Theorem mat1rhmelval 20286

Description: The value of the ring homomorphism F. (Contributed by AV, 22-Dec-2019.)

Hypotheses

Ref	Expression
mat1rhmval.k	|- K = ( Base ` R )
mat1rhmval.a	|- A = ( { E } Mat R )
mat1rhmval.b	|- B = ( Base ` A )
mat1rhmval.o	|- O = <. E , E >.
mat1rhmval.f	|- F = ( x e. K |-> { <. O , x >. } )

Assertion

Ref	Expression
mat1rhmelval	|- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( E ( F ` X ) E ) = X )

Distinct variable groups:   x, K   x, O   x, E   x, R   x, V   x, X

Allowed substitution hints:   A( x)   B( x)   F( x)

Proof of Theorem mat1rhmelval

Step	Hyp	Ref	Expression
1		df-ov 6653	. 2 |- ( E ( F ` X ) E ) = ( ( F ` X ) ` <. E , E >. )
2		mat1rhmval.k	. . . . 5 |- K = ( Base ` R )
3		mat1rhmval.a	. . . . 5 |- A = ( { E } Mat R )
4		mat1rhmval.b	. . . . 5 |- B = ( Base ` A )
5		mat1rhmval.o	. . . . 5 |- O = <. E , E >.
6		mat1rhmval.f	. . . . 5 |- F = ( x e. K |-> { <. O , x >. } )
7	2, 3, 4, 5, 6	mat1rhmval 20285	. . . 4 |- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( F ` X ) = { <. O , X >. } )
8	7	fveq1d 6193	. . 3 |- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( ( F ` X ) ` <. E , E >. ) = ( { <. O , X >. } ` <. E , E >. ) )
9	5	eqcomi 2631	. . . . 5 |- <. E , E >. = O
10	9	fveq2i 6194	. . . 4 |- ( { <. O , X >. } ` <. E , E >. ) = ( { <. O , X >. } ` O )
11		opex 4932	. . . . . 6 |- <. E , E >. e. _V
12	5, 11	eqeltri 2697	. . . . 5 |- O e. _V
13		simp3 1063	. . . . 5 |- ( ( R e. Ring /\ E e. V /\ X e. K ) -> X e. K )
14		fvsng 6447	. . . . 5 |- ( ( O e. _V /\ X e. K ) -> ( { <. O , X >. } ` O ) = X )
15	12, 13, 14	sylancr 695	. . . 4 |- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( { <. O , X >. } ` O ) = X )
16	10, 15	syl5eq 2668	. . 3 |- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( { <. O , X >. } ` <. E , E >. ) = X )
17	8, 16	eqtrd 2656	. 2 |- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( ( F ` X ) ` <. E , E >. ) = X )
18	1, 17	syl5eq 2668	1 |- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( E ( F ` X ) E ) = X )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Ringcrg 18547   Mat cmat 20213

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-mpt 4730  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

This theorem is referenced by:  mat1ghm  20289  mat1mhm  20290