chmaidscmat - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > chmaidscmat		Structured version Visualization version GIF version

Theorem chmaidscmat 23072

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 20383	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		chmaidscmat.p	. . . . . . 7 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 22471	. . . . . 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 23056	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸)
12		chmaidscmat.y	. . . . . . 7 ⊢ 𝑌 = (𝑁 Mat 𝑃)
13	2, 12	pmatring 22916	. . . . . 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 20405	. . . . 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 22652	. . 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 3239	. . . 4 ⊢ ((𝐶‘𝑀) ∈ 𝐸 ↔ ∃𝑐 ∈ 𝐸 𝑐 = (𝐶‘𝑀))
24		oveq1 7423	. . . . . . . 8 ⊢ ((𝐶‘𝑀) = 𝑐 → ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 ))
25	24	eqcoms 2770	. . . . . . 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 3177	. . . . 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 22726	. . 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 3088 ‘cfv 6537 (class class class)co 7416 Fincfn 8955 Basecbs 17303 ·_𝑠 cvsca 17348 1_rcur 20319 Ringcrg 20371 CRingccrg 20372 Poly₁cpl1 22401 Mat cmat 22628 ScMat cscmat 22710 CharPlyMat cchpmat 23050

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-addf 11204 ax-mulf 11205

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 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 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-ofr 7682 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-oi 9485 df-card 9947 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-xnn0 12603 df-z 12617 df-dec 12738 df-uz 12889 df-rp 13043 df-fz 13562 df-fzo 13710 df-seq 14066 df-exp 14126 df-hash 14395 df-word 14579 df-lsw 14628 df-concat 14636 df-s1 14663 df-substr 14709 df-pfx 14741 df-splice 14819 df-reverse 14828 df-s2 14919 df-struct 17241 df-sets 17258 df-slot 17276 df-ndx 17288 df-base 17304 df-ress 17325 df-plusg 17357 df-mulr 17358 df-starv 17359 df-sca 17360 df-vsca 17361 df-ip 17362 df-tset 17363 df-ple 17364 df-ds 17366 df-unif 17367 df-hom 17368 df-cco 17369 df-0g 17528 df-gsum 17529 df-prds 17534 df-pws 17536 df-mre 17672 df-mrc 17673 df-acs 17675 df-mgm 18732 df-sgrp 18821 df-mnd 18837 df-mhm 18890 df-submnd 18891 df-efmnd 18977 df-grp 19059 df-minusg 19060 df-sbg 19061 df-mulg 19190 df-subg 19245 df-ghm 19340 df-gim 19385 df-cntz 19443 df-oppg 19472 df-symg 19496 df-pmtr 19568 df-psgn 19617 df-cmn 19908 df-abl 19909 df-mgp 20273 df-rng 20287 df-ur 20320 df-ring 20373 df-cring 20374 df-oppr 20477 df-dvdsr 20497 df-unit 20498 df-invr 20528 df-dvr 20541 df-rhm 20612 df-subrng 20707 df-subrg 20731 df-drng 20891 df-lmod 21045 df-lss 21115 df-sra 21356 df-rgmod 21357 df-cnfld 21585 df-zring 21659 df-zrh 21715 df-dsmm 21944 df-frlm 21959 df-ascl 22069 df-psr 22123 df-mvr 22124 df-mpl 22125 df-opsr 22127 df-psr1 22404 df-vr1 22405 df-ply1 22406 df-mamu 22612 df-mat 22629 df-scmat 22712 df-mdet 22806 df-mat2pmat 22931 df-chpmat 23051

This theorem is used by: (None)

W3C validator