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Theorem idpm2idmp 20520
 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 20412 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
4 pm2mpval.b . . . . 5 𝐵 = (Base‘𝐶)
5 eqid 2626 . . . . 5 (1r𝐶) = (1r𝐶)
64, 5ringidcl 18484 . . . 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 20515 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (1r𝐶) ∈ 𝐵) → (𝑇‘(1r𝐶)) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (((1r𝐶) decompPMat 𝑘) (𝑘 𝑋)))))
157, 14mpd3an3 1422 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r𝐶)) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (((1r𝐶) decompPMat 𝑘) (𝑘 𝑋)))))
16 eqid 2626 . . . . . . 7 (0g𝐴) = (0g𝐴)
17 eqid 2626 . . . . . . 7 (1r𝐴) = (1r𝐴)
181, 2, 5, 11, 16, 17decpmatid 20489 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑘 ∈ ℕ0) → ((1r𝐶) decompPMat 𝑘) = if(𝑘 = 0, (1r𝐴), (0g𝐴)))
19183expa 1262 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r𝐶) decompPMat 𝑘) = if(𝑘 = 0, (1r𝐴), (0g𝐴)))
2019oveq1d 6620 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (((1r𝐶) decompPMat 𝑘) (𝑘 𝑋)) = (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋)))
2120mpteq2dva 4709 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑘 ∈ ℕ0 ↦ (((1r𝐶) decompPMat 𝑘) (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋))))
2221oveq2d 6621 . 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 6691 . . . . . 6 (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋)) = if(𝑘 = 0, ((1r𝐴) (𝑘 𝑋)), ((0g𝐴) (𝑘 𝑋)))
2411matring 20163 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
2512ply1sca 19537 . . . . . . . . . . . 12 (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
2624, 25syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 = (Scalar‘𝑄))
2726adantr 481 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → 𝐴 = (Scalar‘𝑄))
2827fveq2d 6154 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (1r𝐴) = (1r‘(Scalar‘𝑄)))
2928oveq1d 6620 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r𝐴) (𝑘 𝑋)) = ((1r‘(Scalar‘𝑄)) (𝑘 𝑋)))
3012ply1lmod 19536 . . . . . . . . . . 11 (𝐴 ∈ Ring → 𝑄 ∈ LMod)
3124, 30syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
3231adantr 481 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → 𝑄 ∈ LMod)
33 eqid 2626 . . . . . . . . . . 11 (mulGrp‘𝑄) = (mulGrp‘𝑄)
34 eqid 2626 . . . . . . . . . . 11 (Base‘𝑄) = (Base‘𝑄)
3512, 10, 33, 9, 34ply1moncl 19555 . . . . . . . . . 10 ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ0) → (𝑘 𝑋) ∈ (Base‘𝑄))
3624, 35sylan 488 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (𝑘 𝑋) ∈ (Base‘𝑄))
37 eqid 2626 . . . . . . . . . 10 (Scalar‘𝑄) = (Scalar‘𝑄)
38 eqid 2626 . . . . . . . . . 10 (1r‘(Scalar‘𝑄)) = (1r‘(Scalar‘𝑄))
3934, 37, 8, 38lmodvs1 18807 . . . . . . . . 9 ((𝑄 ∈ LMod ∧ (𝑘 𝑋) ∈ (Base‘𝑄)) → ((1r‘(Scalar‘𝑄)) (𝑘 𝑋)) = (𝑘 𝑋))
4032, 36, 39syl2anc 692 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r‘(Scalar‘𝑄)) (𝑘 𝑋)) = (𝑘 𝑋))
4129, 40eqtrd 2660 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r𝐴) (𝑘 𝑋)) = (𝑘 𝑋))
4227fveq2d 6154 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (0g𝐴) = (0g‘(Scalar‘𝑄)))
4342oveq1d 6620 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((0g𝐴) (𝑘 𝑋)) = ((0g‘(Scalar‘𝑄)) (𝑘 𝑋)))
44 eqid 2626 . . . . . . . . . 10 (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
45 eqid 2626 . . . . . . . . . 10 (0g𝑄) = (0g𝑄)
4634, 37, 8, 44, 45lmod0vs 18812 . . . . . . . . 9 ((𝑄 ∈ LMod ∧ (𝑘 𝑋) ∈ (Base‘𝑄)) → ((0g‘(Scalar‘𝑄)) (𝑘 𝑋)) = (0g𝑄))
4732, 36, 46syl2anc 692 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((0g‘(Scalar‘𝑄)) (𝑘 𝑋)) = (0g𝑄))
4843, 47eqtrd 2660 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((0g𝐴) (𝑘 𝑋)) = (0g𝑄))
