Theorem marepveval 20374

Description: An entry of a matrix with a replaced column. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)

Hypotheses

Ref	Expression
marepvfval.a	|- A = ( N Mat R )
marepvfval.b	|- B = ( Base ` A )
marepvfval.q	|- Q = ( N matRepV R )
marepvfval.v	|- V = ( ( Base ` R ) ^m N )

Assertion

Ref	Expression
marepveval	|- ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> ( I ( ( M Q C ) ` K ) J ) = if ( J = K , ( C ` I ) , ( I M J ) ) )

Proof of Theorem marepveval

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

Step	Hyp	Ref	Expression
1		marepvfval.a	. . . 4 |- A = ( N Mat R )
2		marepvfval.b	. . . 4 |- B = ( Base ` A )
3		marepvfval.q	. . . 4 |- Q = ( N matRepV R )
4		marepvfval.v	. . . 4 |- V = ( ( Base ` R ) ^m N )
5	1, 2, 3, 4	marepvval 20373	. . 3 |- ( ( M e. B /\ C e. V /\ K e. N ) -> ( ( M Q C ) ` K ) = ( i e. N , j e. N |-> if ( j = K , ( C ` i ) , ( i M j ) ) ) )
6	5	adantr 481	. 2 |- ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> ( ( M Q C ) ` K ) = ( i e. N , j e. N |-> if ( j = K , ( C ` i ) , ( i M j ) ) ) )
7		simprl 794	. . 3 |- ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> I e. N )
8		simplrr 801	. . 3 |- ( ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) /\ i = I ) -> J e. N )
9		fvexd 6203	. . . . 5 |- ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> ( C ` i ) e. _V )
10		ovexd 6680	. . . . 5 |- ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> ( i M j ) e. _V )
11	9, 10	ifcld 4131	. . . 4 |- ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> if ( j = K , ( C ` i ) , ( i M j ) ) e. _V )
12	11	adantr 481	. . 3 |- ( ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) /\ ( i = I /\ j = J ) ) -> if ( j = K , ( C ` i ) , ( i M j ) ) e. _V )
13		eqeq1 2626	. . . . . 6 |- ( j = J -> ( j = K <-> J = K ) )
14	13	adantl 482	. . . . 5 |- ( ( i = I /\ j = J ) -> ( j = K <-> J = K ) )
15		fveq2 6191	. . . . . 6 |- ( i = I -> ( C ` i ) = ( C ` I ) )
16	15	adantr 481	. . . . 5 |- ( ( i = I /\ j = J ) -> ( C ` i ) = ( C ` I ) )
17		oveq12 6659	. . . . 5 |- ( ( i = I /\ j = J ) -> ( i M j ) = ( I M J ) )
18	14, 16, 17	ifbieq12d 4113	. . . 4 |- ( ( i = I /\ j = J ) -> if ( j = K , ( C ` i ) , ( i M j ) ) = if ( J = K , ( C ` I ) , ( I M J ) ) )
19	18	adantl 482	. . 3 |- ( ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) /\ ( i = I /\ j = J ) ) -> if ( j = K , ( C ` i ) , ( i M j ) ) = if ( J = K , ( C ` I ) , ( I M J ) ) )
20	7, 8, 12, 19	ovmpt2dv2 6794	. 2 |- ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> ( ( ( M Q C ) ` K ) = ( i e. N , j e. N |-> if ( j = K , ( C ` i ) , ( i M j ) ) ) -> ( I ( ( M Q C ) ` K ) J ) = if ( J = K , ( C ` I ) , ( I M J ) ) ) )
21	6, 20	mpd 15	1 |- ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> ( I ( ( M Q C ) ` K ) J ) = if ( J = K , ( C ` I ) , ( I M J ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   Basecbs 15857   Mat cmat 20213   matRepV cmatrepV 20363

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-pw 4160  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-1st 7168  df-2nd 7169  df-slot 15861  df-base 15863  df-mat 20214  df-marepv 20365

This theorem is referenced by:  ma1repveval  20377  1marepvsma1  20389