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Theorem evls1val 20459
 Description: Value of the univariate polynomial evaluation map. (Contributed by AV, 10-Sep-2019.)
Hypotheses
Ref Expression
evls1fval.q 𝑄 = (𝑆 evalSub1 𝑅)
evls1fval.e 𝐸 = (1o evalSub 𝑆)
evls1fval.b 𝐵 = (Base‘𝑆)
evls1val.m 𝑀 = (1o mPoly (𝑆s 𝑅))
evls1val.k 𝐾 = (Base‘𝑀)
Assertion
Ref Expression
evls1val ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝑄𝐴) = (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
Distinct variable group:   𝑦,𝐵
Allowed substitution hints:   𝐴(𝑦)   𝑄(𝑦)   𝑅(𝑦)   𝑆(𝑦)   𝐸(𝑦)   𝐾(𝑦)   𝑀(𝑦)

Proof of Theorem evls1val
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 evls1fval.b . . . . . . . 8 𝐵 = (Base‘𝑆)
21subrgss 19512 . . . . . . 7 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
32adantl 484 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅𝐵)
4 elpwg 4518 . . . . . . 7 (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵𝑅𝐵))
54adantl 484 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑅 ∈ 𝒫 𝐵𝑅𝐵))
63, 5mpbird 259 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ 𝒫 𝐵)
7 evls1fval.q . . . . . 6 𝑄 = (𝑆 evalSub1 𝑅)
8 evls1fval.e . . . . . 6 𝐸 = (1o evalSub 𝑆)
97, 8, 1evls1fval 20458 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸𝑅)))
106, 9syldan 593 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸𝑅)))
1110fveq1d 6648 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄𝐴) = (((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸𝑅))‘𝐴))
12113adant3 1128 . 2 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝑄𝐴) = (((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸𝑅))‘𝐴))
13 1on 8087 . . . . 5 1o ∈ On
14 simp1 1132 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → 𝑆 ∈ CRing)
15 simp2 1133 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → 𝑅 ∈ (SubRing‘𝑆))
168fveq1i 6647 . . . . . 6 (𝐸𝑅) = ((1o evalSub 𝑆)‘𝑅)
17 evls1val.m . . . . . 6 𝑀 = (1o mPoly (𝑆s 𝑅))
18 eqid 2820 . . . . . 6 (𝑆s 𝑅) = (𝑆s 𝑅)
19 eqid 2820 . . . . . 6 (𝑆s (𝐵m 1o)) = (𝑆s (𝐵m 1o))
2016, 17, 18, 19, 1evlsrhm 20277 . . . . 5 ((1o ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝐸𝑅) ∈ (𝑀 RingHom (𝑆s (𝐵m 1o))))
2113, 14, 15, 20mp3an2i 1462 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝐸𝑅) ∈ (𝑀 RingHom (𝑆s (𝐵m 1o))))
22 evls1val.k . . . . 5 𝐾 = (Base‘𝑀)
23 eqid 2820 . . . . 5 (Base‘(𝑆s (𝐵m 1o))) = (Base‘(𝑆s (𝐵m 1o)))
2422, 23rhmf 19457 . . . 4 ((𝐸𝑅) ∈ (𝑀 RingHom (𝑆s (𝐵m 1o))) → (𝐸𝑅):𝐾⟶(Base‘(𝑆s (𝐵m 1o))))
2521, 24syl 17 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝐸𝑅):𝐾⟶(Base‘(𝑆s (𝐵m 1o))))
26 simp3 1134 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → 𝐴𝐾)
27 fvco3 6736 . . 3 (((𝐸𝑅):𝐾⟶(Base‘(𝑆s (𝐵m 1o))) ∧ 𝐴𝐾) → (((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸𝑅))‘𝐴) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))‘((𝐸𝑅)‘𝐴)))
2825, 26, 27syl2anc 586 . 2 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸𝑅))‘𝐴) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))‘((𝐸𝑅)‘𝐴)))
2925, 26ffvelrnd 6828 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → ((𝐸𝑅)‘𝐴) ∈ (Base‘(𝑆s (𝐵m 1o))))
30 ovex 7166 . . . . 5 (𝐵m 1o) ∈ V
