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Mirrors > Home > MPE Home > Th. List > cpmidg2sum | Structured version Visualization version GIF version |
Description: Equality of two sums representing the identity matrix multiplied with the characteristic polynomial of a matrix. (Contributed by AV, 11-Nov-2019.) |
Ref | Expression |
---|---|
cpmadugsum.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
cpmadugsum.b | ⊢ 𝐵 = (Base‘𝐴) |
cpmadugsum.p | ⊢ 𝑃 = (Poly1‘𝑅) |
cpmadugsum.y | ⊢ 𝑌 = (𝑁 Mat 𝑃) |
cpmadugsum.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
cpmadugsum.x | ⊢ 𝑋 = (var1‘𝑅) |
cpmadugsum.e | ⊢ ↑ = (.g‘(mulGrp‘𝑃)) |
cpmadugsum.m | ⊢ · = ( ·𝑠 ‘𝑌) |
cpmadugsum.r | ⊢ × = (.r‘𝑌) |
cpmadugsum.1 | ⊢ 1 = (1r‘𝑌) |
cpmadugsum.g | ⊢ + = (+g‘𝑌) |
cpmadugsum.s | ⊢ − = (-g‘𝑌) |
cpmadugsum.i | ⊢ 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) |
cpmadugsum.j | ⊢ 𝐽 = (𝑁 maAdju 𝑃) |
cpmadugsum.0 | ⊢ 0 = (0g‘𝑌) |
cpmadugsum.g2 | ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) |
cpmidgsum2.c | ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) |
cpmidgsum2.k | ⊢ 𝐾 = (𝐶‘𝑀) |
cpmidg2sum.u | ⊢ 𝑈 = (algSc‘𝑃) |
Ref | Expression |
---|---|
cpmidg2sum | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))(𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · ((𝑈‘((coe1‘𝐾)‘𝑖)) · 1 )))) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpmadugsum.a | . . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | cpmadugsum.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
3 | cpmadugsum.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | cpmadugsum.y | . . . . . 6 ⊢ 𝑌 = (𝑁 Mat 𝑃) | |
5 | cpmadugsum.x | . . . . . 6 ⊢ 𝑋 = (var1‘𝑅) | |
6 | cpmadugsum.e | . . . . . 6 ⊢ ↑ = (.g‘(mulGrp‘𝑃)) | |
7 | cpmadugsum.m | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑌) | |
8 | cpmadugsum.1 | . . . . . 6 ⊢ 1 = (1r‘𝑌) | |
9 | cpmidg2sum.u | . . . . . 6 ⊢ 𝑈 = (algSc‘𝑃) | |
10 | cpmidgsum2.c | . . . . . 6 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) | |
11 | cpmidgsum2.k | . . . . . 6 ⊢ 𝐾 = (𝐶‘𝑀) | |
12 | eqid 2778 | . . . . . 6 ⊢ (𝐾 · 1 ) = (𝐾 · 1 ) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | cpmidgsum 21091 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐾 · 1 ) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · ((𝑈‘((coe1‘𝐾)‘𝑖)) · 1 ))))) |
14 | 13 | eqcomd 2784 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · ((𝑈‘((coe1‘𝐾)‘𝑖)) · 1 )))) = (𝐾 · 1 )) |
15 | 14 | ad3antrrr 720 | . . 3 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ (𝐾 · 1 ) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · ((𝑈‘((coe1‘𝐾)‘𝑖)) · 1 )))) = (𝐾 · 1 )) |
16 | simpr 479 | . . 3 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ (𝐾 · 1 ) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) → (𝐾 · 1 ) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) | |
17 | 15, 16 | eqtrd 2814 | . 2 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ (𝐾 · 1 ) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · ((𝑈‘((coe1‘𝐾)‘𝑖)) · 1 )))) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) |
18 | cpmadugsum.t | . . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
19 | cpmadugsum.r | . . 3 ⊢ × = (.r‘𝑌) | |
20 | cpmadugsum.g | . . 3 ⊢ + = (+g‘𝑌) | |
21 | cpmadugsum.s | . . 3 ⊢ − = (-g‘𝑌) | |
22 | cpmadugsum.i | . . 3 ⊢ 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) | |
23 | cpmadugsum.j | . . 3 ⊢ 𝐽 = (𝑁 maAdju 𝑃) | |
24 | cpmadugsum.0 | . . 3 ⊢ 0 = (0g‘𝑌) | |
25 | cpmadugsum.g2 | . . 3 ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) | |
26 | 1, 2, 3, 4, 18, 5, 6, 7, 19, 8, 20, 21, 22, 23, 24, 25, 10, 11, 12 | cpmidgsum2 21102 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))(𝐾 · 1 ) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) |
27 | 17, 26 | reximddv2 3202 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))(𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · ((𝑈‘((coe1‘𝐾)‘𝑖)) · 1 )))) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 ifcif 4307 class class class wbr 4888 ↦ cmpt 4967 ‘cfv 6137 (class class class)co 6924 ↑𝑚 cmap 8142 Fincfn 8243 0cc0 10274 1c1 10275 + caddc 10277 < clt 10413 − cmin 10608 ℕcn 11379 ℕ0cn0 11647 ...cfz 12648 Basecbs 16266 +gcplusg 16349 .rcmulr 16350 ·𝑠 cvsca 16353 0gc0g 16497 Σg cgsu 16498 -gcsg 17822 .gcmg 17938 mulGrpcmgp 18887 1rcur 18899 CRingccrg 18946 algSccascl 19719 var1cv1 19953 Poly1cpl1 19954 coe1cco1 19955 Mat cmat 20628 maAdju cmadu 20854 matToPolyMat cmat2pmat 20927 CharPlyMat cchpmat 21049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-xor 1583 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-ofr 7177 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-tpos 7636 df-cur 7677 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-sup 8638 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-xnn0 11720 df-z 11734 df-dec 11851 df-uz 11998 df-rp 12143 df-fz 12649 df-fzo 12790 df-seq 13125 df-exp 13184 df-hash 13442 df-word 13606 df-lsw 13659 df-concat 13667 df-s1 13692 df-substr 13737 df-pfx 13786 df-splice 13893 df-reverse 13911 df-s2 14005 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-hom 16373 df-cco 16374 df-0g 16499 df-gsum 16500 df-prds 16505 df-pws 16507 df-mre 16643 df-mrc 16644 df-acs 16646 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-mhm 17732 df-submnd 17733 df-grp 17823 df-minusg 17824 df-sbg 17825 df-mulg 17939 df-subg 17986 df-ghm 18053 df-gim 18096 df-cntz 18144 df-oppg 18170 df-symg 18192 df-pmtr 18256 df-psgn 18305 df-evpm 18306 df-cmn 18592 df-abl 18593 df-mgp 18888 df-ur 18900 df-srg 18904 df-ring 18947 df-cring 18948 df-oppr 19021 df-dvdsr 19039 df-unit 19040 df-invr 19070 df-dvr 19081 df-rnghom 19115 df-drng 19152 df-subrg 19181 df-lmod 19268 df-lss 19336 df-sra 19580 df-rgmod 19581 df-assa 19720 df-ascl 19722 df-psr 19764 df-mvr 19765 df-mpl 19766 df-opsr 19768 df-psr1 19957 df-vr1 19958 df-ply1 19959 df-coe1 19960 df-cnfld 20154 df-zring 20226 df-zrh 20259 df-dsmm 20486 df-frlm 20501 df-mamu 20605 df-mat 20629 df-mdet 20807 df-madu 20856 df-mat2pmat 20930 df-decpmat 20986 df-chpmat 21050 |
This theorem is referenced by: (None) |
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