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Theorem evlsval2 21497
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 21496 . . 3 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)))
11 eqid 2736 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
12 simp1 1136 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝐼𝑍)
134subrgcrng 20226 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
14133adant1 1130 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
15 simp2 1137 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ CRing)
16 ovex 7390 . . . . . 6 (𝐵m 𝐼) ∈ V
175pwscrng 20041 . . . . . 6 ((𝑆 ∈ CRing ∧ (𝐵m 𝐼) ∈ V) → 𝑇 ∈ CRing)
1815, 16, 17sylancl 586 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑇 ∈ CRing)
196subrgss 20223 . . . . . . . . 9 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
20193ad2ant3 1135 . . . . . . . 8 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅𝐵)
2120resmptd 5994 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ↾ 𝑅) = (𝑥𝑅 ↦ ((𝐵m 𝐼) × {𝑥})))
2221, 8eqtr4di 2794 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) ↾ 𝑅) = 𝑋)
23 crngring 19976 . . . . . . . . 9 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
24233ad2ant2 1134 . . . . . . . 8 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ Ring)
25 eqid 2736 . . . . . . . . 9 (𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥})) = (𝑥𝐵 ↦ ((𝐵m 𝐼) × {𝑥}))
265, 6, 25pwsdiagrhm 20256 . . . . . . . 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 1138 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ (SubRing‘𝑆))
294resrhm 20251 . . . . . . 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 2839 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑋 ∈ (𝑈 RingHom 𝑇))
326fvexi 6856 . . . . . . . . . . 11 𝐵 ∈ V
33 simpl1 1191 . . . . . . . . . . 11 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → 𝐼𝑍)
34 elmapg 8778 . . . . . . . . . . 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 477 . . . . . . . . 9 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵m 𝐼)) → 𝑔:𝐼𝐵)
37 simplr 767 . . . . . . . . 9 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵m 𝐼)) → 𝑥𝐼)
3836, 37ffvelcdmd 7036 . . . . . . . 8 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵m 𝐼)) → (𝑔𝑥) ∈ 𝐵)
3938fmpttd 7063 . . . . . . 7 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)):(𝐵m 𝐼)⟶𝐵)
40 simpl2 1192 . . . . . . . 8 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → 𝑆 ∈ CRing)
415, 6, 11pwselbasb 17370 . . . . . . . 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 256 . . . . . 6 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇))
4443, 9fmptd 7062 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑌:𝐼⟶(Base‘𝑇))
452, 11, 7, 3, 12, 14, 18, 31, 44evlseu 21493 . . . 4 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ∃!𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌))
46 riotacl2 7330 . . . 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 2838 . 2 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)})
49 coeq1 5813 . . . . 5 (𝑚 = 𝑄 → (𝑚𝐴) = (𝑄𝐴))
5049eqeq1d 2738 . . . 4 (𝑚 = 𝑄 → ((𝑚𝐴) = 𝑋 ↔ (𝑄𝐴) = 𝑋))
51 coeq1 5813 . . . . 5 (𝑚 = 𝑄 → (𝑚𝑉) = (𝑄𝑉))
5251eqeq1d 2738 . . . 4 (𝑚 = 𝑄 → ((𝑚𝑉) = 𝑌 ↔ (𝑄𝑉) = 𝑌))
5350, 52anbi12d 631 . . 3 (𝑚 = 𝑄 → (((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌) ↔ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
5453elrab 3645 . 2 (𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)} ↔ (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
5548, 54sylib 217 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  ∃!wreu 3351  {crab 3407  Vcvv 3445  wss 3910  {csn 4586  cmpt 5188   × cxp 5631  cres 5635  ccom 5637  wf 6492  cfv 6496  crio 7312  (class class class)co 7357  m cmap 8765  Basecbs 17083  s cress 17112  s cpws 17328  Ringcrg 19964  CRingccrg 19965   RingHom crh 20143  SubRingcsubrg 20218  algSccascl 21258   mVar cmvr 21307   mPoly cmpl 21308   evalSub ces 21480
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-srg 19918  df-ring 19966  df-cring 19967  df-rnghom 20146  df-subrg 20220  df-lmod 20324  df-lss 20393  df-lsp 20433  df-assa 21259  df-asp 21260  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-evls 21482
This theorem is referenced by:  evlsrhm  21498  evlssca  21499  evlsvar  21500
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