MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmatmat Structured version   Visualization version   GIF version

Theorem dmatmat 21987
Description: An 𝑁 x 𝑁 diagonal matrix over (the ring) 𝑅 is an 𝑁 x 𝑁 matrix over (the ring) 𝑅. (Contributed by AV, 18-Dec-2019.)
Hypotheses
Ref Expression
dmatval.a 𝐴 = (𝑁 Mat 𝑅)
dmatval.b 𝐵 = (Base‘𝐴)
dmatval.0 0 = (0g𝑅)
dmatval.d 𝐷 = (𝑁 DMat 𝑅)
Assertion
Ref Expression
dmatmat ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷𝑀𝐵))

Proof of Theorem dmatmat
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmatval.a . . 3 𝐴 = (𝑁 Mat 𝑅)
2 dmatval.b . . 3 𝐵 = (Base‘𝐴)
3 dmatval.0 . . 3 0 = (0g𝑅)
4 dmatval.d . . 3 𝐷 = (𝑁 DMat 𝑅)
51, 2, 3, 4dmatel 21986 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷 ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ))))
6 simpl 483 . 2 ((𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 )) → 𝑀𝐵)
75, 6syl6bi 252 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  cfv 6540  (class class class)co 7405  Fincfn 8935  Basecbs 17140  0gc0g 17381   Mat cmat 21898   DMat cdmat 21981
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 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-dmat 21983
This theorem is referenced by:  dmatmul  21990  dmatsubcl  21991  dmatsgrp  21992  dmatmulcl  21993  dmatcrng  21995  dmatscmcl  21996
  Copyright terms: Public domain W3C validator