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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 21613 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring) |
5 | ringgrp 19591 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Grp) |
7 | 6 | 3adant3 1134 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp) |
8 | 2 | ply1ring 21193 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
9 | 8 | anim2i 620 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
10 | 9 | 3adant3 1134 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
11 | chmatcl.x | . . . . . 6 ⊢ 𝑋 = (var1‘𝑅) | |
12 | eqid 2738 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
13 | 11, 2, 12 | vr1cl 21162 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
14 | 13 | 3ad2ant2 1136 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
15 | 4 | 3adant3 1134 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring) |
16 | eqid 2738 | . . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
17 | chmatcl.1 | . . . . . 6 ⊢ 1 = (1r‘𝑌) | |
18 | 16, 17 | ringidcl 19610 | . . . . 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 21352 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ (𝑋 ∈ (Base‘𝑃) ∧ 1 ∈ (Base‘𝑌))) → (𝑋 · 1 ) ∈ (Base‘𝑌)) |
22 | 10, 14, 19, 21 | syl12anc 837 | . . 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 21647 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
27 | chmatcl.s | . . . 4 ⊢ − = (-g‘𝑌) | |
28 | 16, 27 | grpsubcl 18467 | . . 3 ⊢ ((𝑌 ∈ Grp ∧ (𝑋 · 1 ) ∈ (Base‘𝑌) ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → ((𝑋 · 1 ) − (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
29 | 7, 22, 26, 28 | syl3anc 1373 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 ) − (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
30 | 1, 29 | eqeltrid 2843 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐻 ∈ (Base‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 ‘cfv 6397 (class class class)co 7231 Fincfn 8646 Basecbs 16784 ·𝑠 cvsca 16830 Grpcgrp 18389 -gcsg 18391 1rcur 19540 Ringcrg 19586 var1cv1 21121 Poly1cpl1 21122 Mat cmat 21328 matToPolyMat cmat2pmat 21625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-ot 4564 df-uni 4834 df-int 4874 df-iun 4920 df-iin 4921 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-se 5524 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-isom 6406 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-of 7487 df-ofr 7488 df-om 7663 df-1st 7779 df-2nd 7780 df-supp 7924 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-er 8411 df-map 8530 df-pm 8531 df-ixp 8599 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-fsupp 9010 df-sup 9082 df-oi 9150 df-card 9579 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-nn 11855 df-2 11917 df-3 11918 df-4 11919 df-5 11920 df-6 11921 df-7 11922 df-8 11923 df-9 11924 df-n0 12115 df-z 12201 df-dec 12318 df-uz 12463 df-fz 13120 df-fzo 13263 df-seq 13599 df-hash 13921 df-struct 16724 df-sets 16741 df-slot 16759 df-ndx 16769 df-base 16785 df-ress 16809 df-plusg 16839 df-mulr 16840 df-sca 16842 df-vsca 16843 df-ip 16844 df-tset 16845 df-ple 16846 df-ds 16848 df-hom 16850 df-cco 16851 df-0g 16970 df-gsum 16971 df-prds 16976 df-pws 16978 df-mre 17113 df-mrc 17114 df-acs 17116 df-mgm 18138 df-sgrp 18187 df-mnd 18198 df-mhm 18242 df-submnd 18243 df-grp 18392 df-minusg 18393 df-sbg 18394 df-mulg 18513 df-subg 18564 df-ghm 18644 df-cntz 18735 df-cmn 19196 df-abl 19197 df-mgp 19529 df-ur 19541 df-ring 19588 df-subrg 19822 df-lmod 19925 df-lss 19993 df-sra 20233 df-rgmod 20234 df-dsmm 20718 df-frlm 20733 df-ascl 20841 df-psr 20892 df-mvr 20893 df-mpl 20894 df-opsr 20896 df-psr1 21125 df-vr1 21126 df-ply1 21127 df-mamu 21307 df-mat 21329 df-mat2pmat 21628 |
This theorem is referenced by: chpmatply1 21753 chpmatval2 21754 cpmadurid 21788 cpmadugsumfi 21798 |
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