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Theorem dmatmat 22216
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 22215 . 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 6543  (class class class)co 7411  Fincfn 8941  Basecbs 17148  0gc0g 17389   Mat cmat 22127   DMat cdmat 22210
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 5299  ax-nul 5306  ax-pr 5427
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 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-dmat 22212
This theorem is referenced by:  dmatmul  22219  dmatsubcl  22220  dmatsgrp  22221  dmatmulcl  22222  dmatcrng  22224  dmatscmcl  22225
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