Theorem cayleyhamilton 22753

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 5106	. . . . . 6 ⊢ (𝑙 = 𝑛 → ((𝑥 + 1) < 𝑙 ↔ (𝑥 + 1) < 𝑛))
20		fvoveq1 7392	. . . . . . . 8 ⊢ (𝑙 = 𝑛 → (𝑦‘(𝑙 − 1)) = (𝑦‘(𝑛 − 1)))
21	20	fveq2d 6844	. . . . . . 7 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑙 − 1))) = ((𝑁 matToPolyMat 𝑅)‘(𝑦‘(𝑛 − 1))))
22		2fveq3 6845	. . . . . . . 8 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑙)) = ((𝑁 matToPolyMat 𝑅)‘(𝑦‘𝑛)))
23	22	oveq2d 7385	. . . . . . 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 7387	. . . . . 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 4511	. . . . 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 4511	. . . 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 4511	. . 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 5206	. 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 22752	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 4484   class class class wbr 5102   ↦ cmpt 5183  ‘cfv 6499  (class class class)co 7369  Fincfn 8895  0cc0 11044  1c1 11045   + caddc 11047   < clt 11184   − cmin 11381  ℕ₀cn0 12418  Basecbs 17155  ._rcmulr 17197   ·_𝑠 cvsca 17200  0_gc0g 17378   Σ_g cgsu 17379  -_gcsg 18843  ._gcmg 18975  mulGrpcmgp 20025  1_rcur 20066  CRingccrg 20119  Poly₁cpl1 22037  coe₁cco1 22038   Mat cmat 22270   matToPolyMat cmat2pmat 22567   cPolyMatToMat ccpmat2mat 22568   CharPlyMat cchpmat 22689

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-addf 11123  ax-mulf 11124

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-ofr 7634  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-tpos 8182  df-cur 8223  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-sup 9369  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-xnn0 12492  df-z 12506  df-dec 12626  df-uz 12770  df-rp 12928  df-fz 13445  df-fzo 13592  df-seq 13943  df-exp 14003  df-hash 14272  df-word 14455  df-lsw 14504  df-concat 14512  df-s1 14537  df-substr 14582  df-pfx 14612  df-splice 14691  df-reverse 14700  df-s2 14790  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-0g 17380  df-gsum 17381  df-prds 17386  df-pws 17388  df-mre 17523  df-mrc 17524  df-acs 17526  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-mhm 18686  df-submnd 18687  df-efmnd 18772  df-grp 18844  df-minusg 18845  df-sbg 18846  df-mulg 18976  df-subg 19031  df-ghm 19121  df-gim 19167  df-cntz 19225  df-oppg 19254  df-symg 19276  df-pmtr 19348  df-psgn 19397  df-evpm 19398  df-cmn 19688  df-abl 19689  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 20744  df-lss 20814  df-sra 21056  df-rgmod 21057  df-cnfld 21241  df-zring 21333  df-zrh 21389  df-dsmm 21617  df-frlm 21632  df-assa 21738  df-ascl 21740  df-psr 21794  df-mvr 21795  df-mpl 21796  df-opsr 21798  df-psr1 22040  df-vr1 22041  df-ply1 22042  df-coe1 22043  df-mamu 22254  df-mat 22271  df-mdet 22448  df-madu 22497  df-cpmat 22569  df-mat2pmat 22570  df-cpmat2mat 22571  df-decpmat 22626  df-pm2mp 22656  df-chpmat 22690

This theorem is referenced by:  cayleyhamilton1  22755