MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  evlsval2 Structured version   Visualization version   GIF version

Theorem evlsval2 19460
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 19459 . . 3 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)))
11 eqid 2621 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
12 elex 3202 . . . . . 6 (𝐼𝑍𝐼 ∈ V)
13123ad2ant1 1080 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝐼 ∈ V)
144subrgcrng 18724 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
15143adant1 1077 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
16 simp2 1060 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ CRing)
17 ovex 6643 . . . . . 6 (𝐵𝑚 𝐼) ∈ V
185pwscrng 18557 . . . . . 6 ((𝑆 ∈ CRing ∧ (𝐵𝑚 𝐼) ∈ V) → 𝑇 ∈ CRing)
1916, 17, 18sylancl 693 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑇 ∈ CRing)
206subrgss 18721 . . . . . . . . 9 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
21203ad2ant3 1082 . . . . . . . 8 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅𝐵)
2221resmptd 5421 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ↾ 𝑅) = (𝑥𝑅 ↦ ((𝐵𝑚 𝐼) × {𝑥})))
2322, 8syl6eqr 2673 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ↾ 𝑅) = 𝑋)
24 crngring 18498 . . . . . . . . 9 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
25243ad2ant2 1081 . . . . . . . 8 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ Ring)
26 eqid 2621 . . . . . . . . 9 (𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) = (𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥}))
275, 6, 26pwsdiagrhm 18753 . . . . . . . 8 ((𝑆 ∈ Ring ∧ (𝐵𝑚 𝐼) ∈ V) → (𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
2825, 17, 27sylancl 693 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
29 simp3 1061 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ (SubRing‘𝑆))
304resrhm 18749 . . . . . . 7 (((𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇) ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ↾ 𝑅) ∈ (𝑈 RingHom 𝑇))
3128, 29, 30syl2anc 692 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ↾ 𝑅) ∈ (𝑈 RingHom 𝑇))
3223, 31eqeltrrd 2699 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑋 ∈ (𝑈 RingHom 𝑇))
33 fvex 6168 . . . . . . . . . . . 12 (Base‘𝑆) ∈ V
346, 33eqeltri 2694 . . . . . . . . . . 11 𝐵 ∈ V
35 simpl1 1062 . . . . . . . . . . 11 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → 𝐼𝑍)
36 elmapg 7830 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ 𝐼𝑍) → (𝑔 ∈ (𝐵𝑚 𝐼) ↔ 𝑔:𝐼𝐵))
3734, 35, 36sylancr 694 . . . . . . . . . 10 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵𝑚 𝐼) ↔ 𝑔:𝐼𝐵))
3837biimpa 501 . . . . . . . . 9 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵𝑚 𝐼)) → 𝑔:𝐼𝐵)
39 simplr 791 . . . . . . . . 9 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵𝑚 𝐼)) → 𝑥𝐼)
4038, 39ffvelrnd 6326 . . . . . . . 8 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵𝑚 𝐼)) → (𝑔𝑥) ∈ 𝐵)
41 eqid 2621 . . . . . . . 8 (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)) = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥))
4240, 41fmptd 6351 . . . . . . 7 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)):(𝐵𝑚 𝐼)⟶𝐵)
43 simpl2 1063 . . . . . . . 8 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → 𝑆 ∈ CRing)
445, 6, 11pwselbasb 16088 . . . . . . . 8 ((𝑆 ∈ CRing ∧ (𝐵𝑚 𝐼) ∈ V) → ((𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇) ↔ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)):(𝐵𝑚 𝐼)⟶𝐵))
4543, 17, 44sylancl 693 . . . . . . 7 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → ((𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇) ↔ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)):(𝐵𝑚 𝐼)⟶𝐵))
4642, 45mpbird 247 . . . . . 6 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇))
4746, 9fmptd 6351 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑌:𝐼⟶(Base‘𝑇))
482, 11, 7, 3, 13, 15, 19, 32, 47evlseu 19456 . . . 4 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ∃!𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌))
49 riotacl2 6589 . . . 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)})
5048, 49syl 17 . . 3 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)) ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)})
5110, 50eqeltrd 2698 . 2 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)})
52 coeq1 5249 . . . . 5 (𝑚 = 𝑄 → (𝑚𝐴) = (𝑄𝐴))
5352eqeq1d 2623 . . . 4 (𝑚 = 𝑄 → ((𝑚𝐴) = 𝑋 ↔ (𝑄𝐴) = 𝑋))
54 coeq1 5249 . . . . 5 (𝑚 = 𝑄 → (𝑚𝑉) = (𝑄𝑉))
5554eqeq1d 2623 . . . 4 (𝑚 = 𝑄 → ((𝑚𝑉) = 𝑌 ↔ (𝑄𝑉) = 𝑌))
5653, 55anbi12d 746 . . 3 (𝑚 = 𝑄 → (((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌) ↔ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
5756elrab 3351 . 2 (𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)} ↔ (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
5851, 57sylib 208 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  ∃!wreu 2910  {crab 2912  Vcvv 3190  wss 3560  {csn 4155  cmpt 4683   × cxp 5082  cres 5086  ccom 5088  wf 5853  cfv 5857  crio 6575  (class class class)co 6615  𝑚 cmap 7817  Basecbs 15800  s cress 15801  s cpws 16047  Ringcrg 18487  CRingccrg 18488   RingHom crh 18652  SubRingcsubrg 18716  algSccascl 19251   mVar cmvr 19292   mPoly cmpl 19293   evalSub ces 19444
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
This theorem is referenced by:  evlsrhm  19461  evlssca  19462  evlsvar  19463
  Copyright terms: Public domain W3C validator