Theorem uvcvval 18339

Description: Value of a unit vector coordinate 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
uvcvval	|- ( ( ( R e. V /\ I e. W /\ J e. I ) /\ K e. I ) -> ( ( U ` J ) ` K ) = if ( K = J , .1. , .0. ) )

Proof of Theorem uvcvval

Dummy variable k 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	uvcval 18338	. . . 4 |- ( ( R e. V /\ I e. W /\ J e. I ) -> ( U ` J ) = ( k e. I |-> if ( k = J , .1. , .0. ) ) )
5	4	fveq1d 5804	. . 3 |- ( ( R e. V /\ I e. W /\ J e. I ) -> ( ( U ` J ) ` K ) = ( ( k e. I |-> if ( k = J , .1. , .0. ) ) ` K ) )
6	5	adantr 465	. 2 |- ( ( ( R e. V /\ I e. W /\ J e. I ) /\ K e. I ) -> ( ( U ` J ) ` K ) = ( ( k e. I |-> if ( k = J , .1. , .0. ) ) ` K ) )
7		simpr 461	. . 3 |- ( ( ( R e. V /\ I e. W /\ J e. I ) /\ K e. I ) -> K e. I )
8		fvex 5812	. . . . 5 |- ( 1r ` R ) e. _V
9	2, 8	eqeltri 2538	. . . 4 |- .1. e. _V
10		fvex 5812	. . . . 5 |- ( 0g ` R ) e. _V
11	3, 10	eqeltri 2538	. . . 4 |- .0. e. _V
12	9, 11	ifex 3969	. . 3 |- if ( K = J , .1. , .0. ) e. _V
13		eqeq1 2458	. . . . 5 |- ( k = K -> ( k = J <-> K = J ) )
14	13	ifbid 3922	. . . 4 |- ( k = K -> if ( k = J , .1. , .0. ) = if ( K = J , .1. , .0. ) )
15		eqid 2454	. . . 4 |- ( k e. I |-> if ( k = J , .1. , .0. ) ) = ( k e. I |-> if ( k = J , .1. , .0. ) )
16	14, 15	fvmptg 5884	. . 3 |- ( ( K e. I /\ if ( K = J , .1. , .0. ) e. _V ) -> ( ( k e. I |-> if ( k = J , .1. , .0. ) ) ` K ) = if ( K = J , .1. , .0. ) )
17	7, 12, 16	sylancl 662	. 2 |- ( ( ( R e. V /\ I e. W /\ J e. I ) /\ K e. I ) -> ( ( k e. I |-> if ( k = J , .1. , .0. ) ) ` K ) = if ( K = J , .1. , .0. ) )
18	6, 17	eqtrd 2495	1 |- ( ( ( R e. V /\ I e. W /\ J e. I ) /\ K e. I ) -> ( ( U ` J ) ` K ) = if ( K = J , .1. , .0. ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   _Vcvv 3078   ifcif 3902   |-> cmpt 4461   ` cfv 5529  (class class class)co 6203   0gc0g 14500   1rcur 16728   unitVec cuvc 18335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-uvc 18336

This theorem is referenced by:  uvcvvcl  18340  uvcvvcl2  18341  uvcvv1  18342  uvcvv0  18343