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Theorem evlsval2 20796
 Description: Characterizing properties of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 12-Mar-2015.) (Revised by AV, 18-Sep-2021.)
Hypotheses
Ref Expression
evlsval.q 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
evlsval.w 𝑊 = (𝐼 mPoly 𝑈)
evlsval.v 𝑉 = (𝐼 mVar 𝑈)
evlsval.u 𝑈 = (𝑆s 𝑅)
evlsval.t 𝑇 = (𝑆s (𝐵m 𝐼))
evlsval.b 𝐵 = (Base‘𝑆)
evlsval.a 𝐴 = (algSc‘𝑊)
evlsval.x 𝑋 = (𝑥𝑅 ↦ ((𝐵m 𝐼) × {𝑥}))
evlsval.y 𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)))
Assertion
Ref Expression
evlsval2 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
Distinct variable groups:   𝑔,𝐼,𝑥   𝑥,𝑅   𝑆,𝑔,𝑥   𝐵,𝑔,𝑥   𝑅,𝑔   𝑥,𝑇   𝑔,𝑍,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑔)   𝑄(𝑥,𝑔)   𝑇(𝑔)   𝑈(𝑥,𝑔)   𝑉(𝑥,𝑔)   𝑊(𝑥,𝑔)   𝑋(𝑥,𝑔)   𝑌(𝑥,𝑔)

Proof of Theorem evlsval2
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 evlsval.q . . . 4 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
2 evlsval.w . . . 4 𝑊 = (𝐼 mPoly 𝑈)
3 evlsval.v . . . 4 𝑉 = (𝐼 mVar 𝑈)
4 evlsval.u . . . 4 𝑈 = (𝑆s 𝑅)
5 evlsval.t . . . 4 𝑇 = (𝑆s (𝐵m 𝐼))
6 evlsval.b . . . 4 𝐵 = (Base‘𝑆)
7 evlsval.a . . . 4 𝐴 = (algSc‘𝑊)
8 evlsval.x . . . 4 𝑋 = (𝑥𝑅 ↦ ((𝐵m 𝐼) × {𝑥}))
9 evlsval.y . . . 4 𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)))
101, 2, 3, 4, 5, 6, 7, 8, 9evlsval 20795 . . 3 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)))
11 eqid 2798 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
12 simp1 1133 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝐼𝑍)
134subrgcrng 19553 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
14133adant1 1127 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
15 simp2 1134 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ CRing)
16 ovex 7178 . . . . . 6 (𝐵m 𝐼) ∈ V
175pwscrng 19384 . . . . . 6 ((𝑆 ∈ CRing ∧ (𝐵m 𝐼) ∈ V) → 𝑇 ∈ CRing)
1815, 16, 17sylancl 589 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑇 ∈ CRing)
196subrgss 19550 . . . . . . . . 9 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
20193ad2ant3 1132 . . . . . . . 8 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅𝐵)
2120resmptd 5879 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ↾ 𝑅) = (𝑥𝑅 ↦ ((𝐵m 𝐼) × {𝑥})))
2221, 8eqtr4di 2851 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ↾ 𝑅) = 𝑋)
23 crngring 19323 . . . . . . . . 9 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
24233ad2ant2 1131 . . . . . . . 8 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ Ring)
25 eqid 2798 . . . . . . . . 9 (𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) = (𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥}))
265, 6, 25pwsdiagrhm 19583 . . . . . . . 8 ((𝑆 ∈ Ring ∧ (𝐵m 𝐼) ∈ V) → (𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
2724, 16, 26sylancl 589 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
28 simp3 1135 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ (SubRing‘𝑆))
294resrhm 19578 . . . . . . 7 (((𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇) ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ↾ 𝑅) ∈ (𝑈 RingHom 𝑇))
3027, 28, 29syl2anc 587 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ↾ 𝑅) ∈ (𝑈 RingHom 𝑇))
3122, 30eqeltrrd 2891 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑋 ∈ (𝑈 RingHom 𝑇))
326fvexi 6669 . . . . . . . . . . 11 𝐵 ∈ V
33 simpl1 1188 . . . . . . . . . . 11 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → 𝐼𝑍)
34 elmapg 8420 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ 𝐼𝑍) → (𝑔 ∈ (𝐵m 𝐼) ↔ 𝑔:𝐼𝐵))
3532, 33, 34sylancr 590 . . . . . . . . . 10 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵m 𝐼) ↔ 𝑔:𝐼𝐵))
3635biimpa 480 . . . . . . . . 9 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵m 𝐼)) → 𝑔:𝐼𝐵)
37 simplr 768 . . . . . . . . 9 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵m 𝐼)) → 𝑥𝐼)
3836, 37ffvelrnd 6839 . . . . . . . 8 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵m 𝐼)) → (𝑔𝑥) ∈ 𝐵)
