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Mirrors > Home > MPE Home > Th. List > scmatscmid | Structured version Visualization version GIF version |
Description: A scalar matrix can be expressed as a multiplication of a scalar with the identity matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 18-Dec-2019.) |
Ref | Expression |
---|---|
scmatval.k | โข ๐พ = (Baseโ๐ ) |
scmatval.a | โข ๐ด = (๐ Mat ๐ ) |
scmatval.b | โข ๐ต = (Baseโ๐ด) |
scmatval.1 | โข 1 = (1rโ๐ด) |
scmatval.t | โข ยท = ( ยท๐ โ๐ด) |
scmatval.s | โข ๐ = (๐ ScMat ๐ ) |
Ref | Expression |
---|---|
scmatscmid | โข ((๐ โ Fin โง ๐ โ ๐ โง ๐ โ ๐) โ โ๐ โ ๐พ ๐ = (๐ ยท 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scmatval.k | . . . 4 โข ๐พ = (Baseโ๐ ) | |
2 | scmatval.a | . . . 4 โข ๐ด = (๐ Mat ๐ ) | |
3 | scmatval.b | . . . 4 โข ๐ต = (Baseโ๐ด) | |
4 | scmatval.1 | . . . 4 โข 1 = (1rโ๐ด) | |
5 | scmatval.t | . . . 4 โข ยท = ( ยท๐ โ๐ด) | |
6 | scmatval.s | . . . 4 โข ๐ = (๐ ScMat ๐ ) | |
7 | 1, 2, 3, 4, 5, 6 | scmatel 21870 | . . 3 โข ((๐ โ Fin โง ๐ โ ๐) โ (๐ โ ๐ โ (๐ โ ๐ต โง โ๐ โ ๐พ ๐ = (๐ ยท 1 )))) |
8 | 7 | simplbda 501 | . 2 โข (((๐ โ Fin โง ๐ โ ๐) โง ๐ โ ๐) โ โ๐ โ ๐พ ๐ = (๐ ยท 1 )) |
9 | 8 | 3impa 1111 | 1 โข ((๐ โ Fin โง ๐ โ ๐ โง ๐ โ ๐) โ โ๐ โ ๐พ ๐ = (๐ ยท 1 )) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 397 โง w3a 1088 = wceq 1542 โ wcel 2107 โwrex 3074 โcfv 6501 (class class class)co 7362 Fincfn 8890 Basecbs 17090 ยท๐ cvsca 17144 1rcur 19920 Mat cmat 21770 ScMat cscmat 21854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-scmat 21856 |
This theorem is referenced by: scmate 21875 scmatscm 21878 scmataddcl 21881 scmatsubcl 21882 scmatfo 21895 |
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