Theorem scmatel 20311

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

Hypotheses

Ref	Expression
scmatval.k	|- K = ( Base ` R )
scmatval.a	|- A = ( N Mat R )
scmatval.b	|- B = ( Base ` A )
scmatval.1	|- .1. = ( 1r ` A )
scmatval.t	|- .x. = ( .s ` A )
scmatval.s	|- S = ( N ScMat R )

Assertion

Ref	Expression
scmatel	|- ( ( N e. Fin /\ R e. V ) -> ( M e. S <-> ( M e. B /\ E. c e. K M = ( c .x. .1. ) ) ) )

Distinct variable groups:   K, c   N, c   R, c   M, c

Allowed substitution hints:   A( c)   B( c)   S( c)   .x. ( c)   .1. ( c)   V( c)

Proof of Theorem scmatel

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		scmatval.k	. . . 4 |- K = ( Base ` R )
2		scmatval.a	. . . 4 |- A = ( N Mat R )
3		scmatval.b	. . . 4 |- B = ( Base ` A )
4		scmatval.1	. . . 4 |- .1. = ( 1r ` A )
5		scmatval.t	. . . 4 |- .x. = ( .s ` A )
6		scmatval.s	. . . 4 |- S = ( N ScMat R )
7	1, 2, 3, 4, 5, 6	scmatval 20310	. . 3 |- ( ( N e. Fin /\ R e. V ) -> S = { m e. B | E. c e. K m = ( c .x. .1. ) } )
8	7	eleq2d 2687	. 2 |- ( ( N e. Fin /\ R e. V ) -> ( M e. S <-> M e. { m e. B | E. c e. K m = ( c .x. .1. ) } ) )
9		eqeq1 2626	. . . 4 |- ( m = M -> ( m = ( c .x. .1. ) <-> M = ( c .x. .1. ) ) )
10	9	rexbidv 3052	. . 3 |- ( m = M -> ( E. c e. K m = ( c .x. .1. ) <-> E. c e. K M = ( c .x. .1. ) ) )
11	10	elrab 3363	. 2 |- ( M e. { m e. B | E. c e. K m = ( c .x. .1. ) } <-> ( M e. B /\ E. c e. K M = ( c .x. .1. ) ) )
12	8, 11	syl6bb 276	1 |- ( ( N e. Fin /\ R e. V ) -> ( M e. S <-> ( M e. B /\ E. c e. K M = ( c .x. .1. ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   ` 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:  scmatscmid  20312  scmatmat  20315  scmatid  20320  scmataddcl  20322  scmatsubcl  20323  scmatmulcl  20324  smatvscl  20330  scmatrhmcl  20334  mat0scmat  20344  mat1scmat  20345  chmaidscmat  20653