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Mirrors > Home > MPE Home > Th. List > chmatcl | Structured version Visualization version GIF version |
Description: Closure of the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.) (Proof shortened by AV, 29-Nov-2019.) |
Ref | Expression |
---|---|
chmatcl.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
chmatcl.b | ⊢ 𝐵 = (Base‘𝐴) |
chmatcl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
chmatcl.y | ⊢ 𝑌 = (𝑁 Mat 𝑃) |
chmatcl.x | ⊢ 𝑋 = (var1‘𝑅) |
chmatcl.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
chmatcl.s | ⊢ − = (-g‘𝑌) |
chmatcl.m | ⊢ · = ( ·𝑠 ‘𝑌) |
chmatcl.1 | ⊢ 1 = (1r‘𝑌) |
chmatcl.h | ⊢ 𝐻 = ((𝑋 · 1 ) − (𝑇‘𝑀)) |
Ref | Expression |
---|---|
chmatcl | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐻 ∈ (Base‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chmatcl.h | . 2 ⊢ 𝐻 = ((𝑋 · 1 ) − (𝑇‘𝑀)) | |
2 | chmatcl.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | chmatcl.y | . . . . . 6 ⊢ 𝑌 = (𝑁 Mat 𝑃) | |
4 | 2, 3 | pmatring 21300 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring) |
5 | ringgrp 19301 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Grp) |
7 | 6 | 3adant3 1128 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp) |
8 | 2 | ply1ring 20415 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
9 | 8 | anim2i 618 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
10 | 9 | 3adant3 1128 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
11 | chmatcl.x | . . . . . 6 ⊢ 𝑋 = (var1‘𝑅) | |
12 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
13 | 11, 2, 12 | vr1cl 20384 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
14 | 13 | 3ad2ant2 1130 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
15 | 4 | 3adant3 1128 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring) |
16 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
17 | chmatcl.1 | . . . . . 6 ⊢ 1 = (1r‘𝑌) | |
18 | 16, 17 | ringidcl 19317 | . . . . 5 ⊢ (𝑌 ∈ Ring → 1 ∈ (Base‘𝑌)) |
19 | 15, 18 | syl 17 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 1 ∈ (Base‘𝑌)) |
20 | chmatcl.m | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑌) | |
21 | 12, 3, 16, 20 | matvscl 21039 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ (𝑋 ∈ (Base‘𝑃) ∧ 1 ∈ (Base‘𝑌))) → (𝑋 · 1 ) ∈ (Base‘𝑌)) |
22 | 10, 14, 19, 21 | syl12anc 834 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑋 · 1 ) ∈ (Base‘𝑌)) |
23 | chmatcl.t | . . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
24 | chmatcl.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
25 | chmatcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
26 | 23, 24, 25, 2, 3 | mat2pmatbas 21333 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
27 | chmatcl.s | . . . 4 ⊢ − = (-g‘𝑌) | |
28 | 16, 27 | grpsubcl 18178 | . . 3 ⊢ ((𝑌 ∈ Grp ∧ (𝑋 · 1 ) ∈ (Base‘𝑌) ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → ((𝑋 · 1 ) − (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
29 | 7, 22, 26, 28 | syl3anc 1367 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 ) − (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
30 | 1, 29 | eqeltrid 2917 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐻 ∈ (Base‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 Fincfn 8508 Basecbs 16482 ·𝑠 cvsca 16568 Grpcgrp 18102 -gcsg 18104 1rcur 19250 Ringcrg 19296 var1cv1 20343 Poly1cpl1 20344 Mat cmat 21015 matToPolyMat cmat2pmat 21311 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-ofr 7409 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-hom 16588 df-cco 16589 df-0g 16714 df-gsum 16715 df-prds 16720 df-pws 16722 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-ghm 18355 df-cntz 18446 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-subrg 19532 df-lmod 19635 df-lss 19703 df-sra 19943 df-rgmod 19944 df-ascl 20086 df-psr 20135 df-mvr 20136 df-mpl 20137 df-opsr 20139 df-psr1 20347 df-vr1 20348 df-ply1 20349 df-dsmm 20875 df-frlm 20890 df-mamu 20994 df-mat 21016 df-mat2pmat 21314 |
This theorem is referenced by: chpmatply1 21439 chpmatval2 21440 cpmadurid 21474 cpmadugsumfi 21484 |
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