![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > chmaidscmat | Structured version Visualization version GIF version |
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.) |
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 = (1r‘𝑌) |
chmaidscmat.d | ⊢ 𝑆 = (𝑁 ScMat 𝑃) |
Ref | Expression |
---|---|
chmaidscmat | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐶‘𝑀) · 1 ) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 20263 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | chmaidscmat.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | 2 | ply1ring 22265 | . . . . . 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 1131 | . . 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 22854 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸) |
12 | chmaidscmat.y | . . . . . . 7 ⊢ 𝑌 = (𝑁 Mat 𝑃) | |
13 | 2, 12 | pmatring 22714 | . . . . . 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 = (1r‘𝑌) | |
17 | 15, 16 | ringidcl 20280 | . . . . 5 ⊢ (𝑌 ∈ Ring → 1 ∈ 𝐾) |
18 | 14, 17 | syl 17 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 1 ∈ 𝐾) |
19 | 18 | 3adant3 1131 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ 𝐾) |
20 | chmaidscmat.m | . . . 4 ⊢ · = ( ·𝑠 ‘𝑌) | |
21 | 10, 12, 15, 20 | matvscl 22453 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝐶‘𝑀) ∈ 𝐸 ∧ 1 ∈ 𝐾)) → ((𝐶‘𝑀) · 1 ) ∈ 𝐾) |
22 | 6, 11, 19, 21 | syl12anc 837 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐶‘𝑀) · 1 ) ∈ 𝐾) |
23 | risset 3231 | . . . 4 ⊢ ((𝐶‘𝑀) ∈ 𝐸 ↔ ∃𝑐 ∈ 𝐸 𝑐 = (𝐶‘𝑀)) | |
24 | oveq1 7438 | . . . . . . . 8 ⊢ ((𝐶‘𝑀) = 𝑐 → ((𝐶‘𝑀) · 1 ) = (𝑐 · 1 )) | |
25 | 24 | eqcoms 2743 | . . . . . . 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 3166 | . . . . 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 22527 | . . 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 1537 ∈ wcel 2106 ∃wrex 3068 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 Basecbs 17245 ·𝑠 cvsca 17302 1rcur 20199 Ringcrg 20251 CRingccrg 20252 Poly1cpl1 22194 Mat cmat 22427 ScMat cscmat 22511 CharPlyMat cchpmat 22848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1509 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-word 14550 df-lsw 14598 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-splice 14785 df-reverse 14794 df-s2 14884 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-efmnd 18895 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-gim 19290 df-cntz 19348 df-oppg 19377 df-symg 19402 df-pmtr 19475 df-psgn 19524 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-dsmm 21770 df-frlm 21785 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-mamu 22411 df-mat 22428 df-scmat 22513 df-mdet 22607 df-mat2pmat 22729 df-chpmat 22849 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |