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Theorem evlsval2 19992
 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 19991 . . 3 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)))
11 eqid 2795 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
12 elex 3455 . . . . . 6 (𝐼𝑍𝐼 ∈ V)
13123ad2ant1 1126 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝐼 ∈ V)
144subrgcrng 19234 . . . . . 6 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
15143adant1 1123 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
16 simp2 1130 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ CRing)
17 ovex 7053 . . . . . 6 (𝐵𝑚 𝐼) ∈ V
185pwscrng 19062 . . . . . 6 ((𝑆 ∈ CRing ∧ (𝐵𝑚 𝐼) ∈ V) → 𝑇 ∈ CRing)
1916, 17, 18sylancl 586 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑇 ∈ CRing)
206subrgss 19231 . . . . . . . . 9 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
21203ad2ant3 1128 . . . . . . . 8 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅𝐵)
2221resmptd 5794 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ↾ 𝑅) = (𝑥𝑅 ↦ ((𝐵𝑚 𝐼) × {𝑥})))
2322, 8syl6eqr 2849 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ↾ 𝑅) = 𝑋)
24 crngring 19003 . . . . . . . . 9 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
25243ad2ant2 1127 . . . . . . . 8 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑆 ∈ Ring)
26 eqid 2795 . . . . . . . . 9 (𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) = (𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥}))
275, 6, 26pwsdiagrhm 19264 . . . . . . . 8 ((𝑆 ∈ Ring ∧ (𝐵𝑚 𝐼) ∈ V) → (𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
2825, 17, 27sylancl 586 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇))
29 simp3 1131 . . . . . . 7 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ (SubRing‘𝑆))
304resrhm 19259 . . . . . . 7 (((𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ∈ (𝑆 RingHom 𝑇) ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ↾ 𝑅) ∈ (𝑈 RingHom 𝑇))
3128, 29, 30syl2anc 584 . . . . . 6 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥𝐵 ↦ ((𝐵𝑚 𝐼) × {𝑥})) ↾ 𝑅) ∈ (𝑈 RingHom 𝑇))
3223, 31eqeltrrd 2884 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑋 ∈ (𝑈 RingHom 𝑇))
336fvexi 6557 . . . . . . . . . . 11 𝐵 ∈ V
34 simpl1 1184 . . . . . . . . . . 11 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → 𝐼𝑍)
35 elmapg 8274 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ 𝐼𝑍) → (𝑔 ∈ (𝐵𝑚 𝐼) ↔ 𝑔:𝐼𝐵))
3633, 34, 35sylancr 587 . . . . . . . . . 10 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵𝑚 𝐼) ↔ 𝑔:𝐼𝐵))
3736biimpa 477 . . . . . . . . 9 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵𝑚 𝐼)) → 𝑔:𝐼𝐵)
38 simplr 765 . . . . . . . . 9 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵𝑚 𝐼)) → 𝑥𝐼)
3937, 38ffvelrnd 6722 . . . . . . . 8 ((((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) ∧ 𝑔 ∈ (𝐵𝑚 𝐼)) → (𝑔𝑥) ∈ 𝐵)
4039fmpttd 6747 . . . . . . 7 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)):(𝐵𝑚 𝐼)⟶𝐵)
41 simpl2 1185 . . . . . . . 8 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → 𝑆 ∈ CRing)
425, 6, 11pwselbasb 16595 . . . . . . . 8 ((𝑆 ∈ CRing ∧ (𝐵𝑚 𝐼) ∈ V) → ((𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇) ↔ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)):(𝐵𝑚 𝐼)⟶𝐵))
4341, 17, 42sylancl 586 . . . . . . 7 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → ((𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇) ↔ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)):(𝐵𝑚 𝐼)⟶𝐵))
