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Theorem idpm2idmp 21412
Description: The transformation of the identity polynomial matrix into polynomials over matrices results in the identity of the polynomials over matrices. (Contributed by AV, 18-Oct-2019.) (Revised by AV, 5-Dec-2019.)
Hypotheses
Ref Expression
pm2mpval.p 𝑃 = (Poly1𝑅)
pm2mpval.c 𝐶 = (𝑁 Mat 𝑃)
pm2mpval.b 𝐵 = (Base‘𝐶)
pm2mpval.m = ( ·𝑠𝑄)
pm2mpval.e = (.g‘(mulGrp‘𝑄))
pm2mpval.x 𝑋 = (var1𝐴)
pm2mpval.a 𝐴 = (𝑁 Mat 𝑅)
pm2mpval.q 𝑄 = (Poly1𝐴)
pm2mpval.t 𝑇 = (𝑁 pMatToMatPoly 𝑅)
Assertion
Ref Expression
idpm2idmp ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r𝐶)) = (1r𝑄))

Proof of Theorem idpm2idmp
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 pm2mpval.p . . . . 5 𝑃 = (Poly1𝑅)
2 pm2mpval.c . . . . 5 𝐶 = (𝑁 Mat 𝑃)
31, 2pmatring 21304 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
4 pm2mpval.b . . . . 5 𝐵 = (Base‘𝐶)
5 eqid 2824 . . . . 5 (1r𝐶) = (1r𝐶)
64, 5ringidcl 19324 . . . 4 (𝐶 ∈ Ring → (1r𝐶) ∈ 𝐵)
73, 6syl 17 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐶) ∈ 𝐵)
8 pm2mpval.m . . . 4 = ( ·𝑠𝑄)
9 pm2mpval.e . . . 4 = (.g‘(mulGrp‘𝑄))
10 pm2mpval.x . . . 4 𝑋 = (var1𝐴)
11 pm2mpval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
12 pm2mpval.q . . . 4 𝑄 = (Poly1𝐴)
13 pm2mpval.t . . . 4 𝑇 = (𝑁 pMatToMatPoly 𝑅)
141, 2, 4, 8, 9, 10, 11, 12, 13pm2mpfval 21407 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (1r𝐶) ∈ 𝐵) → (𝑇‘(1r𝐶)) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (((1r𝐶) decompPMat 𝑘) (𝑘 𝑋)))))
157, 14mpd3an3 1459 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r𝐶)) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (((1r𝐶) decompPMat 𝑘) (𝑘 𝑋)))))
16 eqid 2824 . . . . . . 7 (0g𝐴) = (0g𝐴)
17 eqid 2824 . . . . . . 7 (1r𝐴) = (1r𝐴)
181, 2, 5, 11, 16, 17decpmatid 21381 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑘 ∈ ℕ0) → ((1r𝐶) decompPMat 𝑘) = if(𝑘 = 0, (1r𝐴), (0g𝐴)))
19183expa 1115 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r𝐶) decompPMat 𝑘) = if(𝑘 = 0, (1r𝐴), (0g𝐴)))
2019oveq1d 7165 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (((1r𝐶) decompPMat 𝑘) (𝑘 𝑋)) = (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋)))
2120mpteq2dva 5148 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑘 ∈ ℕ0 ↦ (((1r𝐶) decompPMat 𝑘) (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋))))
2221oveq2d 7166 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (((1r𝐶) decompPMat 𝑘) (𝑘 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋)))))
23 ovif 7245 . . . . . 6 (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋)) = if(𝑘 = 0, ((1r𝐴) (𝑘 𝑋)), ((0g𝐴) (𝑘 𝑋)))
2411matring 21055 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
2512ply1sca 20424 . . . . . . . . . . . 12 (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
2624, 25syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 = (Scalar‘𝑄))
2726adantr 484 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → 𝐴 = (Scalar‘𝑄))
