Theorem uvcval 20124

Description: Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)

Hypotheses

Ref	Expression
uvcfval.u	|- U = ( R unitVec I )
uvcfval.o	|- .1. = ( 1r ` R )
uvcfval.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
uvcval	|- ( ( R e. V /\ I e. W /\ J e. I ) -> ( U ` J ) = ( k e. I |-> if ( k = J , .1. , .0. ) ) )

Distinct variable groups:   .1. , k   R, k   k, I   .0. , k   k, J

Allowed substitution hints:   U( k)   V( k)   W( k)

Proof of Theorem uvcval

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvcfval.u	. . . . 5 |- U = ( R unitVec I )
2		uvcfval.o	. . . . 5 |- .1. = ( 1r ` R )
3		uvcfval.z	. . . . 5 |- .0. = ( 0g ` R )
4	1, 2, 3	uvcfval 20123	. . . 4 |- ( ( R e. V /\ I e. W ) -> U = ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) )
5	4	fveq1d 6193	. . 3 |- ( ( R e. V /\ I e. W ) -> ( U ` J ) = ( ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) ` J ) )
6	5	3adant3 1081	. 2 |- ( ( R e. V /\ I e. W /\ J e. I ) -> ( U ` J ) = ( ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) ` J ) )
7		simp3 1063	. . 3 |- ( ( R e. V /\ I e. W /\ J e. I ) -> J e. I )
8		mptexg 6484	. . . 4 |- ( I e. W -> ( k e. I |-> if ( k = J , .1. , .0. ) ) e. _V )
9	8	3ad2ant2 1083	. . 3 |- ( ( R e. V /\ I e. W /\ J e. I ) -> ( k e. I |-> if ( k = J , .1. , .0. ) ) e. _V )
10		eqeq2 2633	. . . . . 6 |- ( j = J -> ( k = j <-> k = J ) )
11	10	ifbid 4108	. . . . 5 |- ( j = J -> if ( k = j , .1. , .0. ) = if ( k = J , .1. , .0. ) )
12	11	mpteq2dv 4745	. . . 4 |- ( j = J -> ( k e. I |-> if ( k = j , .1. , .0. ) ) = ( k e. I |-> if ( k = J , .1. , .0. ) ) )
13		eqid 2622	. . . 4 |- ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) = ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) )
14	12, 13	fvmptg 6280	. . 3 |- ( ( J e. I /\ ( k e. I |-> if ( k = J , .1. , .0. ) ) e. _V ) -> ( ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) ` J ) = ( k e. I |-> if ( k = J , .1. , .0. ) ) )
15	7, 9, 14	syl2anc 693	. 2 |- ( ( R e. V /\ I e. W /\ J e. I ) -> ( ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) ` J ) = ( k e. I |-> if ( k = J , .1. , .0. ) ) )
16	6, 15	eqtrd 2656	1 |- ( ( R e. V /\ I e. W /\ J e. I ) -> ( U ` J ) = ( k e. I |-> if ( k = J , .1. , .0. ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0gc0g 16100   1rcur 18501   unitVec cuvc 20121

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-rep 4771  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-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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  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-uvc 20122

This theorem is referenced by:  uvcvval  20125