 16711  CRingccrg 19289   RingHom crh 19458  SubRingcsubrg 19522   mPoly cmpl 20589   evalSub ces 20741  PwSer1cps1 20802  Poly1cpl1 20804   evalSub1 ces1 20935 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-of 7394  df-ofr 7395  df-om 7566  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-struct 16476  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-plusg 16569  df-mulr 16570  df-sca 16572  df-vsca 16573  df-ip 16574  df-tset 16575  df-ple 16576  df-ds 16578  df-hom 16580  df-cco 16581  df-0g 16706  df-gsum 16707  df-prds 16712  df-pws 16714  df-mre 16848  df-mrc 16849  df-acs 16851  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-mhm 17947  df-submnd 17948  df-grp 18097  df-minusg 18098  df-sbg 18099  df-mulg 18216  df-subg 18267  df-ghm 18347  df-cntz 18438  df-cmn 18899  df-abl 18900  df-mgp 19231  df-ur 19243  df-srg 19247  df-ring 19290  df-cring 19291  df-rnghom 19461  df-subrg 19524  df-lmod 19627  df-lss 19695  df-lsp 19735  df-assa 20540  df-asp 20541  df-ascl 20542  df-psr 20592  df-mvr 20593  df-mpl 20594  df-opsr 20596  df-evls 20743  df-psr1 20807  df-ply1 20809  df-evls1 20937 This theorem is referenced by:  evls1gsumadd  20946  evls1gsummul  20947  evls1pw  20948
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