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Theorem evls1sca 20486
Description: Univariate polynomial evaluation maps scalars to constant functions. (Contributed by AV, 8-Sep-2019.)
Hypotheses
Ref Expression
evls1sca.q 𝑄 = (𝑆 evalSub1 𝑅)
evls1sca.w 𝑊 = (Poly1𝑈)
evls1sca.u 𝑈 = (𝑆s 𝑅)
evls1sca.b 𝐵 = (Base‘𝑆)
evls1sca.a 𝐴 = (algSc‘𝑊)
evls1sca.s (𝜑𝑆 ∈ CRing)
evls1sca.r (𝜑𝑅 ∈ (SubRing‘𝑆))
evls1sca.x (𝜑𝑋𝑅)
Assertion
Ref Expression
evls1sca (𝜑 → (𝑄‘(𝐴𝑋)) = (𝐵 × {𝑋}))

Proof of Theorem evls1sca
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1on 8105 . . . . . 6 1o ∈ On
2 evls1sca.s . . . . . 6 (𝜑𝑆 ∈ CRing)
3 evls1sca.r . . . . . 6 (𝜑𝑅 ∈ (SubRing‘𝑆))
4 eqid 2824 . . . . . . 7 ((1o evalSub 𝑆)‘𝑅) = ((1o evalSub 𝑆)‘𝑅)
5 eqid 2824 . . . . . . 7 (1o mPoly 𝑈) = (1o mPoly 𝑈)
6 evls1sca.u . . . . . . 7 𝑈 = (𝑆s 𝑅)
7 eqid 2824 . . . . . . 7 (𝑆s (𝐵m 1o)) = (𝑆s (𝐵m 1o))
8 evls1sca.b . . . . . . 7 𝐵 = (Base‘𝑆)
94, 5, 6, 7, 8evlsrhm 20301 . . . . . 6 ((1o ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆s (𝐵m 1o))))
101, 2, 3, 9mp3an2i 1463 . . . . 5 (𝜑 → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆s (𝐵m 1o))))
11 eqid 2824 . . . . . 6 (Base‘(1o mPoly 𝑈)) = (Base‘(1o mPoly 𝑈))
12 eqid 2824 . . . . . 6 (Base‘(𝑆s (𝐵m 1o))) = (Base‘(𝑆s (𝐵m 1o)))
1311, 12rhmf 19481 . . . . 5 (((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆s (𝐵m 1o))) → ((1o evalSub 𝑆)‘𝑅):(Base‘(1o mPoly 𝑈))⟶(Base‘(𝑆s (𝐵m 1o))))
1410, 13syl 17 . . . 4 (𝜑 → ((1o evalSub 𝑆)‘𝑅):(Base‘(1o mPoly 𝑈))⟶(Base‘(𝑆s (𝐵m 1o))))
15 evls1sca.a . . . . . . 7 𝐴 = (algSc‘𝑊)
16 eqid 2824 . . . . . . 7 (Scalar‘𝑊) = (Scalar‘𝑊)
176subrgring 19538 . . . . . . . . 9 (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
183, 17syl 17 . . . . . . . 8 (𝜑𝑈 ∈ Ring)
19 evls1sca.w . . . . . . . . 9 𝑊 = (Poly1𝑈)
2019ply1ring 20416 . . . . . . . 8 (𝑈 ∈ Ring → 𝑊 ∈ Ring)
2118, 20syl 17 . . . . . . 7 (𝜑𝑊 ∈ Ring)
2219ply1lmod 20420 . . . . . . . 8 (𝑈 ∈ Ring → 𝑊 ∈ LMod)
2318, 22syl 17 . . . . . . 7 (𝜑𝑊 ∈ LMod)
24 eqid 2824 . . . . . . 7 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
25 eqid 2824 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
2615, 16, 21, 23, 24, 25asclf 20111 . . . . . 6 (𝜑𝐴:(Base‘(Scalar‘𝑊))⟶(Base‘𝑊))
278subrgss 19536 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
283, 27syl 17 . . . . . . . . 9 (𝜑𝑅𝐵)
