Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1smat1 Structured version   Unicode version

Theorem 1smat1 28626
Description: The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 19595 (Contributed by Thierry Arnoux, 19-Aug-2020.)
Hypotheses
Ref Expression
1smat1.1  |-  .1.  =  ( 1r `  ( ( 1 ... N ) Mat 
R ) )
1smat1.r  |-  ( ph  ->  R  e.  Ring )
1smat1.n  |-  ( ph  ->  N  e.  NN )
1smat1.i  |-  ( ph  ->  I  e.  ( 1 ... N ) )
Assertion
Ref Expression
1smat1  |-  ( ph  ->  ( I (subMat1 `  .1.  ) I )  =  ( 1r `  (
( 1 ... ( N  -  1 ) ) Mat  R ) ) )

Proof of Theorem 1smat1
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2422 . . . . 5  |-  ( I (subMat1 `  .1.  ) I )  =  ( I (subMat1 `  .1.  ) I )
2 1smat1.n . . . . . 6  |-  ( ph  ->  N  e.  NN )
32adantr 466 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  ->  N  e.  NN )
4 1smat1.i . . . . . 6  |-  ( ph  ->  I  e.  ( 1 ... N ) )
54adantr 466 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  ->  I  e.  ( 1 ... N ) )
6 1smat1.r . . . . . . . 8  |-  ( ph  ->  R  e.  Ring )
7 fzfi 12185 . . . . . . . 8  |-  ( 1 ... N )  e. 
Fin
8 eqid 2422 . . . . . . . . 9  |-  ( ( 1 ... N ) Mat 
R )  =  ( ( 1 ... N
) Mat  R )
9 eqid 2422 . . . . . . . . 9  |-  ( Base `  ( ( 1 ... N ) Mat  R ) )  =  ( Base `  ( ( 1 ... N ) Mat  R ) )
10 1smat1.1 . . . . . . . . 9  |-  .1.  =  ( 1r `  ( ( 1 ... N ) Mat 
R ) )
118, 9, 10mat1bas 19461 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
1 ... N )  e. 
Fin )  ->  .1.  e.  ( Base `  (
( 1 ... N
) Mat  R ) ) )
126, 7, 11sylancl 666 . . . . . . 7  |-  ( ph  ->  .1.  e.  ( Base `  ( ( 1 ... N ) Mat  R ) ) )
13 eqid 2422 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
148, 13matbas2 19433 . . . . . . . 8  |-  ( ( ( 1 ... N
)  e.  Fin  /\  R  e.  Ring )  -> 
( ( Base `  R
)  ^m  ( (
1 ... N )  X.  ( 1 ... N
) ) )  =  ( Base `  (
( 1 ... N
) Mat  R ) ) )
157, 6, 14sylancr 667 . . . . . . 7  |-  ( ph  ->  ( ( Base `  R
)  ^m  ( (
1 ... N )  X.  ( 1 ... N
) ) )  =  ( Base `  (
( 1 ... N
) Mat  R ) ) )
1612, 15eleqtrrd 2513 . . . . . 6  |-  ( ph  ->  .1.  e.  ( (
Base `  R )  ^m  ( ( 1 ... N )  X.  (
1 ... N ) ) ) )
1716adantr 466 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  ->  .1.  e.  ( ( Base `  R )  ^m  (
( 1 ... N
)  X.  ( 1 ... N ) ) ) )
18 fz1ssnn 11831 . . . . . 6  |-  ( 1 ... ( N  - 
1 ) )  C_  NN
19 simprl 762 . . . . . 6  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  ( 1 ... ( N  - 
1 ) ) )
2018, 19sseldi 3462 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  NN )
21 simprr 764 . . . . . 6  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  ( 1 ... ( N  - 
1 ) ) )
2218, 21sseldi 3462 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  NN )
23 eqidd 2423 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  ->  if ( i  <  I ,  i ,  ( i  +  1 ) )  =  if ( i  <  I ,  i ,  ( i  +  1 ) ) )
24 eqidd 2423 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  ->  if ( j  <  I ,  j ,  ( j  +  1 ) )  =  if ( j  <  I ,  j ,  ( j  +  1 ) ) )
251, 3, 3, 5, 5, 17, 20, 22, 23, 24smatlem 28619 . . . 4  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( i ( I (subMat1 `  .1.  ) I ) j )  =  ( if ( i  <  I ,  i ,  ( i  +  1 ) )  .1. 
