MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  evls1rhm Structured version   Visualization version   GIF version

Theorem evls1rhm 22326
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 20572 . . . . 5 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
32adantl 481 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅𝐵)
4 elpwg 4603 . . . . 5 (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵𝑅𝐵))
54adantl 481 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑅 ∈ 𝒫 𝐵𝑅𝐵))
63, 5mpbird 257 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ 𝒫 𝐵)
7 evls1rhm.q . . . 4 𝑄 = (𝑆 evalSub1 𝑅)
8 eqid 2737 . . . 4 (1o evalSub 𝑆) = (1o evalSub 𝑆)
97, 8, 1evls1fval 22323 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)))
106, 9syldan 591 . 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 2737 . . . 4 (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
131, 11, 12evls1rhmlem 22325 . . 3 (𝑆 ∈ CRing → (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑆s (𝐵m 1o)) RingHom 𝑇))
14 1on 8518 . . . . 5 1o ∈ On
15 eqid 2737 . . . . . 6 ((1o evalSub 𝑆)‘𝑅) = ((1o evalSub 𝑆)‘𝑅)
16 eqid 2737 . . . . . 6 (1o mPoly 𝑈) = (1o mPoly 𝑈)
17 evls1rhm.u . . . . . 6 𝑈 = (𝑆s 𝑅)
18 eqid 2737 . . . . . 6 (𝑆s (𝐵m 1o)) = (𝑆s (𝐵m 1o))
1915, 16, 17, 18, 1evlsrhm 22112 . . . . 5 ((1o ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆s (𝐵m 1o))))
2014, 19mp3an1 1450 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆s (𝐵m 1o))))
21 eqidd 2738 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘𝑊))
22 eqidd 2738 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘(𝑆s (𝐵m 1o))) = (Base‘(𝑆s (𝐵m 1o))))
23 evls1rhm.w . . . . . . 7 𝑊 = (Poly1𝑈)
24 eqid 2737 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
2523, 24ply1bas 22196 . . . . . 6 (Base‘𝑊) = (Base‘(1o mPoly 𝑈))
2625a1i 11 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘(1o mPoly 𝑈)))
27 eqid 2737 . . . . . . . 8 (+g𝑊) = (+g𝑊)
2823, 16, 27ply1plusg 22225 . . . . . . 7 (+g𝑊) = (+g‘(1o mPoly 𝑈))
2928a1i 11 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (+g𝑊) = (+g‘(1o mPoly 𝑈)))
3029oveqdr 7459 . . . . 5 (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g‘(1o mPoly 𝑈))𝑦))
31 eqidd 2738 . . . . 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))
32 eqid 2737 . . . . . . . 8 (.r𝑊) = (.r𝑊)
3323, 16, 32ply1mulr 22227 . . . . . . 7 (.r𝑊) = (.r‘(1o mPoly 𝑈))
3433a1i 11 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (.r𝑊) = (.r‘(1o mPoly 𝑈)))
3534oveqdr 7459 . . . . 5 (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(.r𝑊)𝑦) = (𝑥(.r‘(1o mPoly 𝑈))𝑦))
36 eqidd 2738 . . . . 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))
3721, 22, 26, 22, 30, 31, 35, 36rhmpropd 20609 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑊 RingHom (𝑆s (𝐵m 1o))) = ((1o mPoly 𝑈) RingHom (𝑆s (𝐵m 1o))))
3820, 37eleqtrrd 2844 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ (𝑊 RingHom (𝑆s (𝐵m 1o))))
39 rhmco 20501 . . 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))
4013, 38, 39syl2an2r 685 . 2 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)) ∈ (𝑊 RingHom 𝑇))
4110, 40eqeltrd 2841 1 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951  𝒫 cpw 4600  {csn 4626  cmpt 5225   × cxp 5683  ccom 5689  Oncon0 6384  cfv 6561  (class class class)co 7431  1oc1o 8499  m cmap 8866  Basecbs 17247  s cress 17274  +gcplusg 17297  .rcmulr 17298  s cpws 17491  CRingccrg 20231   RingHom crh 20469  SubRingcsubrg 20569   mPoly cmpl 21926   evalSub ces 22096  Poly1cpl1 22178   evalSub1 ces1 22317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-rhm 20472  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-lsp 20970  df-assa 21873  df-asp 21874  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-evls 22098  df-psr1 22181  df-ply1 22183  df-evls1 22319
This theorem is referenced by:  evls1gsumadd  22328  evls1gsummul  22329  evls1pw  22330  evls1expd  22371  evls1fpws  22373  ressply1evl  22374  evls1fn  33586  evls1dm  33587  evls1fvf  33588  elirng  33736  irngnzply1lem  33740  irngnzply1  33741
  Copyright terms: Public domain W3C validator