296, 8ressbas2 16555 . . . . . . . . 9 (𝑅𝐵𝑅 = (Base‘𝑈))
3028, 29syl 17 . . . . . . . 8 (𝜑𝑅 = (Base‘𝑈))
3119ply1sca 20421 . . . . . . . . . 10 (𝑈 ∈ Ring → 𝑈 = (Scalar‘𝑊))
3218, 31syl 17 . . . . . . . . 9 (𝜑𝑈 = (Scalar‘𝑊))
3332fveq2d 6665 . . . . . . . 8 (𝜑 → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
3430, 33eqtrd 2859 . . . . . . 7 (𝜑𝑅 = (Base‘(Scalar‘𝑊)))
35 eqid 2824 . . . . . . . . . 10 (PwSer1𝑈) = (PwSer1𝑈)
3619, 35, 25ply1bas 20363 . . . . . . . . 9 (Base‘𝑊) = (Base‘(1o mPoly 𝑈))
3736a1i 11 . . . . . . . 8 (𝜑 → (Base‘𝑊) = (Base‘(1o mPoly 𝑈)))
3837eqcomd 2830 . . . . . . 7 (𝜑 → (Base‘(1o mPoly 𝑈)) = (Base‘𝑊))
3934, 38feq23d 6498 . . . . . 6 (𝜑 → (𝐴:𝑅⟶(Base‘(1o mPoly 𝑈)) ↔ 𝐴:(Base‘(Scalar‘𝑊))⟶(Base‘𝑊)))
4026, 39mpbird 260 . . . . 5 (𝜑𝐴:𝑅⟶(Base‘(1o mPoly 𝑈)))
41 evls1sca.x . . . . 5 (𝜑𝑋𝑅)
4240, 41ffvelrnd 6843 . . . 4 (𝜑 → (𝐴𝑋) ∈ (Base‘(1o mPoly 𝑈)))
43 fvco3 6751 . . . 4 ((((1o evalSub 𝑆)‘𝑅):(Base‘(1o mPoly 𝑈))⟶(Base‘(𝑆s (𝐵m 1o))) ∧ (𝐴𝑋) ∈ (Base‘(1o mPoly 𝑈))) → (((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅))‘(𝐴𝑋)) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))‘(((1o evalSub 𝑆)‘𝑅)‘(𝐴𝑋))))
4414, 42, 43syl2anc 587 . . 3 (𝜑 → (((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅))‘(𝐴𝑋)) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))‘(((1o evalSub 𝑆)‘𝑅)‘(𝐴𝑋))))
4515a1i 11 . . . . . . . 8 (𝜑𝐴 = (algSc‘𝑊))
46 eqid 2824 . . . . . . . . 9 (algSc‘𝑊) = (algSc‘𝑊)
4719, 46ply1ascl 20426 . . . . . . . 8 (algSc‘𝑊) = (algSc‘(1o mPoly 𝑈))
4845, 47syl6eq 2875 . . . . . . 7 (𝜑𝐴 = (algSc‘(1o mPoly 𝑈)))
4948fveq1d 6663 . . . . . 6 (𝜑 → (𝐴𝑋) = ((algSc‘(1o mPoly 𝑈))‘𝑋))
5049fveq2d 6665 . . . . 5 (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘(𝐴𝑋)) = (((1o evalSub 𝑆)‘𝑅)‘((algSc‘(1o mPoly 𝑈))‘𝑋)))
51 eqid 2824 . . . . . 6 (algSc‘(1o mPoly 𝑈)) = (algSc‘(1o mPoly 𝑈))
521a1i 11 . . . . . 6 (𝜑 → 1o ∈ On)
534, 5, 6, 8, 51, 52, 2, 3, 41evlssca 20302 . . . . 5 (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘((algSc‘(1o mPoly 𝑈))‘𝑋)) = ((𝐵m 1o) × {𝑋}))
5450, 53eqtrd 2859 . . . 4 (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘(𝐴𝑋)) = ((𝐵m 1o) × {𝑋}))
5554fveq2d 6665 . . 3 (𝜑 → ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))‘(((1o evalSub 𝑆)‘𝑅)‘(𝐴𝑋))) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))‘((𝐵m 1o) × {𝑋})))