if ( j  < 
I ,  j ,  ( j  +  1 ) ) ) )
26 eqid 2422 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
27 eqid 2422 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
287a1i 11 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( 1 ... N
)  e.  Fin )
296adantr 466 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  ->  R  e.  Ring )
30 nnuz 11195 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
3120, 30syl6eleq 2520 . . . . . . . 8  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  ( ZZ>= ` 
1 ) )
32 fznatpl1 11851 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  i  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( i  +  1 )  e.  ( 1 ... N ) )
333, 19, 32syl2anc 665 . . . . . . . 8  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( i  +  1 )  e.  ( 1 ... N ) )
34 peano2fzr 11813 . . . . . . . 8  |-  ( ( i  e.  ( ZZ>= ` 
1 )  /\  (
i  +  1 )  e.  ( 1 ... N ) )  -> 
i  e.  ( 1 ... N ) )
3531, 33, 34syl2anc 665 . . . . . . 7  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  ( 1 ... N ) )
3635, 33jca 534 . . . . . 6  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( i  e.  ( 1 ... N )  /\  ( i  +  1 )  e.  ( 1 ... N ) ) )
37 eleq1 2494 . . . . . . 7  |-  ( i  =  if ( i  <  I ,  i ,  ( i  +  1 ) )  -> 
( i  e.  ( 1 ... N )  <-> 
if ( i  < 
I ,  i ,  ( i  +  1 ) )  e.  ( 1 ... N ) ) )
38 eleq1 2494 . . . . . . 7  |-  ( ( i  +  1 )  =  if ( i  <  I ,  i ,  ( i  +  1 ) )  -> 
( ( i  +  1 )  e.  ( 1 ... N )  <-> 
if ( i  < 
I ,  i ,  ( i  +  1 ) )  e.  ( 1 ... N ) ) )
3937, 38ifboth 3945 . . . . . 6  |-  ( ( i  e.  ( 1 ... N )  /\  ( i  +  1 )  e.  ( 1 ... N ) )  ->  if ( i  <  I ,  i ,  ( i  +  1 ) )  e.  ( 1 ... N
) )
4036, 39syl 17 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  ->  if ( i  <  I ,  i ,  ( i  +  1 ) )  e.  ( 1 ... N ) )
4122, 30syl6eleq 2520 . . . . . . . 8  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  ( ZZ>= ` 
1 ) )
42 fznatpl1 11851 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  j  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( j  +  1 )  e.  ( 1 ... N ) )
433, 21, 42syl2anc 665 . . . . . . . 8  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( j  +  1 )  e.  ( 1 ... N ) )
44 peano2fzr 11813 . . . . . . . 8  |-  ( ( j  e.  ( ZZ>= ` 
1 )  /\  (
j  +  1 )  e.  ( 1 ... N ) )  -> 
j  e.  ( 1 ... N ) )
4541, 43, 44syl2anc 665 . . . . . . 7  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  ( 1 ... N ) )
4645, 43jca 534 . . . . . 6  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( j  e.  ( 1 ... N )  /\  ( j  +  1 )  e.  ( 1 ... N ) ) )
47 eleq1 2494 . . . . . . 7  |-  ( j  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  -> 
( j  e.  ( 1 ... N )  <-> 
if ( j  < 
I ,  j ,  ( j  +  1 ) )  e.  ( 1 ... N ) ) )
48 eleq1 2494 . . . . . . 7  |-  ( ( j  +  1 )  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  -> 
( ( j  +  1 )  e.  ( 1 ... N )  <-> 
if ( j  < 
I ,  j ,  ( j  +  1 ) )  e.  ( 1 ... N ) ) )
4947, 48ifboth 3945 . . . . . 6  |-  ( ( j  e.  ( 1 ... N )  /\  ( j  +  1 )  e.  ( 1 ... N ) )  ->  if ( j  <  I ,  j ,  ( j  +  1 ) )  e.  ( 1 ... N
) )
5046, 49syl 17 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  ->  if ( j  <  I ,  j ,  ( j  +  1 ) )  e.  ( 1 ... N ) )
518, 26, 27, 28, 29, 40, 50, 10mat1ov 19460 . . . 4  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( if ( i  <  I ,  i ,  ( i  +  1 ) )  .1. 
