chmaidscmat - Metamath Proof Explorer

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Theorem chmaidscmat 22743

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 20178	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		chmaidscmat.p	. . . . . . 7 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 22159	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 17	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
5	4	anim2i 616	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
6	5	3adant3 1130	. . 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 22727	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸)
12		chmaidscmat.y	. . . . . . 7 ⊢ 𝑌 = (𝑁 Mat 𝑃)
13	2, 12	pmatring 22587	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
14	1, 13	sylan2 592	. . . . 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 20195	. . . . 5 ⊢ (𝑌 ∈ Ring → 1 ∈ 𝐾)
18	14, 17	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 1 ∈ 𝐾)
19	18	3adant3 1130	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ 𝐾)
20		chmaidscmat.m	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑌)
21	10, 12, 15, 20	matvscl 22326	. . 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 3225	. . . 4 ⊢ ((𝐶‘𝑀) ∈ 𝐸 ↔ ∃𝑐 ∈ 𝐸 𝑐 = (𝐶‘𝑀))
24		oveq1 7421	. . . . . . . 8 ⊢ ((𝐶‘𝑀) = 𝑐 → ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 ))
25	24	eqcoms 2735	. . . . . . 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 3163	. . . . 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 216	. . 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 22400	. . 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 712	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐶‘𝑀) · 1 ) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 ‘cfv 6542 (class class class)co 7414 Fincfn 8957 Basecbs 17173 ·_𝑠 cvsca 17230 1_rcur 20114 Ringcrg 20166 CRingccrg 20167 Poly₁cpl1 22089 Mat cmat 22300 ScMat cscmat 22384 CharPlyMat cchpmat 22721

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-addf 11211 ax-mulf 11212

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-xor 1506 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-xnn0 12569 df-z 12583 df-dec 12702 df-uz 12847 df-rp 13001 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-word 14491 df-lsw 14539 df-concat 14547 df-s1 14572 df-substr 14617 df-pfx 14647 df-splice 14726 df-reverse 14735 df-s2 14825 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-0g 17416 df-gsum 17417 df-prds 17422 df-pws 17424 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-efmnd 18814 df-grp 18886 df-minusg 18887 df-sbg 18888 df-mulg 19017 df-subg 19071 df-ghm 19161 df-gim 19206 df-cntz 19261 df-oppg 19290 df-symg 19315 df-pmtr 19390 df-psgn 19439 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-dvr 20333 df-rhm 20404 df-subrng 20476 df-subrg 20501 df-drng 20619 df-lmod 20738 df-lss 20809 df-sra 21051 df-rgmod 21052 df-cnfld 21273 df-zring 21366 df-zrh 21422 df-dsmm 21659 df-frlm 21674 df-ascl 21782 df-psr 21835 df-mvr 21836 df-mpl 21837 df-opsr 21839 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-mamu 22279 df-mat 22301 df-scmat 22386 df-mdet 22480 df-mat2pmat 22602 df-chpmat 22722

This theorem is referenced by: (None)

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