56 eqidd 2825 . . . . 5 (𝜑 → (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))))
57 coeq1 5715 . . . . . 6 (𝑥 = ((𝐵m 1o) × {𝑋}) → (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))) = (((𝐵m 1o) × {𝑋}) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
5857adantl 485 . . . . 5 ((𝜑𝑥 = ((𝐵m 1o) × {𝑋})) → (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))) = (((𝐵m 1o) × {𝑋}) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
5928, 41sseldd 3954 . . . . . . 7 (𝜑𝑋𝐵)
60 fconst6g 6558 . . . . . . 7 (𝑋𝐵 → ((𝐵m 1o) × {𝑋}):(𝐵m 1o)⟶𝐵)
6159, 60syl 17 . . . . . 6 (𝜑 → ((𝐵m 1o) × {𝑋}):(𝐵m 1o)⟶𝐵)
628fvexi 6675 . . . . . . . 8 𝐵 ∈ V
6362a1i 11 . . . . . . 7 (𝜑𝐵 ∈ V)
64 ovex 7182 . . . . . . . 8 (𝐵m 1o) ∈ V
6564a1i 11 . . . . . . 7 (𝜑 → (𝐵m 1o) ∈ V)
6663, 65elmapd 8416 . . . . . 6 (𝜑 → (((𝐵m 1o) × {𝑋}) ∈ (𝐵m (𝐵m 1o)) ↔ ((𝐵m 1o) × {𝑋}):(𝐵m 1o)⟶𝐵))
6761, 66mpbird 260 . . . . 5 (𝜑 → ((𝐵m 1o) × {𝑋}) ∈ (𝐵m (𝐵m 1o)))
68 snex 5319 . . . . . . . 8 {𝑋} ∈ V
6964, 68xpex 7470 . . . . . . 7 ((𝐵m 1o) × {𝑋}) ∈ V
7069a1i 11 . . . . . 6 (𝜑 → ((𝐵m 1o) × {𝑋}) ∈ V)
7163mptexd 6978 . . . . . 6 (𝜑 → (𝑦𝐵 ↦ (1o × {𝑦})) ∈ V)
72 coexg 7629 . . . . . 6 ((((𝐵m 1o) × {𝑋}) ∈ V ∧ (𝑦𝐵 ↦ (1o × {𝑦})) ∈ V) → (((𝐵m 1o) × {𝑋}) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))) ∈ V)
7370, 71, 72syl2anc 587 . . . . 5 (𝜑 → (((𝐵m 1o) × {𝑋}) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))) ∈ V)
7456, 58, 67, 73fvmptd 6766 . . . 4 (𝜑 → ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))‘((𝐵m 1o) × {𝑋})) = (((𝐵m 1o) × {𝑋}) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
75 fconst6g 6558 . . . . . . 7 (𝑦𝐵 → (1o × {𝑦}):1o𝐵)
7675adantl 485 . . . . . 6 ((𝜑𝑦𝐵) → (1o × {𝑦}):1o𝐵)
7762, 1pm3.2i 474 . . . . . . . 8 (𝐵 ∈ V ∧ 1o ∈ On)
7877a1i 11 . . . . . . 7 ((𝜑𝑦𝐵) → (𝐵 ∈ V ∧ 1o ∈ On))
79 elmapg 8415 . . . . . . 7 ((𝐵 ∈ V ∧ 1o ∈ On) → ((1o × {𝑦}) ∈ (𝐵m 1o) ↔ (1o × {𝑦}):1o𝐵))
8078, 79syl 17 . . . . . 6 ((𝜑𝑦𝐵) → ((1o × {𝑦}) ∈ (𝐵m 1o) ↔ (1o × {𝑦}):1o𝐵))
8176, 80mpbird 260 . . . . 5 ((𝜑𝑦𝐵) → (1o × {𝑦}) ∈ (𝐵m 1o))
82 eqidd 2825 . . . . 5 (𝜑 → (𝑦𝐵 ↦ (1o × {𝑦})) = (𝑦𝐵 ↦ (1o × {𝑦})))
83 fconstmpt 5601 . . . . . 6 ((𝐵m 1o) × {𝑋}) = (𝑧 ∈ (𝐵m 1o) ↦ 𝑋)
8483a1i 11 . . . . 5 (𝜑 → ((𝐵m 1o) × {𝑋}) = (𝑧 ∈ (𝐵m 1o) ↦ 𝑋))
85 eqidd 2825 . . . . 5 (𝑧 = (1o × {𝑦}) → 𝑋 = 𝑋)
8681, 82, 84, 85fmptco 6882 . . . 4 (𝜑 → (((𝐵m 1o) × {𝑋}) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))) = (𝑦𝐵𝑋))
