Theorem scmatmat 22232

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 2731	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		scmatmat.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		scmatmat.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
4		eqid 2731	. . 3 ⊢ (1r‘𝐴) = (1r‘𝐴)
5		eqid 2731	. . 3 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
6		scmatmat.s	. . 3 ⊢ 𝑆 = (𝑁 ScMat 𝑅)
7	1, 2, 3, 4, 5, 6	scmatel 22228	. 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	syl6bi 253	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∃wrex 3069  ‘cfv 6543  (class class class)co 7412  Fincfn 8943  Basecbs 17149   ·_𝑠 cvsca 17206  1_rcur 20076   Mat cmat 22128   ScMat cscmat 22212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  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 7415  df-oprab 7416  df-mpo 7417  df-scmat 22214

This theorem is referenced by:  scmatsgrp  22242  scmatcrng  22244