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Theorem evlsval2 22129
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 22128 . . 3 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)))
11 eqid 2735 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
12 simp1 1135 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝐼𝑍)
134subrgcrng 20592 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
14133adant1 1129 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
15 simp2 1136 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ CRing)
16 ovex 7464 . . . . . 6 (𝐵m 𝐼) ∈ V
175pwscrng 20340 . . . . . 6 ((𝑆 ∈ CRing ∧ (𝐵m 𝐼) ∈ V) → 𝑇 ∈ CRing)
1815, 16, 17sylancl 586 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑇 ∈ CRing)
196subrgss 20589 . . . . . . . . 9 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
20193ad2ant3 1134 . . . . . . . 8 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅𝐵)
2120resmptd 6060 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ↾ 𝑅) = (𝑥𝑅 ↦ ((𝐵m 𝐼) × {𝑥})))
2221, 8eqtr4di 2793 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ↾ 𝑅) = 𝑋)
23 crngring 20263 . . . . . . . . 9 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
24233ad2ant2 1133 . . . . . . . 8 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ Ring)
25 eqid 2735 . . . . . . . . 9 (𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) = (𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥}))
265, 6, 25pwsdiagrhm 20624 . . . . . . . 8 ((𝑆 ∈ Ring ∧ (𝐵m 𝐼) ∈ V) → (𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
2724, 16, 26sylancl 586 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
28 simp3 1137 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ (SubRing‘𝑆))
294resrhm 20618 . . . . . . 7 (((𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇) ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ↾ 𝑅) ∈ (𝑈 RingHom 𝑇))
3027, 28, 29syl2anc 584 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ↾ 𝑅) ∈ (𝑈 RingHom 𝑇))
3122, 30eqeltrrd 2840 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑋 ∈ (𝑈 RingHom 𝑇))
326fvexi 6921 . . . . . . . . . . 11 𝐵 ∈ V
33 simpl1 1190 . . . . . . . . . . 11 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → 𝐼𝑍)
34 elmapg 8878 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ 𝐼𝑍) → (𝑔 ∈ (𝐵m 𝐼) ↔ 𝑔:𝐼𝐵))
3532, 33, 34sylancr 587 . . . . . . . . . 10 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵m 𝐼) ↔ 𝑔:𝐼𝐵))
3635biimpa 476 . . . . . . . . 9 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵m 𝐼)) → 𝑔:𝐼𝐵)
37 simplr 769 . . . . . . . . 9 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵m 𝐼)) → 𝑥𝐼)
3836, 37ffvelcdmd 7105 . . . . . . . 8 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵m 𝐼)) → (𝑔𝑥) ∈ 𝐵)
3938fmpttd 7135 . . . . . . 7 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)):(𝐵m 𝐼)⟶𝐵)
40 simpl2 1191 . . . . . . . 8 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → 𝑆 ∈ CRing)
415, 6, 11pwselbasb 17535 . . . . . . . 8 ((𝑆 ∈ CRing ∧ (𝐵m 𝐼) ∈ V) → ((𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇) ↔ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)):(𝐵m 𝐼)⟶𝐵))
4240, 16, 41sylancl 586 . . . . . . 7 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → ((𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇) ↔ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)):(𝐵m 𝐼)⟶𝐵))
4339, 42mpbird 257 . . . . . 6 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇))
4443, 9fmptd 7134 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑌:𝐼⟶(Base‘𝑇))
452, 11, 7, 3, 12, 14, 18, 31, 44evlseu 22125 . . . 4 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ∃!𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌))
46 riotacl2 7404 . . . 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 2839 . 2 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)})
49 coeq1 5871 . . . . 5 (𝑚 = 𝑄 → (𝑚𝐴) = (𝑄𝐴))
5049eqeq1d 2737 . . . 4 (𝑚 = 𝑄 → ((𝑚𝐴) = 𝑋 ↔ (𝑄𝐴) = 𝑋))
51 coeq1 5871 . . . . 5 (𝑚 = 𝑄 → (𝑚𝑉) = (𝑄𝑉))
5251eqeq1d 2737 . . . 4 (𝑚 = 𝑄 → ((𝑚𝑉) = 𝑌 ↔ (𝑄𝑉) = 𝑌))
5350, 52anbi12d 632 . . 3 (𝑚 = 𝑄 → (((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌) ↔ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
5453elrab 3695 . 2 (𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)} ↔ (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
5548, 54sylib 218 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  ∃!wreu 3376  {crab 3433  Vcvv 3478  wss 3963  {csn 4631  cmpt 5231   × cxp 5687  cres 5691  ccom 5693  wf 6559  cfv 6563  crio 7387  (class class class)co 7431  m cmap 8865  Basecbs 17245  s cress 17274  s cpws 17493  Ringcrg 20251  CRingccrg 20252   RingHom crh 20486  SubRingcsubrg 20586  algSccascl 21890   mVar cmvr 21943   mPoly cmpl 21944   evalSub ces 22114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-srg 20205  df-ring 20253  df-cring 20254  df-rhm 20489  df-subrng 20563  df-subrg 20587  df-lmod 20877  df-lss 20948  df-lsp 20988  df-assa 21891  df-asp 21892  df-ascl 21893  df-psr 21947  df-mvr 21948  df-mpl 21949  df-evls 22116
This theorem is referenced by:  evlsrhm  22130  evlssca  22131  evlsvar  22132
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