8774, 86eqtrd 2859 . . 3 (𝜑 → ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))‘((𝐵m 1o) × {𝑋})) = (𝑦𝐵𝑋))
8844, 55, 873eqtrd 2863 . 2 (𝜑 → (((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅))‘(𝐴𝑋)) = (𝑦𝐵𝑋))
89 elpwg 4525 . . . . . 6 (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵𝑅𝐵))
9027, 89mpbird 260 . . . . 5 (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ∈ 𝒫 𝐵)
913, 90syl 17 . . . 4 (𝜑𝑅 ∈ 𝒫 𝐵)
92 evls1sca.q . . . . 5 𝑄 = (𝑆 evalSub1 𝑅)
93 eqid 2824 . . . . 5 (1o evalSub 𝑆) = (1o evalSub 𝑆)
9492, 93, 8evls1fval 20482 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)))
952, 91, 94syl2anc 587 . . 3 (𝜑𝑄 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)))
9695fveq1d 6663 . 2 (𝜑 → (𝑄‘(𝐴𝑋)) = (((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅))‘(𝐴𝑋)))
97 fconstmpt 5601 . . 3 (𝐵 × {𝑋}) = (𝑦𝐵𝑋)
9897a1i 11 . 2 (𝜑 → (𝐵 × {𝑋}) = (𝑦𝐵𝑋))
9988, 96, 983eqtr4d 2869 1 (𝜑 → (𝑄‘(𝐴𝑋)) = (𝐵 × {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  wss 3919  𝒫 cpw 4522  {csn 4550  cmpt 5132   × cxp 5540  ccom 5546  Oncon0 6178  wf 6339  cfv 6343  (class class class)co 7149  1oc1o 8091  m cmap 8402  Basecbs 16483  s cress 16484  Scalarcsca 16568  s cpws 16720  Ringcrg 19297  CRingccrg 19298   RingHom crh 19467  SubRingcsubrg 19531  LModclmod 19634  algSccascl 20084   mPoly cmpl 20133   evalSub ces 20284  PwSer1cps1 20343  Poly1cpl1 20345   evalSub1 ces1 20476
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-sup 8903  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-dec 12096  df-uz 12241  df-fz 12895  df-fzo 13038  df-seq 13374  df-hash 13696  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-hom 16589  df-cco 16590  df-0g 16715  df-gsum 16716  df-prds 16721  df-pws 16723  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-mulg 18225  df-subg 18276  df-ghm 18356  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-srg 19256  df-ring 19299  df-cring 19300  df-rnghom 19470  df-subrg 19533  df-lmod 19636  df-lss 19704  df-lsp 19744  df-assa 20085  df-asp 20086  df-ascl 20087  df-psr 20136  df-mvr 20137  df-mpl 20138  df-opsr 20140  df-evls 20286  df-psr1 20348  df-ply1 20350  df-evls1 20478
This theorem is referenced by:  evls1scasrng  20502
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