3938fmpttd 6866 . . . . . . 7 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)):(𝐵m 𝐼)⟶𝐵)
40 simpl2 1189 . . . . . . . 8 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → 𝑆 ∈ CRing)
415, 6, 11pwselbasb 16773 . . . . . . . 8 ((𝑆 ∈ CRing ∧ (𝐵m 𝐼) ∈ V) → ((𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇) ↔ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)):(𝐵m 𝐼)⟶𝐵))
4240, 16, 41sylancl 589 . . . . . . 7 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → ((𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇) ↔ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)):(𝐵m 𝐼)⟶𝐵))
4339, 42mpbird 260 . . . . . 6 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇))
4443, 9fmptd 6865 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑌:𝐼⟶(Base‘𝑇))
452, 11, 7, 3, 12, 14, 18, 31, 44evlseu 20792 . . . 4 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ∃!𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌))
46 riotacl2 7119 . . . 4 (∃!𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌) → (𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)) ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)})
4745, 46syl 17 . . 3 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)) ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)})
4810, 47eqeltrd 2890 . 2 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)})
49 coeq1 5696 . . . . 5 (𝑚 = 𝑄 → (𝑚𝐴) = (𝑄𝐴))
5049eqeq1d 2800 . . . 4 (𝑚 = 𝑄 → ((𝑚𝐴) = 𝑋 ↔ (𝑄𝐴) = 𝑋))
51 coeq1 5696 . . . . 5 (𝑚 = 𝑄 → (𝑚𝑉) = (𝑄𝑉))
5251eqeq1d 2800 . . . 4 (𝑚 = 𝑄 → ((𝑚𝑉) = 𝑌 ↔ (𝑄𝑉) = 𝑌))
5350, 52anbi12d 633 . . 3 (𝑚 = 𝑄 → (((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌) ↔ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
5453elrab 3630 . 2 (𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)} ↔ (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
5548, 54sylib 221 1 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃!wreu 3108  {crab 3110  Vcvv 3442   ⊆ wss 3883  {csn 4528   ↦ cmpt 5114   × cxp 5521   ↾ cres 5525   ∘ ccom 5527  ⟶wf 6328  ‘cfv 6332  ℩crio 7102  (class class class)co 7145   ↑m cmap 8407  Basecbs 16495   ↾s cress 16496   ↑s cpws 16732  Ringcrg 19311  CRingccrg 19312   RingHom crh 19481  SubRingcsubrg 19545  algSccascl 20563   mVar cmvr 20613   mPoly cmpl 20614   evalSub ces 20779 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-iin 4888  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-isom 6341  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7400  df-ofr 7401  df-om 7574  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-2o 8104  df-oadd 8107  df-er 8290  df-map 8409  df-pm 8410  df-ixp 8463  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-fsupp 8836  df-sup 8908  df-oi 8976  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-2 11706  df-3 11707  df-4 11708  df-5 11709  df-6 11710  df-7 11711  df-8 11712  df-9 11713  df-n0 11904  df-z 11990  df-dec 12107  df-uz 12252  df-fz 12906  df-fzo 13049  df-seq 13385  df-hash 13707  df-struct 16497  df-ndx 16498  df-slot 16499  df-base 16501  df-sets 16502  df-ress 16503  df-plusg 16590  df-mulr 16591  df-sca 16593  df-vsca 16594  df-ip 16595  df-tset 16596  df-ple 16597  df-ds 16599  df-hom 16601  df-cco 16602  df-0g 16727  df-gsum 16728  df-prds 16733  df-pws 16735  df-mre 16869  df-mrc 16870  df-acs 16872  df-mgm 17864  df-sgrp 17913  df-mnd 17924  df-mhm 17968  df-submnd 17969  df-grp 18118  df-minusg 18119  df-sbg 18120  df-mulg 18238  df-subg 18289  df-ghm 18369  df-cntz 18460  df-cmn 18921  df-abl 18922  df-mgp 19254  df-ur 19266  df-srg 19270  df-ring 19313  df-cring 19314  df-rnghom 19484  df-subrg 19547  df-lmod 19650  df-lss 19718  df-lsp 19758  df-assa 20564  df-asp 20565  df-ascl 20566  df-psr 20617  df-mvr 20618  df-mpl 20619  df-evls 20781 This theorem is referenced by:  evlsrhm  20797  evlssca  20798  evlsvar  20799
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