Theorem uvcvval 18991

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 18990	. . . 4 |- ( ( R e. V /\ I e. W /\ J e. I ) -> ( U ` J ) = ( k e. I |-> if ( k = J , .1. , .0. ) ) )
5	4	fveq1d 5850	. . 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 463	. 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 459	. . 3 |- ( ( ( R e. V /\ I e. W /\ J e. I ) /\ K e. I ) -> K e. I )
8		fvex 5858	. . . . 5 |- ( 1r ` R ) e. _V
9	2, 8	eqeltri 2538	. . . 4 |- .1. e. _V
10		fvex 5858	. . . . 5 |- ( 0g ` R ) e. _V
11	3, 10	eqeltri 2538	. . . 4 |- .0. e. _V
12	9, 11	ifex 3997	. . 3 |- if ( K = J , .1. , .0. ) e. _V
13		eqeq1 2458	. . . . 5 |- ( k = K -> ( k = J <-> K = J ) )
14	13	ifbid 3951	. . . 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 5929	. . 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 660	. 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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   _Vcvv 3106   ifcif 3929   |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   0gc0g 14932   1rcur 17351   unitVec cuvc 18987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-uvc 18988

This theorem is referenced by:  uvcvvcl  18992  uvcvvcl2  18993  uvcvv1  18994  uvcvv0  18995