chmaidscmat - Metamath Proof Explorer

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Theorem chmaidscmat 22735

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 20154	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		chmaidscmat.p	. . . . . . 7 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 22132	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 17	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
5	4	anim2i 617	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
6	5	3adant3 1132	. . 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 22719	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸)
12		chmaidscmat.y	. . . . . . 7 ⊢ 𝑌 = (𝑁 Mat 𝑃)
13	2, 12	pmatring 22579	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
14	1, 13	sylan2 593	. . . . 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 20174	. . . . 5 ⊢ (𝑌 ∈ Ring → 1 ∈ 𝐾)
18	14, 17	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 1 ∈ 𝐾)
19	18	3adant3 1132	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ 𝐾)
20		chmaidscmat.m	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑌)
21	10, 12, 15, 20	matvscl 22318	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝐶‘𝑀) ∈ 𝐸 ∧ 1 ∈ 𝐾)) → ((𝐶‘𝑀) · 1 ) ∈ 𝐾)
22	6, 11, 19, 21	syl12anc 836	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐶‘𝑀) · 1 ) ∈ 𝐾)
23		risset 3212	. . . 4 ⊢ ((𝐶‘𝑀) ∈ 𝐸 ↔ ∃𝑐 ∈ 𝐸 𝑐 = (𝐶‘𝑀))
24		oveq1 7394	. . . . . . . 8 ⊢ ((𝐶‘𝑀) = 𝑐 → ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 ))
25	24	eqcoms 2737	. . . . . . 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 3146	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑐 ∈ 𝐸 𝑐 = (𝐶‘𝑀) → ∃𝑐 ∈ 𝐸 ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 )))
28	27	com12 32	. . . 4 ⊢ (∃𝑐 ∈ 𝐸 𝑐 = (𝐶‘𝑀) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑐 ∈ 𝐸 ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 )))
29	23, 28	sylbi 217	. . 3 ⊢ ((𝐶‘𝑀) ∈ 𝐸 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑐 ∈ 𝐸 ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 )))
30	11, 29	mpcom 38	. 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 22392	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → (((𝐶‘𝑀) · 1 ) ∈ 𝑆 ↔ (((𝐶‘𝑀) · 1 ) ∈ 𝐾 ∧ ∃𝑐 ∈ 𝐸 ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 ))))
33	6, 32	syl 17	. 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 713	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐶‘𝑀) · 1 ) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 ‘cfv 6511 (class class class)co 7387 Fincfn 8918 Basecbs 17179 ·_𝑠 cvsca 17224 1_rcur 20090 Ringcrg 20142 CRingccrg 20143 Poly₁cpl1 22061 Mat cmat 22294 ScMat cscmat 22376 CharPlyMat cchpmat 22713

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-addf 11147 ax-mulf 11148

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-ot 4598 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-rp 12952 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-word 14479 df-lsw 14528 df-concat 14536 df-s1 14561 df-substr 14606 df-pfx 14636 df-splice 14715 df-reverse 14724 df-s2 14814 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-efmnd 18796 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-ghm 19145 df-gim 19191 df-cntz 19249 df-oppg 19278 df-symg 19300 df-pmtr 19372 df-psgn 19421 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-rhm 20381 df-subrng 20455 df-subrg 20479 df-drng 20640 df-lmod 20768 df-lss 20838 df-sra 21080 df-rgmod 21081 df-cnfld 21265 df-zring 21357 df-zrh 21413 df-dsmm 21641 df-frlm 21656 df-ascl 21764 df-psr 21818 df-mvr 21819 df-mpl 21820 df-opsr 21822 df-psr1 22064 df-vr1 22065 df-ply1 22066 df-mamu 22278 df-mat 22295 df-scmat 22378 df-mdet 22472 df-mat2pmat 22594 df-chpmat 22714

This theorem is referenced by: (None)

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