Theorem cayleyhamilton 22775

Description: The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", see theorem 7.8 in [Roman] p. 170 (without proof!), or theorem 3.1 in [Lang] p. 561. In other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. This is Metamath 100 proof #49. (Contributed by Alexander van der Vekens, 25-Nov-2019.)

Hypotheses

Ref	Expression
cayleyhamilton.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cayleyhamilton.b	⊢ 𝐵 = (Base‘𝐴)
cayleyhamilton.0	⊢ 0 = (0g‘𝐴)
cayleyhamilton.c	⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
cayleyhamilton.k	⊢ 𝐾 = (coe1‘(𝐶‘𝑀))
cayleyhamilton.m	⊢ ∗ = ( ·𝑠 ‘𝐴)
cayleyhamilton.e	⊢ ↑ = (.g‘(mulGrp‘𝐴))

Assertion

Ref	Expression
cayleyhamilton	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )

Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝐶,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   ∗ ,𝑛   ↑ ,𝑛

Allowed substitution hints:   𝐾(𝑛)   0 (𝑛)

Proof of Theorem cayleyhamilton

Dummy variables 𝑥 𝑦 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cayleyhamilton.a	. 2 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		cayleyhamilton.b	. 2 ⊢ 𝐵 = (Base‘𝐴)
3		cayleyhamilton.0	. 2 ⊢ 0 = (0g‘𝐴)
4		eqid 2729	. 2 ⊢ (1r‘𝐴) = (1r‘𝐴)
5		cayleyhamilton.m	. 2 ⊢ ∗ = ( ·𝑠 ‘𝐴)
6		cayleyhamilton.e	. 2 ⊢ ↑ = (.g‘(mulGrp‘𝐴))
7		cayleyhamilton.c	. 2 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
8		cayleyhamilton.k	. 2 ⊢ 𝐾 = (coe1‘(𝐶‘𝑀))
9		eqid 2729	. 2 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
10		eqid 2729	. 2 ⊢ (𝑁 Mat (Poly1‘𝑅)) = (𝑁 Mat (Poly1‘𝑅))
11		eqid 2729	. 2 ⊢ (.r‘(𝑁 Mat (Poly1‘𝑅))) = (.r‘(𝑁 Mat (Poly1‘𝑅)))
12		eqid 2729	. 2 ⊢ (-g‘(𝑁 Mat (Poly1‘𝑅))) = (-g‘(𝑁 Mat (Poly1‘𝑅)))
13		eqid 2729	. 2 ⊢ (0g‘(𝑁 Mat (Poly1‘𝑅))) = (0g‘(𝑁 Mat (Poly1‘𝑅)))
14		eqid 2729	. 2 ⊢ (Base‘(𝑁 Mat (Poly1‘𝑅))) = (Base‘(𝑁 Mat (Poly1‘𝑅)))
15		eqid 2729	. 2 ⊢ (.g‘(mulGrp‘(𝑁 Mat (Poly1‘𝑅)))) = (.g‘(mulGrp‘(𝑁 Mat (Poly1‘𝑅))))
16		eqid 2729	. 2 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅)
17		eqeq1 2733	. . . 4 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
18		eqeq1 2733	. . . . 5 ⊢ (𝑙 = 𝑛 → (𝑙 = (𝑥 + 1) ↔ 𝑛 = (𝑥 + 1)))
19		breq2 5096	. . . . . 6 ⊢ (𝑙 = 𝑛 → ((𝑥 + 1) < 𝑙 ↔ (𝑥 + 1) < 𝑛))
20		fvoveq1 7372	. . . . . . . 8 ⊢ (𝑙 = 𝑛 → (𝑦‘(𝑙 − 1)) = (𝑦‘(𝑛 − 1)))
21	20	fveq2d 6826	. . . . . . 7 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑙 − 1))) = ((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑛 − 1))))
22		2fveq3 6827	. . . . . . . 8 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑙)) = ((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑛)))
23	22	oveq2d 7365	. . . . . . 7 ⊢ (𝑙 = 𝑛 → (((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑙))) = (((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑛))))
24	21, 23	oveq12d 7367	. . . . . 6 ⊢ (𝑙 = 𝑛 → (((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑙)))) = (((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑛 − 1)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑛)))))
25	19, 24	ifbieq2d 4503	. . . . 5 ⊢ (𝑙 = 𝑛 → if((𝑥 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑙))))) = if((𝑥 + 1) < 𝑛, (0g‘(𝑁 Mat (Poly1‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑛 − 1)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑛))))))
26	18, 25	ifbieq2d 4503	. . . 4 ⊢ (𝑙 = 𝑛 → if(𝑙 = (𝑥 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑥)), if((𝑥 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑙)))))) = if(𝑛 = (𝑥 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑥)), if((𝑥 + 1) < 𝑛, (0g‘(𝑁 Mat (Poly1‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑛 − 1)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑛)))))))
