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Mirrors > Home > MPE Home > Th. List > matsca2 | Structured version Visualization version GIF version |
Description: The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
matsca2.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
Ref | Expression |
---|---|
matsca2 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 = (Scalar‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpfi 8778 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin) | |
2 | 1 | anidms 567 | . . 3 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin) |
3 | eqid 2821 | . . . . 5 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) = (𝑅 freeLMod (𝑁 × 𝑁)) | |
4 | 3 | frlmsca 20827 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 × 𝑁) ∈ Fin) → 𝑅 = (Scalar‘(𝑅 freeLMod (𝑁 × 𝑁)))) |
5 | 4 | ancoms 459 | . . 3 ⊢ (((𝑁 × 𝑁) ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 = (Scalar‘(𝑅 freeLMod (𝑁 × 𝑁)))) |
6 | 2, 5 | sylan 580 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 = (Scalar‘(𝑅 freeLMod (𝑁 × 𝑁)))) |
7 | matsca2.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
8 | 7, 3 | matsca 20954 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Scalar‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Scalar‘𝐴)) |
9 | 6, 8 | eqtrd 2856 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 = (Scalar‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 × cxp 5547 ‘cfv 6349 (class class class)co 7145 Fincfn 8498 Scalarcsca 16558 freeLMod cfrlm 20820 Mat cmat 20946 |
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 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 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 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-ot 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-map 8398 df-ixp 8451 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-fz 12883 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-sca 16571 df-vsca 16572 df-ip 16573 df-tset 16574 df-ple 16575 df-ds 16577 df-hom 16579 df-cco 16580 df-prds 16711 df-pws 16713 df-sra 19875 df-rgmod 19876 df-dsmm 20806 df-frlm 20821 df-mat 20947 |
This theorem is referenced by: matvscl 20970 matassa 20983 mat0dimscm 21008 scmatid 21053 scmataddcl 21055 scmatsubcl 21056 smatvscl 21063 scmatlss 21064 scmatghm 21072 scmatmhm 21073 matinv 21216 pmatcollpwfi 21320 pmatcollpw3fi1lem1 21324 pm2mp 21363 chpmat1dlem 21373 chpmat1d 21374 chpdmatlem0 21375 chfacfscmulcl 21395 chfacfscmul0 21396 chfacfscmulgsum 21398 cpmidpmatlem3 21410 cpmadugsumlemB 21412 cpmadugsumlemC 21413 cpmadugsumlemF 21414 cpmadugsumfi 21415 cpmidgsum2 21417 cayhamlem2 21422 chcoeffeqlem 21423 matdim 30913 |
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