Theorem d1mat2pmat 20544

Description: The transformation of a matrix of dimenson 1. (Contributed by AV, 4-Aug-2019.)

Hypotheses

Ref	Expression
d1mat2pmat.t	|- T = ( N matToPolyMat R )
d1mat2pmat.b	|- B = ( Base ` ( N Mat R ) )
d1mat2pmat.p	|- P = (Poly1 ` R )
d1mat2pmat.s	|- S = (algSc ` P )

Assertion

Ref	Expression
d1mat2pmat	|- ( ( R e. V /\ ( N = { A } /\ A e. V ) /\ M e. B ) -> ( T ` M ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } )

Proof of Theorem d1mat2pmat

Dummy variables i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snfi 8038	. . . . . 6 |- { A } e. Fin
2		eleq1 2689	. . . . . 6 |- ( N = { A } -> ( N e. Fin <-> { A } e. Fin ) )
3	1, 2	mpbiri 248	. . . . 5 |- ( N = { A } -> N e. Fin )
4	3	adantr 481	. . . 4 |- ( ( N = { A } /\ A e. V ) -> N e. Fin )
5	4	3ad2ant2 1083	. . 3 |- ( ( R e. V /\ ( N = { A } /\ A e. V ) /\ M e. B ) -> N e. Fin )
6		simp1 1061	. . 3 |- ( ( R e. V /\ ( N = { A } /\ A e. V ) /\ M e. B ) -> R e. V )
7		simp3 1063	. . 3 |- ( ( R e. V /\ ( N = { A } /\ A e. V ) /\ M e. B ) -> M e. B )
8		d1mat2pmat.t	. . . 4 |- T = ( N matToPolyMat R )
9		eqid 2622	. . . 4 |- ( N Mat R ) = ( N Mat R )
10		d1mat2pmat.b	. . . 4 |- B = ( Base ` ( N Mat R ) )
11		d1mat2pmat.p	. . . 4 |- P = (Poly1 ` R )
12		d1mat2pmat.s	. . . 4 |- S = (algSc ` P )
13	8, 9, 10, 11, 12	mat2pmatval 20529	. . 3 |- ( ( N e. Fin /\ R e. V /\ M e. B ) -> ( T ` M ) = ( i e. N , j e. N |-> ( S ` ( i M j ) ) ) )
14	5, 6, 7, 13	syl3anc 1326	. 2 |- ( ( R e. V /\ ( N = { A } /\ A e. V ) /\ M e. B ) -> ( T ` M ) = ( i e. N , j e. N |-> ( S ` ( i M j ) ) ) )
15		id 22	. . . . . . 7 |- ( A e. V -> A e. V )
16		fvexd 6203	. . . . . . 7 |- ( A e. V -> ( S ` ( A M A ) ) e. _V )
17	15, 15, 16	3jca 1242	. . . . . 6 |- ( A e. V -> ( A e. V /\ A e. V /\ ( S ` ( A M A ) ) e. _V ) )
18	17	adantl 482	. . . . 5 |- ( ( N = { A } /\ A e. V ) -> ( A e. V /\ A e. V /\ ( S ` ( A M A ) ) e. _V ) )
19	18	3ad2ant2 1083	. . . 4 |- ( ( R e. V /\ ( N = { A } /\ A e. V ) /\ M e. B ) -> ( A e. V /\ A e. V /\ ( S ` ( A M A ) ) e. _V ) )
20		eqid 2622	. . . . 5 |- ( i e. { A } , j e. { A } |-> ( S ` ( i M j ) ) ) = ( i e. { A } , j e. { A } |-> ( S ` ( i M j ) ) )
21		oveq1 6657	. . . . . 6 |- ( i = A -> ( i M j ) = ( A M j ) )
22	21	fveq2d 6195	. . . . 5 |- ( i = A -> ( S ` ( i M j ) ) = ( S ` ( A M j ) ) )
23		oveq2 6658	. . . . . 6 |- ( j = A -> ( A M j ) = ( A M A ) )
24	23	fveq2d 6195	. . . . 5 |- ( j = A -> ( S ` ( A M j ) ) = ( S ` ( A M A ) ) )
25	20, 22, 24	mpt2sn 7268	. . . 4 |- ( ( A e. V /\ A e. V /\ ( S ` ( A M A ) ) e. _V ) -> ( i e. { A } , j e. { A } |-> ( S ` ( i M j ) ) ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } )
26	19, 25	syl 17	. . 3 |- ( ( R e. V /\ ( N = { A } /\ A e. V ) /\ M e. B ) -> ( i e. { A } , j e. { A } |-> ( S ` ( i M j ) ) ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } )
27		mpt2eq12 6715	. . . . . . 7 |- ( ( N = { A } /\ N = { A } ) -> ( i e. N , j e. N |-> ( S ` ( i M j ) ) ) = ( i e. { A } , j e. { A } |-> ( S ` ( i M j ) ) ) )
28	27	eqeq1d 2624	. . . . . 6 |- ( ( N = { A } /\ N = { A } ) -> ( ( i e. N , j e. N |-> ( S ` ( i M j ) ) ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } <-> ( i e. { A } , j e. { A } |-> ( S ` ( i M j ) ) ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } ) )
29	28	anidms 677	. . . . 5 |- ( N = { A } -> ( ( i e. N , j e. N |-> ( S ` ( i M j ) ) ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } <-> ( i e. { A } , j e. { A } |-> ( S ` ( i M j ) ) ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } ) )
30	29	adantr 481	. . . 4 |- ( ( N = { A } /\ A e. V ) -> ( ( i e. N , j e. N |-> ( S ` ( i M j ) ) ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } <-> ( i e. { A } , j e. { A } |-> ( S ` ( i M j ) ) ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } ) )
31	30	3ad2ant2 1083	. . 3 |- ( ( R e. V /\ ( N = { A } /\ A e. V ) /\ M e. B ) -> ( ( i e. N , j e. N |-> ( S ` ( i M j ) ) ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } <-> ( i e. { A } , j e. { A } |-> ( S ` ( i M j ) ) ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } ) )
32	26, 31	mpbird 247	. 2 |- ( ( R e. V /\ ( N = { A } /\ A e. V ) /\ M e. B ) -> ( i e. N , j e. N |-> ( S ` ( i M j ) ) ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } )
33	14, 32	eqtrd 2656	1 |- ( ( R e. V /\ ( N = { A } /\ A e. V ) /\ M e. B ) -> ( T ` M ) = { <. <. A , A >. , ( S ` ( A M A ) ) >. } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   Basecbs 15857  algSccascl 19311  Poly₁cpl1 19547   Mat cmat 20213   matToPolyMat cmat2pmat 20509

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-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-1st 7168  df-2nd 7169  df-1o 7560  df-en 7956  df-fin 7959  df-mat2pmat 20512

This theorem is referenced by: (None)