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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 21297 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring) |
5 | ringgrp 19295 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Grp) |
7 | 6 | 3adant3 1129 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp) |
8 | 2 | ply1ring 20877 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
9 | 8 | anim2i 619 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
10 | 9 | 3adant3 1129 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring)) |
11 | chmatcl.x | . . . . . 6 ⊢ 𝑋 = (var1‘𝑅) | |
12 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
13 | 11, 2, 12 | vr1cl 20846 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
14 | 13 | 3ad2ant2 1131 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
15 | 4 | 3adant3 1129 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring) |
16 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
17 | chmatcl.1 | . . . . . 6 ⊢ 1 = (1r‘𝑌) | |
18 | 16, 17 | ringidcl 19314 | . . . . 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 21036 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ (𝑋 ∈ (Base‘𝑃) ∧ 1 ∈ (Base‘𝑌))) → (𝑋 · 1 ) ∈ (Base‘𝑌)) |
22 | 10, 14, 19, 21 | syl12anc 835 | . . 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 21331 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌)) |
27 | chmatcl.s | . . . 4 ⊢ − = (-g‘𝑌) | |
28 | 16, 27 | grpsubcl 18171 | . . 3 ⊢ ((𝑌 ∈ Grp ∧ (𝑋 · 1 ) ∈ (Base‘𝑌) ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → ((𝑋 · 1 ) − (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
29 | 7, 22, 26, 28 | syl3anc 1368 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 ) − (𝑇‘𝑀)) ∈ (Base‘𝑌)) |
30 | 1, 29 | eqeltrid 2894 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐻 ∈ (Base‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 Basecbs 16475 ·𝑠 cvsca 16561 Grpcgrp 18095 -gcsg 18097 1rcur 19244 Ringcrg 19290 var1cv1 20805 Poly1cpl1 20806 Mat cmat 21012 matToPolyMat cmat2pmat 21309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-subrg 19526 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-dsmm 20421 df-frlm 20436 df-ascl 20544 df-psr 20594 df-mvr 20595 df-mpl 20596 df-opsr 20598 df-psr1 20809 df-vr1 20810 df-ply1 20811 df-mamu 20991 df-mat 21013 df-mat2pmat 21312 |
This theorem is referenced by: chpmatply1 21437 chpmatval2 21438 cpmadurid 21472 cpmadugsumfi 21482 |
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