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Theorem evls1val 19625
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 18721 . . . . . . 7 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
32adantl 482 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅𝐵)
4 elpwg 4144 . . . . . . 7 (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵𝑅𝐵))
54adantl 482 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑅 ∈ 𝒫 𝐵𝑅𝐵))
63, 5mpbird 247 . . . . 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 19624 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅)))
106, 9syldan 487 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅)))
1110fveq1d 6160 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄𝐴) = (((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅))‘𝐴))
12113adant3 1079 . 2 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝑄𝐴) = (((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅))‘𝐴))
13 1on 7527 . . . . . 6 1𝑜 ∈ On
1413a1i 11 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → 1𝑜 ∈ On)
15 simp1 1059 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → 𝑆 ∈ CRing)
16 simp2 1060 . . . . 5 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → 𝑅 ∈ (SubRing‘𝑆))
178fveq1i 6159 . . . . . 6 (𝐸𝑅) = ((1𝑜 evalSub 𝑆)‘𝑅)
18 evls1val.m . . . . . 6 𝑀 = (1𝑜 mPoly (𝑆s 𝑅))
19 eqid 2621 . . . . . 6 (𝑆s 𝑅) = (𝑆s 𝑅)
20 eqid 2621 . . . . . 6 (𝑆s (𝐵𝑚 1𝑜)) = (𝑆s (𝐵𝑚 1𝑜))
2117, 18, 19, 20, 1evlsrhm 19461 . . . . 5 ((1𝑜 ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝐸𝑅) ∈ (𝑀 RingHom (𝑆s (𝐵𝑚 1𝑜))))
2214, 15, 16, 21syl3anc 1323 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝐸𝑅) ∈ (𝑀 RingHom (𝑆s (𝐵𝑚 1𝑜))))
23 evls1val.k . . . . 5 𝐾 = (Base‘𝑀)
24 eqid 2621 . . . . 5 (Base‘(𝑆s (𝐵𝑚 1𝑜))) = (Base‘(𝑆s (𝐵𝑚 1𝑜)))
2523, 24rhmf 18666 . . . 4 ((𝐸𝑅) ∈ (𝑀 RingHom (𝑆s (𝐵𝑚 1𝑜))) → (𝐸𝑅):𝐾⟶(Base‘(𝑆s (𝐵𝑚 1𝑜))))
2622, 25syl 17 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝐸𝑅):𝐾⟶(Base‘(𝑆s (𝐵𝑚 1𝑜))))
27 simp3 1061 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → 𝐴𝐾)
28 fvco3 6242 . . 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)))
2926, 27, 28syl2anc 692 . 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)))
3026, 27ffvelrnd 6326 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → ((𝐸𝑅)‘𝐴) ∈ (Base‘(𝑆s (𝐵𝑚 1𝑜))))
31 ovex 6643 . . . . 5 (𝐵𝑚 1𝑜) ∈ V
3220, 1pwsbas 16087 . . . . 5 ((𝑆 ∈ CRing ∧ (𝐵𝑚 1𝑜) ∈ V) → (𝐵𝑚 (𝐵𝑚 1𝑜)) = (Base‘(𝑆s (𝐵𝑚 1𝑜))))
3315, 31, 32sylancl 693 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝐵𝑚 (𝐵𝑚 1𝑜)) = (Base‘(𝑆s (𝐵𝑚 1𝑜))))
3430, 33eleqtrrd 2701 . . 3 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → ((𝐸𝑅)‘𝐴) ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)))
35 coeq1 5249 . . . 4 (𝑥 = ((𝐸𝑅)‘𝐴) → (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) = (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
36 eqid 2621 . . . 4 (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) = (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
37 fvex 6168 . . . . 5 ((𝐸𝑅)‘𝐴) ∈ V
38 fvex 6168 . . . . . . 7 (Base‘𝑆) ∈ V
391, 38eqeltri 2694 . . . . . 6 𝐵 ∈ V
4039mptex 6451 . . . . 5 (𝑦𝐵 ↦ (1𝑜 × {𝑦})) ∈ V
4137, 40coex 7080 . . . 4 (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) ∈ V
4235, 36, 41fvmpt 6249 . . 3 (((𝐸𝑅)‘𝐴) ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) → ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐸𝑅)‘𝐴)) = (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
4334, 42syl 17 . 2 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐸𝑅)‘𝐴)) = (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
4412, 29, 433eqtrd 2659 1 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝑄𝐴) = (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3190  wss 3560  𝒫 cpw 4136  {csn 4155  cmpt 4683   × cxp 5082  ccom 5088  Oncon0 5692  wf 5853  cfv 5857  (class class class)co 6615  1𝑜c1o 7513  𝑚 cmap 7817  Basecbs 15800  s cress 15801  s cpws 16047  CRingccrg 18488   RingHom crh 18652  SubRingcsubrg 18716   mPoly cmpl 19293   evalSub ces 19444   evalSub1 ces1 19618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-ofr 6863  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-sup 8308  df-oi 8375  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-fz 12285  df-fzo 12423  df-seq 12758  df-hash 13074  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-sca 15897  df-vsca 15898  df-ip 15899  df-tset 15900  df-ple 15901  df-ds 15904  df-hom 15906  df-cco 15907  df-0g 16042  df-gsum 16043  df-prds 16048  df-pws 16050  df-mre 16186  df-mrc 16187  df-acs 16189  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275  df-submnd 17276  df-grp 17365  df-minusg 17366  df-sbg 17367  df-mulg 17481  df-subg 17531  df-ghm 17598  df-cntz 17690  df-cmn 18135  df-abl 18136  df-mgp 18430  df-ur 18442  df-srg 18446  df-ring 18489  df-cring 18490  df-rnghom 18655  df-subrg 18718  df-lmod 18805  df-lss 18873  df-lsp 18912  df-assa 19252  df-asp 19253  df-ascl 19254  df-psr 19296  df-mvr 19297  df-mpl 19298  df-evls 19446  df-evls1 19620
This theorem is referenced by:  evls1var  19642
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