27	17, 26	ifbieq2d 4503	. . 3 ⊢ (𝑙 = 𝑛 → if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1‘𝑅)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘0)))), if(𝑙 = (𝑥 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑥)), if((𝑥 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑙))))))) = if(𝑛 = 0, ((0g‘(𝑁 Mat (Poly1‘𝑅)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘0)))), if(𝑛 = (𝑥 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑥)), if((𝑥 + 1) < 𝑛, (0g‘(𝑁 Mat (Poly1‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑛 − 1)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑛))))))))
28	27	cbvmptv 5196	. 2 ⊢ (𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1‘𝑅)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘0)))), if(𝑙 = (𝑥 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑥)), if((𝑥 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑙)))))))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ((0g‘(𝑁 Mat (Poly1‘𝑅)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘0)))), if(𝑛 = (𝑥 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑥)), if((𝑥 + 1) < 𝑛, (0g‘(𝑁 Mat (Poly1‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑛 − 1)))(-g‘(𝑁 Mat (Poly1‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑛))))))))
29		eqid 2729	. 2 ⊢ (𝑁 cPolyMatToMat 𝑅) = (𝑁 cPolyMatToMat 𝑅)
30	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 28, 29	cayleyhamilton0 22774	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ifcif 4476   class class class wbr 5092   ↦ cmpt 5173  ‘cfv 6482  (class class class)co 7349  Fincfn 8872  0cc0 11009  1c1 11010   + caddc 11012   < clt 11149   − cmin 11347  ℕ₀cn0 12384  Basecbs 17120  ._rcmulr 17162   ·_𝑠 cvsca 17165  0_gc0g 17343   Σ_g cgsu 17344  -_gcsg 18814  ._gcmg 18946  mulGrpcmgp 20025  1_rcur 20066  CRingccrg 20119  Poly₁cpl1 22059  coe₁cco1 22060   Mat cmat 22292   matToPolyMat cmat2pmat 22589   cPolyMatToMat ccpmat2mat 22590   CharPlyMat cchpmat 22711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-addf 11088  ax-mulf 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1512  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-ofr 7614  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-tpos 8159  df-cur 8200  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-sup 9332  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-xnn0 12458  df-z 12472  df-dec 12592  df-uz 12736  df-rp 12894  df-fz 13411  df-fzo 13558  df-seq 13909  df-exp 13969  df-hash 14238  df-word 14421  df-lsw 14470  df-concat 14478  df-s1 14503  df-substr 14548  df-pfx 14578  df-splice 14656  df-reverse 14665  df-s2 14755  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-hom 17185  df-cco 17186  df-0g 17345  df-gsum 17346  df-prds 17351  df-pws 17353  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-submnd 18658  df-efmnd 18743  df-grp 18815  df-minusg 18816  df-sbg 18817  df-mulg 18947  df-subg 19002  df-ghm 19092  df-gim 19138  df-cntz 19196  df-oppg 19225  df-symg 19249  df-pmtr 19321  df-psgn 19370  df-evpm 19371  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-srg 20072  df-ring 20120  df-cring 20121  df-oppr 20222  df-dvdsr 20242  df-unit 20243  df-invr 20273  df-dvr 20286  df-rhm 20357  df-subrng 20431  df-subrg 20455  df-drng 20616  df-lmod 20765  df-lss 20835  df-sra 21077  df-rgmod 21078  df-cnfld 21262  df-zring 21354  df-zrh 21410  df-dsmm 21639  df-frlm 21654  df-assa 21760  df-ascl 21762  df-psr 21816  df-mvr 21817  df-mpl 21818  df-opsr 21820  df-psr1 22062  df-vr1 22063  df-ply1 22064  df-coe1 22065  df-mamu 22276  df-mat 22293  df-mdet 22470  df-madu 22519  df-cpmat 22591  df-mat2pmat 22592  df-cpmat2mat 22593  df-decpmat 22648  df-pm2mp 22678  df-chpmat 22712

This theorem is referenced by:  cayleyhamilton1  22777