Theorem cayleyhamilton 23016

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 2769	. 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 2769	. 2 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
10		eqid 2769	. 2 ⊢ (𝑁 Mat (Poly1‘𝑅)) = (𝑁 Mat (Poly1‘𝑅))
11		eqid 2769	. 2 ⊢ (.r‘(𝑁 Mat (Poly1‘𝑅))) = (.r‘(𝑁 Mat (Poly1‘𝑅)))
12		eqid 2769	. 2 ⊢ (-g‘(𝑁 Mat (Poly1‘𝑅))) = (-g‘(𝑁 Mat (Poly1‘𝑅)))
13		eqid 2769	. 2 ⊢ (0g‘(𝑁 Mat (Poly1‘𝑅))) = (0g‘(𝑁 Mat (Poly1‘𝑅)))
14		eqid 2769	. 2 ⊢ (Base‘(𝑁 Mat (Poly1‘𝑅))) = (Base‘(𝑁 Mat (Poly1‘𝑅)))
15		eqid 2769	. 2 ⊢ (.g‘(mulGrp‘(𝑁 Mat (Poly1‘𝑅)))) = (.g‘(mulGrp‘(𝑁 Mat (Poly1‘𝑅))))
16		eqid 2769	. 2 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅)
17		eqeq1 2773	. . . 4 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
18		eqeq1 2773	. . . . 5 ⊢ (𝑙 = 𝑛 → (𝑙 = (𝑥 + 1) ↔ 𝑛 = (𝑥 + 1)))
19		breq2 5117	. . . . . 6 ⊢ (𝑙 = 𝑛 → ((𝑥 + 1) < 𝑙 ↔ (𝑥 + 1) < 𝑛))
20		fvoveq1 7434	. . . . . . . 8 ⊢ (𝑙 = 𝑛 → (𝑦‘(𝑙 − 1)) = (𝑦‘(𝑛 − 1)))
21	20	fveq2d 6886	. . . . . . 7 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑙 − 1))) = ((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑛 − 1))))
22		2fveq3 6887	. . . . . . . 8 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑙)) = ((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑛)))
23	22	oveq2d 7427	. . . . . . 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 7429	. . . . . 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 4519	. . . . 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 4519	. . . 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 4519	. . 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 5219	. 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 2769	. 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 23015	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ifcif 4492   class class class wbr 5113   ↦ cmpt 5196  ‘cfv 6537  (class class class)co 7411  Fincfn 8943  0cc0 11100  1c1 11101   + caddc 11103   < clt 11243   − cmin 11441  ℕ₀cn0 12504  Basecbs 17269  ._rcmulr 17311   ·_𝑠 cvsca 17314  0_gc0g 17492   Σ_g cgsu 17493  -_gcsg 19002  ._gcmg 19133  mulGrpcmgp 20216  1_rcur 20263  CRingccrg 20316  Poly₁cpl1 22306  coe₁cco1 22307   Mat cmat 22533   matToPolyMat cmat2pmat 22830   cPolyMatToMat ccpmat2mat 22831   CharPlyMat cchpmat 22952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-addf 11179  ax-mulf 11180

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-xor 1539  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-ofr 7676  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-tpos 8222  df-cur 8263  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-map 8826  df-pm 8827  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-sup 9402  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-xnn0 12578  df-z 12592  df-dec 12712  df-uz 12863  df-rp 13017  df-fz 13536  df-fzo 13683  df-seq 14038  df-exp 14098  df-hash 14367  df-word 14551  df-lsw 14600  df-concat 14608  df-s1 14634  df-substr 14679  df-pfx 14709  df-splice 14787  df-reverse 14796  df-s2 14885  df-struct 17207  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-starv 17325  df-sca 17326  df-vsca 17327  df-ip 17328  df-tset 17329  df-ple 17330  df-ds 17332  df-unif 17333  df-hom 17334  df-cco 17335  df-0g 17494  df-gsum 17495  df-prds 17500  df-pws 17502  df-mre 17638  df-mrc 17639  df-acs 17641  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-mhm 18841  df-submnd 18842  df-efmnd 18928  df-grp 19003  df-minusg 19004  df-sbg 19005  df-mulg 19134  df-subg 19189  df-ghm 19284  df-gim 19329  df-cntz 19387  df-oppg 19416  df-symg 19440  df-pmtr 19512  df-psgn 19561  df-evpm 19562  df-cmn 19852  df-abl 19853  df-mgp 20217  df-rng 20231  df-ur 20264  df-srg 20269  df-ring 20317  df-cring 20318  df-oppr 20419  df-dvdsr 20439  df-unit 20440  df-invr 20470  df-dvr 20483  df-rhm 20554  df-subrng 20631  df-subrg 20655  df-drng 20815  df-lmod 20961  df-lss 21031  df-sra 21272  df-rgmod 21273  df-cnfld 21492  df-zring 21566  df-zrh 21622  df-dsmm 21851  df-frlm 21866  df-assa 21972  df-ascl 21974  df-psr 22028  df-mvr 22029  df-mpl 22030  df-opsr 22032  df-psr1 22309  df-vr1 22310  df-ply1 22311  df-coe1 22312  df-mamu 22517  df-mat 22534  df-mdet 22711  df-madu 22760  df-cpmat 22832  df-mat2pmat 22833  df-cpmat2mat 22834  df-decpmat 22889  df-pm2mp 22919  df-chpmat 22953

This theorem is referenced by:  cayleyhamilton1  23018