2827fveq2d 6666 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (1r𝐴) = (1r‘(Scalar‘𝑄)))
2928oveq1d 7165 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r𝐴) (𝑘 𝑋)) = ((1r‘(Scalar‘𝑄)) (𝑘 𝑋)))
3012ply1lmod 20423 . . . . . . . . . 10 (𝐴 ∈ Ring → 𝑄 ∈ LMod)
3124, 30syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
32 eqid 2824 . . . . . . . . . . 11 (mulGrp‘𝑄) = (mulGrp‘𝑄)
33 eqid 2824 . . . . . . . . . . 11 (Base‘𝑄) = (Base‘𝑄)
3412, 10, 32, 9, 33ply1moncl 20442 . . . . . . . . . 10 ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ0) → (𝑘 𝑋) ∈ (Base‘𝑄))
3524, 34sylan 583 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (𝑘 𝑋) ∈ (Base‘𝑄))
36 eqid 2824 . . . . . . . . . 10 (Scalar‘𝑄) = (Scalar‘𝑄)
37 eqid 2824 . . . . . . . . . 10 (1r‘(Scalar‘𝑄)) = (1r‘(Scalar‘𝑄))
3833, 36, 8, 37lmodvs1 19665 . . . . . . . . 9 ((𝑄 ∈ LMod ∧ (𝑘 𝑋) ∈ (Base‘𝑄)) → ((1r‘(Scalar‘𝑄)) (𝑘 𝑋)) = (𝑘 𝑋))
3931, 35, 38syl2an2r 684 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r‘(Scalar‘𝑄)) (𝑘 𝑋)) = (𝑘 𝑋))
4029, 39eqtrd 2859 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r𝐴) (𝑘 𝑋)) = (𝑘 𝑋))
4127fveq2d 6666 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (0g𝐴) = (0g‘(Scalar‘𝑄)))
4241oveq1d 7165 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((0g𝐴) (𝑘 𝑋)) = ((0g‘(Scalar‘𝑄)) (𝑘 𝑋)))
43 eqid 2824 . . . . . . . . . 10 (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
44 eqid 2824 . . . . . . . . . 10 (0g𝑄) = (0g𝑄)
4533, 36, 8, 43, 44lmod0vs 19670 . . . . . . . . 9 ((𝑄 ∈ LMod ∧ (𝑘 𝑋) ∈ (Base‘𝑄)) → ((0g‘(Scalar‘𝑄)) (𝑘 𝑋)) = (0g𝑄))
4631, 35, 45syl2an2r 684 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((0g‘(Scalar‘𝑄)) (𝑘 𝑋)) = (0g𝑄))
4742, 46eqtrd 2859 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((0g𝐴) (𝑘 𝑋)) = (0g𝑄))
4840, 47ifeq12d 4471 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → if(𝑘 = 0, ((1r𝐴) (𝑘 𝑋)), ((0g𝐴) (𝑘 𝑋))) = if(𝑘 = 0, (𝑘 𝑋), (0g𝑄)))
4923, 48syl5eq 2871 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋)) = if(𝑘 = 0, (𝑘 𝑋), (0g𝑄)))
5049mpteq2dva 5148 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 𝑋), (0g𝑄))))
5150oveq2d 7166 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 𝑋), (0g𝑄)))))
5212ply1ring 20419 . . . . 5 (𝐴 ∈ Ring → 𝑄 ∈ Ring)
53 ringmnd 19310 . . . . 5 (𝑄 ∈ Ring → 𝑄 ∈ Mnd)
5424, 52, 533syl 18 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Mnd)
55 nn0ex 11903 . . . . 5 0 ∈ V
5655a1i 11 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ℕ0 ∈ V)
57 0nn0 11912 . . . . 5 0 ∈ ℕ0
5857a1i 11 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ ℕ0)
59 eqid 2824 . . . 4 (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 𝑋), (0g𝑄))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 𝑋), (0g𝑄)))
6035ralrimiva 3177 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑘 ∈ ℕ0 (𝑘 𝑋) ∈ (Base‘𝑄))
6144, 54, 56, 58, 59, 60gsummpt1n0 19088 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 𝑋), (0g𝑄)))) = 0 / 𝑘(𝑘 𝑋))
62 c0ex 10634 . . . . 5 0 ∈ V
63 csbov1g 7195 . . . . 5 (0 ∈ V → 0 / 𝑘(𝑘 𝑋) = (0 / 𝑘𝑘 𝑋))
