chmaidscmat - Metamath Proof Explorer

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Theorem chmaidscmat 23105

Description: The characteristic polynomial of a matrix multiplied with the identity matrix is a scalar matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 5-Jul-2022.)

**Hypotheses**
Ref	Expression
chmaidscmat.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
chmaidscmat.b	⊢ 𝐵 = (Base‘𝐴)
chmaidscmat.c	⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
chmaidscmat.p	⊢ 𝑃 = (Poly₁‘𝑅)
chmaidscmat.e	⊢ 𝐸 = (Base‘𝑃)
chmaidscmat.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
chmaidscmat.k	⊢ 𝐾 = (Base‘𝑌)
chmaidscmat.m	⊢ · = ( ·_𝑠 ‘𝑌)
chmaidscmat.1	⊢ 1 = (1_r‘𝑌)
chmaidscmat.d	⊢ 𝑆 = (𝑁 ScMat 𝑃)

**Assertion**
Ref	Expression
chmaidscmat	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐶‘𝑀) · 1 ) ∈ 𝑆)

Proof of Theorem chmaidscmat Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngring 20411	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		chmaidscmat.p	. . . . . . 7 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 22504	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 18	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
5	4	anim2i 629	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
6	5	3adant3 1150	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
7		chmaidscmat.c	. . . 4 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
8		chmaidscmat.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
9		chmaidscmat.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
10		chmaidscmat.e	. . . 4 ⊢ 𝐸 = (Base‘𝑃)
11	7, 8, 9, 2, 10	chpmatply1 23089	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸)
12		chmaidscmat.y	. . . . . . 7 ⊢ 𝑌 = (𝑁 Mat 𝑃)
13	2, 12	pmatring 22949	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
14	1, 13	sylan2 605	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
15		chmaidscmat.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑌)
16		chmaidscmat.1	. . . . . 6 ⊢ 1 = (1_r‘𝑌)
17	15, 16	ringidcl 20433	. . . . 5 ⊢ (𝑌 ∈ Ring → 1 ∈ 𝐾)
18	14, 17	syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 1 ∈ 𝐾)
19	18	3adant3 1150	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ 𝐾)
20		chmaidscmat.m	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑌)
21	10, 12, 15, 20	matvscl 22685	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝐶‘𝑀) ∈ 𝐸 ∧ 1 ∈ 𝐾)) → ((𝐶‘𝑀) · 1 ) ∈ 𝐾)
22	6, 11, 19, 21	syl12anc 850	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐶‘𝑀) · 1 ) ∈ 𝐾)
23		risset 3237	. . . 4 ⊢ ((𝐶‘𝑀) ∈ 𝐸 ↔ ∃𝑐 ∈ 𝐸 𝑐 = (𝐶‘𝑀))
24		oveq1 7423	. . . . . . . 8 ⊢ ((𝐶‘𝑀) = 𝑐 → ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 ))
25	24	eqcoms 2768	. . . . . . 7 ⊢ (𝑐 = (𝐶‘𝑀) → ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 ))
26	25	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑐 ∈ 𝐸) → (𝑐 = (𝐶‘𝑀) → ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 )))
27	26	reximdva 3175	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑐 ∈ 𝐸 𝑐 = (𝐶‘𝑀) → ∃𝑐 ∈ 𝐸 ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 )))
28	27	com12 33	. . . 4 ⊢ (∃𝑐 ∈ 𝐸 𝑐 = (𝐶‘𝑀) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑐 ∈ 𝐸 ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 )))
29	23, 28	sylbi 220	. . 3 ⊢ ((𝐶‘𝑀) ∈ 𝐸 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑐 ∈ 𝐸 ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 )))
30	11, 29	mpcom 39	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑐 ∈ 𝐸 ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 ))
31		chmaidscmat.d	. . . 4 ⊢ 𝑆 = (𝑁 ScMat 𝑃)
32	10, 12, 15, 16, 20, 31	scmatel 22759	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → (((𝐶‘𝑀) · 1 ) ∈ 𝑆 ↔ (((𝐶‘𝑀) · 1 ) ∈ 𝐾 ∧ ∃𝑐 ∈ 𝐸 ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 ))))
33	6, 32	syl 18	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((𝐶‘𝑀) · 1 ) ∈ 𝑆 ↔ (((𝐶‘𝑀) · 1 ) ∈ 𝐾 ∧ ∃𝑐 ∈ 𝐸 ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 ))))
34	22, 30, 33	mpbir2and 726	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐶‘𝑀) · 1 ) ∈ 𝑆)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∃wrex 3086 ‘cfv 6535 (class class class)co 7416 Fincfn 8959 Basecbs 17326 ·_𝑠 cvsca 17371 1_rcur 20346 Ringcrg 20398 CRingccrg 20399 Poly₁cpl1 22434 Mat cmat 22661 ScMat cscmat 22743 CharPlyMat cchpmat 23083

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-addf 11228 ax-mulf 11229

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1542 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-ofr 7685 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8164 df-tpos 8229 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9339 df-sup 9419 df-oi 9489 df-card 9969 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-xnn0 12627 df-z 12641 df-dec 12762 df-uz 12913 df-rp 13068 df-fz 13587 df-fzo 13735 df-seq 14091 df-exp 14151 df-hash 14420 df-word 14604 df-lsw 14653 df-concat 14661 df-s1 14688 df-substr 14734 df-pfx 14766 df-splice 14844 df-reverse 14853 df-s2 14944 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-starv 17382 df-sca 17383 df-vsca 17384 df-ip 17385 df-tset 17386 df-ple 17387 df-ds 17389 df-unif 17390 df-hom 17391 df-cco 17392 df-0g 17551 df-gsum 17552 df-prds 17557 df-pws 17559 df-mre 17695 df-mrc 17696 df-acs 17698 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-mhm 18917 df-submnd 18918 df-efmnd 19004 df-grp 19086 df-minusg 19087 df-sbg 19088 df-mulg 19217 df-subg 19272 df-ghm 19367 df-gim 19412 df-cntz 19470 df-oppg 19499 df-symg 19523 df-pmtr 19595 df-psgn 19644 df-cmn 19935 df-abl 19936 df-mgp 20300 df-rng 20314 df-ur 20347 df-ring 20400 df-cring 20401 df-oppr 20506 df-dvdsr 20526 df-unit 20527 df-invr 20557 df-dvr 20570 df-rhm 20641 df-subrng 20737 df-subrg 20761 df-drng 20921 df-lmod 21076 df-lss 21146 df-sra 21387 df-rgmod 21388 df-cnfld 21618 df-zring 21692 df-zrh 21748 df-dsmm 21977 df-frlm 21992 df-ascl 22102 df-psr 22156 df-mvr 22157 df-mpl 22158 df-opsr 22160 df-psr1 22437 df-vr1 22438 df-ply1 22439 df-mamu 22645 df-mat 22662 df-scmat 22745 df-mdet 22839 df-mat2pmat 22964 df-chpmat 23084

This theorem is used by: (None)

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