4941, 48ifeq12d 4083 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → if(𝑘 = 0, ((1r𝐴) (𝑘 𝑋)), ((0g𝐴) (𝑘 𝑋))) = if(𝑘 = 0, (𝑘 𝑋), (0g𝑄)))
5023, 49syl5eq 2672 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋)) = if(𝑘 = 0, (𝑘 𝑋), (0g𝑄)))
5150mpteq2dva 4709 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 𝑋), (0g𝑄))))
5251oveq2d 6621 . . 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)))))
5312ply1ring 19532 . . . . 5 (𝐴 ∈ Ring → 𝑄 ∈ Ring)
54 ringmnd 18472 . . . . 5 (𝑄 ∈ Ring → 𝑄 ∈ Mnd)
5524, 53, 543syl 18 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Mnd)
56 nn0ex 11243 . . . . 5 0 ∈ V
5756a1i 11 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ℕ0 ∈ V)
58 0nn0 11252 . . . . 5 0 ∈ ℕ0
5958a1i 11 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ ℕ0)
60 eqid 2626 . . . 4 (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 𝑋), (0g𝑄))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 𝑋), (0g𝑄)))
6136ralrimiva 2965 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑘 ∈ ℕ0 (𝑘 𝑋) ∈ (Base‘𝑄))
6245, 55, 57, 59, 60, 61gsummpt1n0 18280 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 𝑋), (0g𝑄)))) = 0 / 𝑘(𝑘 𝑋))
63 c0ex 9979 . . . . 5 0 ∈ V
64 csbov1g 6644 . . . . 5 (0 ∈ V → 0 / 𝑘(𝑘 𝑋) = (0 / 𝑘𝑘 𝑋))
6563, 64mp1i 13 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 / 𝑘(𝑘 𝑋) = (0 / 𝑘𝑘 𝑋))
66 csbvarg 3980 . . . . . 6 (0 ∈ V → 0 / 𝑘𝑘 = 0)
6763, 66mp1i 13 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 / 𝑘𝑘 = 0)
6867oveq1d 6620 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0 / 𝑘𝑘 𝑋) = (0 𝑋))
6912, 10, 33, 9ply1idvr1 19577 . . . . 5 (𝐴 ∈ Ring → (0 𝑋) = (1r𝑄))
7024, 69syl 17 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0 𝑋) = (1r𝑄))
7165, 68, 703eqtrd 2664 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 / 𝑘(𝑘 𝑋) = (1r𝑄))
7252, 62, 713eqtrd 2664 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r𝐴), (0g𝐴)) (𝑘 𝑋)))) = (1r𝑄))
7315, 22, 723eqtrd 2664 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r𝐶)) = (1r𝑄))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1992  Vcvv 3191  ⦋csb 3519  ifcif 4063   ↦ cmpt 4678  ‘cfv 5850  (class class class)co 6605  Fincfn 7900  0cc0 9881  ℕ0cn0 11237  Basecbs 15776  Scalarcsca 15860   ·𝑠 cvsca 15861  0gc0g 16016   Σg cgsu 16017  Mndcmnd 17210  .gcmg 17456  mulGrpcmgp 18405  1rcur 18417  Ringcrg 18463  LModclmod 18779  var1cv1 19460  Poly1cpl1 19461   Mat cmat 20127   decompPMat cdecpmat 20481   pMatToMatPoly cpm2mp 20511 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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-ofr 6852  df-om 7014  df-1st 7116  df-2nd 7117  df-supp 7242  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-fsupp 8221  df-sup 8293  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-fz 12266  df-fzo 12404  df-seq 12739  df-hash 13055  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-mulr 15871  df-sca 15873  df-vsca 15874  df-ip 15875  df-tset 15876  df-ple 15877  df-ds 15880  df-hom 15882  df-cco 15883  df-0g 16018  df-gsum 16019  df-prds 16024  df-pws 16026  df-mre 16162  df-mrc 16163  df-acs 16165  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-submnd 17252  df-grp 17341  df-minusg 17342  df-sbg 17343  df-mulg 17457  df-subg 17507  df-ghm 17574  df-cntz 17666  df-cmn 18111  df-abl 18112  df-mgp 18406  df-ur 18418  df-ring 18465  df-subrg 18694  df-lmod 18781  df-lss 18847  df-sra 19086  df-rgmod 19087  df-ascl 19228  df-psr 19270  df-mvr 19271  df-mpl 19272  df-opsr 19274  df-psr1 19464  df-vr1 19465  df-ply1 19466  df-coe1 19467  df-dsmm 19990  df-frlm 20005  df-mamu 20104  df-mat 20128  df-decpmat 20482  df-pm2mp 20512 This theorem is referenced by:  pm2mpmhm  20539
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