3119, 1pwsbas 16739 . . . . 5 ((𝑆 ∈ CRing ∧ (𝐵m 1o) ∈ V) → (𝐵m (𝐵m 1o)) = (Base‘(𝑆s (𝐵m 1o))))
3214, 30, 31sylancl 588 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝐵m (𝐵m 1o)) = (Base‘(𝑆s (𝐵m 1o))))
3329, 32eleqtrrd 2914 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → ((𝐸𝑅)‘𝐴) ∈ (𝐵m (𝐵m 1o)))
34 coeq1 5704 . . . 4 (𝑥 = ((𝐸𝑅)‘𝐴) → (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))) = (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
35 eqid 2820 . . . 4 (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
36 fvex 6659 . . . . 5 ((𝐸𝑅)‘𝐴) ∈ V
371fvexi 6660 . . . . . 6 𝐵 ∈ V
3837mptex 6962 . . . . 5 (𝑦𝐵 ↦ (1o × {𝑦})) ∈ V
3936, 38coex 7613 . . . 4 (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))) ∈ V
4034, 35, 39fvmpt 6744 . . 3 (((𝐸𝑅)‘𝐴) ∈ (𝐵m (𝐵m 1o)) → ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))‘((𝐸𝑅)‘𝐴)) = (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
4133, 40syl 17 . 2 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))‘((𝐸𝑅)‘𝐴)) = (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
4212, 28, 413eqtrd 2859 1 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝑄𝐴) = (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3473   ⊆ wss 3913  𝒫 cpw 4515  {csn 4543   ↦ cmpt 5122   × cxp 5529   ∘ ccom 5535  Oncon0 6167  ⟶wf 6327  ‘cfv 6331  (class class class)co 7133  1oc1o 8073   ↑m cmap 8384  Basecbs 16462   ↾s cress 16463   ↑s cpws 16699  CRingccrg 19277   RingHom crh 19443  SubRingcsubrg 19507   mPoly cmpl 20109   evalSub ces 20260   evalSub1 ces1 20452 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-iin 4898  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-of 7387  df-ofr 7388  df-om 7559  df-1st 7667  df-2nd 7668  df-supp 7809  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-2o 8081  df-oadd 8084  df-er 8267  df-map 8386  df-pm 8387  df-ixp 8440  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-fsupp 8812  df-sup 8884  df-oi 8952  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-nn 11617  df-2 11679  df-3 11680  df-4 11681  df-5 11682  df-6 11683  df-7 11684  df-8 11685  df-9 11686  df-n0 11877  df-z 11961  df-dec 12078  df-uz 12223  df-fz 12877  df-fzo 13018  df-seq 13354  df-hash 13676  df-struct 16464  df-ndx 16465  df-slot 16466  df-base 16468  df-sets 16469  df-ress 16470  df-plusg 16557  df-mulr 16558  df-sca 16560  df-vsca 16561  df-ip 16562  df-tset 16563  df-ple 16564  df-ds 16566  df-hom 16568  df-cco 16569  df-0g 16694  df-gsum 16695  df-prds 16700  df-pws 16702  df-mre 16836  df-mrc 16837  df-acs 16839  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-mhm 17935  df-submnd 17936  df-grp 18085  df-minusg 18086  df-sbg 18087  df-mulg 18204  df-subg 18255  df-ghm 18335  df-cntz 18426  df-cmn 18887  df-abl 18888  df-mgp 19219  df-ur 19231  df-srg 19235  df-ring 19278  df-cring 19279  df-rnghom 19446  df-subrg 19509  df-lmod 19612  df-lss 19680  df-lsp 19720  df-assa 20061  df-asp 20062  df-ascl 20063  df-psr 20112  df-mvr 20113  df-mpl 20114  df-evls 20262  df-evls1 20454 This theorem is referenced by:  evls1var  20477
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