if ( j  < 
I ,  j ,  ( j  +  1 ) ) )  =  if ( if ( i  <  I ,  i ,  ( i  +  1 ) )  =  if ( j  <  I ,  j ,  ( j  +  1 ) ) ,  ( 1r `  R
) ,  ( 0g
`  R ) ) )
52 simpr 462 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  i  < 
I )  ->  i  <  I )
5352iftrued 3917 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  i  < 
I )  ->  if ( i  <  I ,  i ,  ( i  +  1 ) )  =  i )
5453eqeq1d 2424 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  i  < 
I )  ->  ( if ( i  <  I ,  i ,  ( i  +  1 ) )  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  <-> 
i  =  if ( j  <  I ,  j ,  ( j  +  1 ) ) ) )
55 simpr 462 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  j  <  I )  ->  j  <  I )
5655iftrued 3917 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  j  <  I )  ->  if ( j  <  I ,  j ,  ( j  +  1 ) )  =  j )
5756eqeq2d 2436 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  j  <  I )  ->  (
i  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  <-> 
i  =  j ) )
58 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  -.  j  <  I )
5958iffalsed 3920 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  if ( j  <  I ,  j ,  ( j  +  1 ) )  =  ( j  +  1 ) )
6059eqeq2d 2436 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
( i  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  <->  i  =  ( j  +  1 ) ) )
6120nnred 10625 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  RR )
6261ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
i  e.  RR )
63 fz1ssnn 11831 . . . . . . . . . . . . . . . . 17  |-  ( 1 ... N )  C_  NN
6463, 4sseldi 3462 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  I  e.  NN )
6564nnred 10625 . . . . . . . . . . . . . . 15  |-  ( ph  ->  I  e.  RR )
6665ad3antrrr 734 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  I  e.  RR )
6722nnred 10625 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  RR )
6867ad2antrr 730 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
j  e.  RR )
69 1red 9659 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
1  e.  RR )
7068, 69readdcld 9671 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
( j  +  1 )  e.  RR )
7152adantr 466 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
i  <  I )
7264nnzd 11040 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  I  e.  ZZ )
7372ad3antrrr 734 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  I  e.  ZZ )
7422nnzd 11040 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  ZZ )
7574ad2antrr 730 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
j  e.  ZZ )
7666, 68, 58nltled 9786 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  I  <_  j )
77 zleltp1 10988 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  ZZ  /\  j  e.  ZZ )  ->  ( I  <_  j  <->  I  <  ( j  +  1 ) ) )
7877biimpa 486 . . . . . . . . . . . . . . 15  |-  ( ( ( I  e.  ZZ  /\  j  e.  ZZ )  /\  I  <_  j
)  ->  I  <  ( j  +  1 ) )
7973, 75, 76, 78syl21anc 1263 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  I  <  ( j  +  1 ) )
8062, 66, 70, 71, 79lttrd 9797 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
i  <  ( j  +  1 ) )
8162, 80ltned 9772 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
i  =/=  ( j  +  1 ) )
8281neneqd 2625 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  -.  i  =  (
j  +  1 ) )
8362, 66, 68, 71, 76ltletrd 9796 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
i  <  j )
8462, 83ltned 9772 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
i  =/=  j )
8584neneqd 2625 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  -.  i  =  j
)
8682, 852falsed 352 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
( i  =  ( j  +  1 )  <-> 
i  =  j ) )
8760, 86bitrd 256 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
( i  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  <->  i  =  j ) )
8857, 87pm2.61dan 798 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  i  < 
I )  ->  (
i  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  <-> 
i  =  j ) )
8954, 88bitrd 256 . . . . . . 7  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  i  < 
I )  ->  ( if ( i  <  I ,  i ,  ( i  +  1 ) )  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  <-> 
i  =  j ) )
90 simpr 462 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  -.  i  <  I )  ->  -.  i  <  I )
9190iffalsed 3920 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  -.  i  <  I )  ->  if ( i  <  I ,  i ,  ( i  +  1 ) )  =  ( i  +  1 ) )
9291eqeq1d 2424 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  -.  i  <  I )  ->  ( if ( i  <  I ,  i ,  ( i  +  1 ) )  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  <-> 
( i  +  1 )  =  if ( j  <  I ,  j ,  ( j  +  1 ) ) ) )
93 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
j  <  I )
9493iftrued 3917 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  if ( j  <  I ,  j ,  ( j  +  1 ) )  =  j )
9594eqeq2d 2436 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
