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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 4340 | . 2 ⊢ (𝑋 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
| 2 | matrcl.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
| 3 | df-mat 22412 | . . . . . 6 ⊢ Mat = (𝑎 ∈ Fin, 𝑏 ∈ V ↦ ((𝑏 freeLMod (𝑎 × 𝑎)) sSet 〈(.r‘ndx), (𝑏 maMul 〈𝑎, 𝑎, 𝑎〉)〉)) | |
| 4 | 3 | mpondm0 7673 | . . . . 5 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅) |
| 5 | 2, 4 | eqtrid 2789 | . . . 4 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐴 = ∅) |
| 6 | 5 | fveq2d 6910 | . . 3 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐴) = (Base‘∅)) |
| 7 | matrcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
| 8 | base0 17252 | . . 3 ⊢ ∅ = (Base‘∅) | |
| 9 | 6, 7, 8 | 3eqtr4g 2802 | . 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 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 〈cotp 4634 × cxp 5683 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 sSet csts 17200 ndxcnx 17230 Basecbs 17247 .rcmulr 17298 freeLMod cfrlm 21766 maMul cmmul 22394 Mat cmat 22411 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-slot 17219 df-ndx 17231 df-base 17248 df-mat 22412 |
| This theorem is referenced by: matbas2i 22428 matecl 22431 matplusg2 22433 matvsca2 22434 matplusgcell 22439 matsubgcell 22440 matinvgcell 22441 matvscacell 22442 matmulcell 22451 mattposcl 22459 mattposvs 22461 mattposm 22465 matgsumcl 22466 madetsumid 22467 madetsmelbas 22470 madetsmelbas2 22471 marrepval0 22567 marrepval 22568 marrepcl 22570 marepvval0 22572 marepvval 22573 marepvcl 22575 ma1repveval 22577 mulmarep1gsum1 22579 mulmarep1gsum2 22580 submabas 22584 submaval0 22586 submaval 22587 mdetleib2 22594 mdetf 22601 mdetrlin 22608 mdetrsca 22609 mdetralt 22614 mdetmul 22629 maduval 22644 maducoeval2 22646 maduf 22647 madutpos 22648 madugsum 22649 madurid 22650 madulid 22651 minmar1val0 22653 minmar1val 22654 marep01ma 22666 smadiadetlem0 22667 smadiadetlem1a 22669 smadiadetlem3 22674 smadiadetlem4 22675 smadiadet 22676 smadiadetglem2 22678 matinv 22683 matunit 22684 slesolvec 22685 slesolinv 22686 slesolinvbi 22687 slesolex 22688 cramerimplem2 22690 cramerimplem3 22691 cramerimp 22692 decpmatcl 22773 decpmataa0 22774 decpmatmul 22778 smatcl 33801 matunitlindflem2 37624 matunitlindf 37625 |
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