Theorem scmatmat 20315

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 e. Fin /\ R e. V ) -> ( M e. S -> M e. B ) )

Proof of Theorem scmatmat

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` R ) = ( Base ` R )
2		scmatmat.a	. . 3 |- A = ( N Mat R )
3		scmatmat.b	. . 3 |- B = ( Base ` A )
4		eqid 2622	. . 3 |- ( 1r ` A ) = ( 1r ` A )
5		eqid 2622	. . 3 |- ( .s ` A ) = ( .s ` A )
6		scmatmat.s	. . 3 |- S = ( N ScMat R )
7	1, 2, 3, 4, 5, 6	scmatel 20311	. 2 |- ( ( N e. Fin /\ R e. V ) -> ( M e. S <-> ( M e. B /\ E. c e. ( Base ` R ) M = ( c ( .s ` A ) ( 1r ` A ) ) ) ) )
8		simpl 473	. 2 |- ( ( M e. B /\ E. c e. ( Base ` R ) M = ( c ( .s ` A ) ( 1r ` A ) ) ) -> M e. B )
9	7, 8	syl6bi 243	1 |- ( ( N e. Fin /\ R e. V ) -> ( M e. S -> M e. B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   .scvsca 15945   1rcur 18501   Mat cmat 20213   ScMat cscmat 20295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-scmat 20297

This theorem is referenced by:  scmatsgrp  20325  scmatcrng  20327