6462, 63mp1i 13 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 / 𝑘(𝑘 𝑋) = (0 / 𝑘𝑘 𝑋))
65 csbvarg 4367 . . . . . 6 (0 ∈ V → 0 / 𝑘𝑘 = 0)
6662, 65mp1i 13 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 / 𝑘𝑘 = 0)
6766oveq1d 7165 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0 / 𝑘𝑘 𝑋) = (0 𝑋))
6812, 10, 32, 9ply1idvr1 20464 . . . . 5 (𝐴 ∈ Ring → (0 𝑋) = (1r𝑄))
6924, 68syl 17 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0 𝑋) = (1r𝑄))
7064, 67, 693eqtrd 2863 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 / 𝑘(𝑘 𝑋) = (1r𝑄))
7151, 61, 703eqtrd 2863 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋)))) = (1r𝑄))
7215, 22, 713eqtrd 2863 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r𝐶)) = (1r𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3481  csb 3867  ifcif 4451  cmpt 5133  cfv 6344  (class class class)co 7150  Fincfn 8506  0cc0 10536  0cn0 11897  Basecbs 16486  Scalarcsca 16571   ·𝑠 cvsca 16572  0gc0g 16716   Σg cgsu 16717  Mndcmnd 17914  .gcmg 18227  mulGrpcmgp 19242  1rcur 19254  Ringcrg 19300  LModclmod 19637  var1cv1 20347  Poly1cpl1 20348   Mat cmat 21019   decompPMat cdecpmat 21373   pMatToMatPoly cpm2mp 21403
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-pss 3939  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-tp 4556  df-op 4558  df-ot 4560  df-uni 4826  df-int 4864  df-iun 4908  df-iin 4909  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-isom 6353  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7404  df-ofr 7405  df-om 7576  df-1st 7685  df-2nd 7686  df-supp 7828  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-1o 8099  df-2o 8100  df-oadd 8103  df-er 8286  df-map 8405  df-pm 8406  df-ixp 8459  df-en 8507  df-dom 8508  df-sdom 8509  df-fin 8510  df-fsupp 8832  df-sup 8904  df-oi 8972  df-card 9366  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11700  df-3 11701  df-4 11702  df-5 11703  df-6 11704  df-7 11705  df-8 11706  df-9 11707  df-n0 11898  df-z 11982  df-dec 12099  df-uz 12244  df-fz 12898  df-fzo 13041  df-seq 13377  df-hash 13699  df-struct 16488  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-mulr 16582  df-sca 16584  df-vsca 16585  df-ip 16586  df-tset 16587  df-ple 16588  df-ds 16590  df-hom 16592  df-cco 16593  df-0g 16718  df-gsum 16719  df-prds 16724  df-pws 16726  df-mre 16860  df-mrc 16861  df-acs 16863  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-mhm 17959  df-submnd 17960  df-grp 18109  df-minusg 18110  df-sbg 18111  df-mulg 18228  df-subg 18279  df-ghm 18359  df-cntz 18450  df-cmn 18911  df-abl 18912  df-mgp 19243  df-ur 19255  df-ring 19302  df-subrg 19536  df-lmod 19639  df-lss 19707  df-sra 19947  df-rgmod 19948  df-ascl 20090  df-psr 20139  df-mvr 20140  df-mpl 20141  df-opsr 20143  df-psr1 20351  df-vr1 20352  df-ply1 20353  df-coe1 20354  df-dsmm 20879  df-frlm 20894  df-mamu 20998  df-mat 21020  df-decpmat 21374  df-pm2mp 21404
This theorem is referenced by:  pm2mpmhm  21431
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