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Theorem dmatmat 21097
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 21096 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷 ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ))))
6 simpl 486 . 2 ((𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 )) → 𝑀𝐵)
75, 6syl6bi 256 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wne 3011  wral 3130  cfv 6334  (class class class)co 7140  Fincfn 8496  Basecbs 16474  0gc0g 16704   Mat cmat 21010   DMat cdmat 21091
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-dmat 21093
This theorem is referenced by:  dmatmul  21100  dmatsubcl  21101  dmatsgrp  21102  dmatmulcl  21103  dmatcrng  21105  dmatscmcl  21106
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