![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4293 | . 2 ⊢ (𝑋 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
2 | matrcl.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | df-mat 21755 | . . . . . 6 ⊢ Mat = (𝑎 ∈ Fin, 𝑏 ∈ V ↦ ((𝑏 freeLMod (𝑎 × 𝑎)) sSet 〈(.r‘ndx), (𝑏 maMul 〈𝑎, 𝑎, 𝑎〉)〉)) | |
4 | 3 | mpondm0 7594 | . . . . 5 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅) |
5 | 2, 4 | eqtrid 2788 | . . . 4 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐴 = ∅) |
6 | 5 | fveq2d 6846 | . . 3 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅)) |
7 | matrcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
8 | base0 17088 | . . 3 ⊢ ∅ = (Base‘∅) | |
9 | 6, 7, 8 | 3eqtr4g 2801 | . 2 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐵 = ∅) |
10 | 1, 9 | nsyl2 141 | 1 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ∅c0 4282 〈cop 4592 〈cotp 4594 × cxp 5631 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 sSet csts 17035 ndxcnx 17065 Basecbs 17083 .rcmulr 17134 freeLMod cfrlm 21152 maMul cmmul 21732 Mat cmat 21754 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-1cn 11109 ax-addcl 11111 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-nn 12154 df-slot 17054 df-ndx 17066 df-base 17084 df-mat 21755 |
This theorem is referenced by: matbas2i 21771 matecl 21774 matplusg2 21776 matvsca2 21777 matplusgcell 21782 matsubgcell 21783 matinvgcell 21784 matvscacell 21785 matmulcell 21794 mattposcl 21802 mattposvs 21804 mattposm 21808 matgsumcl 21809 madetsumid 21810 madetsmelbas 21813 madetsmelbas2 21814 marrepval0 21910 marrepval 21911 marrepcl 21913 marepvval0 21915 marepvval 21916 marepvcl 21918 ma1repveval 21920 mulmarep1gsum1 21922 mulmarep1gsum2 21923 submabas 21927 submaval0 21929 submaval 21930 mdetleib2 21937 mdetf 21944 mdetrlin 21951 mdetrsca 21952 mdetralt 21957 mdetmul 21972 maduval 21987 maducoeval2 21989 maduf 21990 madutpos 21991 madugsum 21992 madurid 21993 madulid 21994 minmar1val0 21996 minmar1val 21997 marep01ma 22009 smadiadetlem0 22010 smadiadetlem1a 22012 smadiadetlem3 22017 smadiadetlem4 22018 smadiadet 22019 smadiadetglem2 22021 matinv 22026 matunit 22027 slesolvec 22028 slesolinv 22029 slesolinvbi 22030 slesolex 22031 cramerimplem2 22033 cramerimplem3 22034 cramerimp 22035 decpmatcl 22116 decpmataa0 22117 decpmatmul 22121 smatcl 32383 matunitlindflem2 36075 matunitlindf 36076 |
Copyright terms: Public domain | W3C validator |