Theorem dmatALTbas 42190

Description: The base set of the algebra of N x N diagonal matrices over a ring R, i.e. the set of all N x N diagonal matrices over the ring R. (Contributed by AV, 8-Dec-2019.)

Hypotheses

Ref	Expression
dmatALTval.a	|- A = ( N Mat R )
dmatALTval.b	|- B = ( Base ` A )
dmatALTval.0	|- .0. = ( 0g ` R )
dmatALTval.d	|- D = ( N DMatALT R )

Assertion

Ref	Expression
dmatALTbas	|- ( ( N e. Fin /\ R e. _V ) -> ( Base ` D ) = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )

Distinct variable groups:   B, m   i, N, j, m   R, i, j, m

Allowed substitution hints:   A( i, j, m)   B( i, j)   D( i, j, m)   .0. ( i, j, m)

Proof of Theorem dmatALTbas

Step	Hyp	Ref	Expression
1		dmatALTval.a	. . . 4 |- A = ( N Mat R )
2		dmatALTval.b	. . . 4 |- B = ( Base ` A )
3		dmatALTval.0	. . . 4 |- .0. = ( 0g ` R )
4		dmatALTval.d	. . . 4 |- D = ( N DMatALT R )
5	1, 2, 3, 4	dmatALTval 42189	. . 3 |- ( ( N e. Fin /\ R e. _V ) -> D = ( A ↾s { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } ) )
6	5	fveq2d 6195	. 2 |- ( ( N e. Fin /\ R e. _V ) -> ( Base ` D ) = ( Base ` ( A ↾s { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } ) ) )
7		fvex 6201	. . . . 5 |- ( Base ` A ) e. _V
8	2, 7	eqeltri 2697	. . . 4 |- B e. _V
9		rabexg 4812	. . . 4 |- ( B e. _V -> { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } e. _V )
10	8, 9	mp1i 13	. . 3 |- ( ( N e. Fin /\ R e. _V ) -> { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } e. _V )
11		eqid 2622	. . . 4 |- ( A ↾s { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } ) = ( A ↾s { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )
12	11, 2	ressbas 15930	. . 3 |- ( { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } e. _V -> ( { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } i^i B ) = ( Base ` ( A ↾s { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } ) ) )
13	10, 12	syl 17	. 2 |- ( ( N e. Fin /\ R e. _V ) -> ( { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } i^i B ) = ( Base ` ( A ↾s { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } ) ) )
14		inrab2 3900	. . 3 |- ( { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } i^i B ) = { m e. ( B i^i B ) | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) }
15		inidm 3822	. . . 4 |- ( B i^i B ) = B
16		rabeq 3192	. . . 4 |- ( ( B i^i B ) = B -> { m e. ( B i^i B ) | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )
17	15, 16	mp1i 13	. . 3 |- ( ( N e. Fin /\ R e. _V ) -> { m e. ( B i^i B ) | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )
18	14, 17	syl5eq 2668	. 2 |- ( ( N e. Fin /\ R e. _V ) -> ( { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } i^i B ) = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )
19	6, 13, 18	3eqtr2d 2662	1 |- ( ( N e. Fin /\ R e. _V ) -> ( Base ` D ) = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   i^i cin 3573   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   ↾_s cress 15858   0gc0g 16100   Mat cmat 20213   DMatALT cdmatalt 42185

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-dmatalt 42187

This theorem is referenced by:  dmatALTbasel  42191  dmatbas  42192