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Mirrors > Home > MPE Home > Th. List > matrcl | Structured version Visualization version GIF version |
Description: Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
matrcl.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
matrcl.b | ⊢ 𝐵 = (Base‘𝐴) |
Ref | Expression |
---|---|
matrcl | ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4296 | . 2 ⊢ (𝑋 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
2 | matrcl.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | df-mat 20945 | . . . . . 6 ⊢ Mat = (𝑎 ∈ Fin, 𝑏 ∈ V ↦ ((𝑏 freeLMod (𝑎 × 𝑎)) sSet 〈(.r‘ndx), (𝑏 maMul 〈𝑎, 𝑎, 𝑎〉)〉)) | |
4 | 3 | mpondm0 7375 | . . . . 5 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅) |
5 | 2, 4 | syl5eq 2865 | . . . 4 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐴 = ∅) |
6 | 5 | fveq2d 6667 | . . 3 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅)) |
7 | matrcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
8 | base0 16524 | . . 3 ⊢ ∅ = (Base‘∅) | |
9 | 6, 7, 8 | 3eqtr4g 2878 | . 2 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐵 = ∅) |
10 | 1, 9 | nsyl2 143 | 1 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 〈cop 4563 〈cotp 4565 × cxp 5546 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 ndxcnx 16468 sSet csts 16469 Basecbs 16471 .rcmulr 16554 freeLMod cfrlm 20818 maMul cmmul 20922 Mat cmat 20944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-slot 16475 df-base 16477 df-mat 20945 |
This theorem is referenced by: matbas2i 20959 matecl 20962 matplusg2 20964 matvsca2 20965 matplusgcell 20970 matsubgcell 20971 matinvgcell 20972 matvscacell 20973 matmulcell 20982 mattposcl 20990 mattposvs 20992 mattposm 20996 matgsumcl 20997 madetsumid 20998 madetsmelbas 21001 madetsmelbas2 21002 marrepval0 21098 marrepval 21099 marrepcl 21101 marepvval0 21103 marepvval 21104 marepvcl 21106 ma1repveval 21108 mulmarep1gsum1 21110 mulmarep1gsum2 21111 submabas 21115 submaval0 21117 submaval 21118 mdetleib2 21125 mdetf 21132 mdetrlin 21139 mdetrsca 21140 mdetralt 21145 mdetmul 21160 maduval 21175 maducoeval2 21177 maduf 21178 madutpos 21179 madugsum 21180 madurid 21181 madulid 21182 minmar1val0 21184 minmar1val 21185 marep01ma 21197 smadiadetlem0 21198 smadiadetlem1a 21200 smadiadetlem3 21205 smadiadetlem4 21206 smadiadet 21207 smadiadetglem2 21209 matinv 21214 matunit 21215 slesolvec 21216 slesolinv 21217 slesolinvbi 21218 slesolex 21219 cramerimplem2 21221 cramerimplem3 21222 cramerimp 21223 decpmatcl 21303 decpmataa0 21304 decpmatmul 21308 smatcl 30966 matunitlindflem2 34770 matunitlindf 34771 |
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