Theorem scmatmat 22396

Description: An 𝑁 x 𝑁 scalar matrix over (the ring) 𝑅 is an 𝑁 x 𝑁 matrix over (the ring) 𝑅. (Contributed by AV, 18-Dec-2019.)

Hypotheses

Ref	Expression
scmatmat.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
scmatmat.b	⊢ 𝐵 = (Base‘𝐴)
scmatmat.s	⊢ 𝑆 = (𝑁 ScMat 𝑅)

Assertion

Ref	Expression
scmatmat	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐵))

Proof of Theorem scmatmat

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		scmatmat.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		scmatmat.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
4		eqid 2729	. . 3 ⊢ (1r‘𝐴) = (1r‘𝐴)
5		eqid 2729	. . 3 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
6		scmatmat.s	. . 3 ⊢ 𝑆 = (𝑁 ScMat 𝑅)
7	1, 2, 3, 4, 5, 6	scmatel 22392	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ (Base‘𝑅)𝑀 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴)))))
8		simpl 482	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ (Base‘𝑅)𝑀 = (𝑐( ·𝑠 ‘𝐴)(1r‘𝐴))) → 𝑀 ∈ 𝐵)
9	7, 8	biimtrdi 253	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  ‘cfv 6511  (class class class)co 7387  Fincfn 8918  Basecbs 17179   ·_𝑠 cvsca 17224  1_rcur 20090   Mat cmat 22294   ScMat cscmat 22376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-scmat 22378

This theorem is referenced by:  scmatsgrp  22406  scmatcrng  22408