Theorem uvcfval 20123

Description: Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem uvcfval

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

Step	Hyp	Ref	Expression
1		uvcfval.u	. 2 |- U = ( R unitVec I )
2		elex 3212	. . 3 |- ( R e. V -> R e. _V )
3		elex 3212	. . 3 |- ( I e. W -> I e. _V )
4		df-uvc 20122	. . . . 5 |- unitVec = ( r e. _V , i e. _V |-> ( j e. i |-> ( k e. i |-> if ( k = j , ( 1r ` r ) , ( 0g ` r ) ) ) ) )
5	4	a1i 11	. . . 4 |- ( ( R e. _V /\ I e. _V ) -> unitVec = ( r e. _V , i e. _V |-> ( j e. i |-> ( k e. i |-> if ( k = j , ( 1r ` r ) , ( 0g ` r ) ) ) ) ) )
6		simpr 477	. . . . . 6 |- ( ( r = R /\ i = I ) -> i = I )
7		fveq2 6191	. . . . . . . . . 10 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
8		uvcfval.o	. . . . . . . . . 10 |- .1. = ( 1r ` R )
9	7, 8	syl6eqr 2674	. . . . . . . . 9 |- ( r = R -> ( 1r ` r ) = .1. )
10		fveq2 6191	. . . . . . . . . 10 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
11		uvcfval.z	. . . . . . . . . 10 |- .0. = ( 0g ` R )
12	10, 11	syl6eqr 2674	. . . . . . . . 9 |- ( r = R -> ( 0g ` r ) = .0. )
13	9, 12	ifeq12d 4106	. . . . . . . 8 |- ( r = R -> if ( k = j , ( 1r ` r ) , ( 0g ` r ) ) = if ( k = j , .1. , .0. ) )
14	13	adantr 481	. . . . . . 7 |- ( ( r = R /\ i = I ) -> if ( k = j , ( 1r ` r ) , ( 0g ` r ) ) = if ( k = j , .1. , .0. ) )
15	6, 14	mpteq12dv 4733	. . . . . 6 |- ( ( r = R /\ i = I ) -> ( k e. i |-> if ( k = j , ( 1r ` r ) , ( 0g ` r ) ) ) = ( k e. I |-> if ( k = j , .1. , .0. ) ) )
16	6, 15	mpteq12dv 4733	. . . . 5 |- ( ( r = R /\ i = I ) -> ( j e. i |-> ( k e. i |-> if ( k = j , ( 1r ` r ) , ( 0g ` r ) ) ) ) = ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) )
17	16	adantl 482	. . . 4 |- ( ( ( R e. _V /\ I e. _V ) /\ ( r = R /\ i = I ) ) -> ( j e. i |-> ( k e. i |-> if ( k = j , ( 1r ` r ) , ( 0g ` r ) ) ) ) = ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) )
18		simpl 473	. . . 4 |- ( ( R e. _V /\ I e. _V ) -> R e. _V )
19		simpr 477	. . . 4 |- ( ( R e. _V /\ I e. _V ) -> I e. _V )
20		mptexg 6484	. . . . 5 |- ( I e. _V -> ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) e. _V )
21	20	adantl 482	. . . 4 |- ( ( R e. _V /\ I e. _V ) -> ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) e. _V )
22	5, 17, 18, 19, 21	ovmpt2d 6788	. . 3 |- ( ( R e. _V /\ I e. _V ) -> ( R unitVec I ) = ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) )
23	2, 3, 22	syl2an 494	. 2 |- ( ( R e. V /\ I e. W ) -> ( R unitVec I ) = ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) )
24	1, 23	syl5eq 2668	1 |- ( ( R e. V /\ I e. W ) -> U = ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   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:  uvcval  20124  uvcff  20130