4440, 43mpbird 258 . . . . . 6 (((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ 𝑥𝐼) → (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)) ∈ (Base‘𝑇))
4544, 9fmptd 6746 . . . . 5 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑌:𝐼⟶(Base‘𝑇))
462, 11, 7, 3, 13, 15, 19, 32, 45evlseu 19988 . . . 4 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ∃!𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌))
47 riotacl2 6995 . . . 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)})
4846, 47syl 17 . . 3 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑚 ∈ (𝑊 RingHom 𝑇)((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)) ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)})
4910, 48eqeltrd 2883 . 2 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)})
50 coeq1 5619 . . . . 5 (𝑚 = 𝑄 → (𝑚𝐴) = (𝑄𝐴))
5150eqeq1d 2797 . . . 4 (𝑚 = 𝑄 → ((𝑚𝐴) = 𝑋 ↔ (𝑄𝐴) = 𝑋))
52 coeq1 5619 . . . . 5 (𝑚 = 𝑄 → (𝑚𝑉) = (𝑄𝑉))
5352eqeq1d 2797 . . . 4 (𝑚 = 𝑄 → ((𝑚𝑉) = 𝑌 ↔ (𝑄𝑉) = 𝑌))
5451, 53anbi12d 630 . . 3 (𝑚 = 𝑄 → (((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌) ↔ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
5554elrab 3619 . 2 (𝑄 ∈ {𝑚 ∈ (𝑊 RingHom 𝑇) ∣ ((𝑚𝐴) = 𝑋 ∧ (𝑚𝑉) = 𝑌)} ↔ (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
5649, 55sylib 219 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 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ∃!wreu 3107  {crab 3109  Vcvv 3437   ⊆ wss 3863  {csn 4476   ↦ cmpt 5045   × cxp 5446   ↾ cres 5450   ∘ ccom 5452  ⟶wf 6226  ‘cfv 6230  ℩crio 6981  (class class class)co 7021   ↑𝑚 cmap 8261  Basecbs 16317   ↾s cress 16318   ↑s cpws 16554  Ringcrg 18992  CRingccrg 18993   RingHom crh 19159  SubRingcsubrg 19226  algSccascl 19778   mVar cmvr 19825   mPoly cmpl 19826   evalSub ces 19976 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-int 4787  df-iun 4831  df-iin 4832  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-se 5408  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-isom 6239  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-of 7272  df-ofr 7273  df-om 7442  df-1st 7550  df-2nd 7551  df-supp 7687  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-1o 7958  df-2o 7959  df-oadd 7962  df-er 8144  df-map 8263  df-pm 8264  df-ixp 8316  df-en 8363  df-dom 8364  df-sdom 8365  df-fin 8366  df-fsupp 8685  df-sup 8757  df-oi 8825  df-card 9219  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-nn 11492  df-2 11553  df-3 11554  df-4 11555  df-5 11556  df-6 11557  df-7 11558  df-8 11559  df-9 11560  df-n0 11751  df-z 11835  df-dec 11953  df-uz 12099  df-fz 12748  df-fzo 12889  df-seq 13225  df-hash 13546  df-struct 16319  df-ndx 16320  df-slot 16321  df-base 16323  df-sets 16324  df-ress 16325  df-plusg 16412  df-mulr 16413  df-sca 16415  df-vsca 16416  df-ip 16417  df-tset 16418  df-ple 16419  df-ds 16421  df-hom 16423  df-cco 16424  df-0g 16549  df-gsum 16550  df-prds 16555  df-pws 16557  df-mre 16691  df-mrc 16692  df-acs 16694  df-mgm 17686  df-sgrp 17728  df-mnd 17739  df-mhm 17779  df-submnd 17780  df-grp 17869  df-minusg 17870  df-sbg 17871  df-mulg 17987  df-subg 18035  df-ghm 18102  df-cntz 18193  df-cmn 18640  df-abl 18641  df-mgp 18935  df-ur 18947  df-srg 18951  df-ring 18994  df-cring 18995  df-rnghom 19162  df-subrg 19228  df-lmod 19331  df-lss 19399  df-lsp 19439  df-assa 19779  df-asp 19780  df-ascl 19781  df-psr 19829  df-mvr 19830  df-mpl 19831  df-evls 19978 This theorem is referenced by:  evlsrhm  19993  evlssca  19994  evlsvar  19995
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