Theorem dmatmat 22620

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 = (0g‘𝑅)
dmatval.d	⊢ 𝐷 = (𝑁 DMat 𝑅)

Assertion

Ref	Expression
dmatmat	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 → 𝑀 ∈ 𝐵))

Proof of Theorem dmatmat

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmatval.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		dmatval.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
3		dmatval.0	. . 3 ⊢ 0 = (0g‘𝑅)
4		dmatval.d	. . 3 ⊢ 𝐷 = (𝑁 DMat 𝑅)
5	1, 2, 3, 4	dmatel 22619	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ))))
6		simpl 487	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) → 𝑀 ∈ 𝐵)
7	5, 6	biimtrdi 256	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 → 𝑀 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ‘cfv 6537  (class class class)co 7411  Fincfn 8943  Basecbs 17269  0_gc0g 17492   Mat cmat 22533   DMat cdmat 22614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-dmat 22616

This theorem is referenced by:  dmatmul  22623  dmatsubcl  22624  dmatsgrp  22625  dmatmulcl  22626  dmatcrng  22628  dmatscmcl  22629