( ( i  +  1 )  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  <->  ( i  +  1 )  =  j ) )
9667ad2antrr 730 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
j  e.  RR )
9765ad3antrrr 734 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  I  e.  RR )
9861ad2antrr 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
i  e.  RR )
99 1red 9659 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
1  e.  RR )
10098, 99readdcld 9671 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
( i  +  1 )  e.  RR )
10172ad3antrrr 734 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  I  e.  ZZ )
10220nnzd 11040 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  ZZ )
103102ad2antrr 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
i  e.  ZZ )
10490adantr 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  -.  i  <  I )
10597, 98, 104nltled 9786 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  I  <_  i )
106 zleltp1 10988 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  ZZ  /\  i  e.  ZZ )  ->  ( I  <_  i  <->  I  <  ( i  +  1 ) ) )
107106biimpa 486 . . . . . . . . . . . . . . . 16  |-  ( ( ( I  e.  ZZ  /\  i  e.  ZZ )  /\  I  <_  i
)  ->  I  <  ( i  +  1 ) )
108101, 103, 105, 107syl21anc 1263 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  I  <  ( i  +  1 ) )
10996, 97, 100, 93, 108lttrd 9797 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
j  <  ( i  +  1 ) )
11096, 109ltned 9772 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
j  =/=  ( i  +  1 ) )
111110necomd 2695 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
( i  +  1 )  =/=  j )
112111neneqd 2625 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  -.  ( i  +  1 )  =  j )
11396, 97, 98, 93, 105ltletrd 9796 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
j  <  i )
11496, 113ltned 9772 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
j  =/=  i )
115114necomd 2695 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
i  =/=  j )
116115neneqd 2625 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  -.  i  =  j
)
117112, 1162falsed 352 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
( ( i  +  1 )  =  j  <-> 
i  =  j ) )
11895, 117bitrd 256 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
( ( i  +  1 )  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  <->  i  =  j ) )
119 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  ->  -.  j  <  I )
120119iffalsed 3920 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  ->  if ( j  <  I ,  j ,  ( j  +  1 ) )  =  ( j  +  1 ) )
121120eqeq2d 2436 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  ->  ( ( i  +  1 )  =  if ( j  < 
I ,  j ,  ( j  +  1 ) )  <->  ( i  +  1 )  =  ( j  +  1 ) ) )
12220nncnd 10626 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  CC )
123122ad3antrrr 734 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  /\  ( i  +  1 )  =  ( j  +  1 ) )  ->  i  e.  CC )
12422nncnd 10626 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  CC )
125124ad3antrrr 734 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  /\  ( i  +  1 )  =  ( j  +  1 ) )  ->  j  e.  CC )
126 1cnd 9660 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  /\  ( i  +  1 )  =  ( j  +  1 ) )  ->  1  e.  CC )
127 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  /\  ( i  +  1 )  =  ( j  +  1 ) )  ->  ( i  +  1 )  =  ( j  +  1 ) )
128123, 125, 126, 127addcan2ad 9840 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  /\  ( i  +  1 )  =  ( j  +  1 ) )  ->  i  =  j )
129 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  /\  i  =  j )  ->  i  =  j )
130129oveq1d 6317 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  /\  i  =  j )  ->  ( i  +  1 )  =  ( j  +  1 ) )
131128, 130impbida 840 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  ->  ( ( i  +  1 )  =  ( j  +  1 )  <->  i  =  j ) )
132121, 131bitrd 256 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  ->  ( ( i  +  1 )  =  if ( j  < 
I ,  j ,  ( j  +  1 ) )  <->  i  =  j ) )
133118, 132pm2.61dan 798 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  -.  i  <  I )  ->  (
( i  +  1 )  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  <-> 
i  =  j ) )
13492, 133bitrd 256 . . . . . . 7  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  -.  i  <  I )  ->  ( if ( i  <  I ,  i ,  ( i  +  1 ) )  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  <-> 
i  =  j ) )
13589, 134pm2.61dan 798 . . . . . 6  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( if ( i  <  I ,  i ,  ( i  +  1 ) )  =  if ( j  < 
I ,  j ,  ( j  +  1 ) )  <->  i  =  j ) )
136135ifbid 3931 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  ->  if ( if ( i  <  I ,  i ,  ( i  +  1 ) )  =  if ( j  < 
I ,  j ,  ( j  +  1 ) ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  =  if ( i  =  j ,  ( 1r
`  R ) ,  ( 0g `  R
) ) )
137 eqid 2422 . . . . . 6  |-  ( ( 1 ... ( N  -  1 ) ) Mat 
R )  =  ( ( 1 ... ( N  -  1 ) ) Mat  R )
138 fzfid 12186 . . . . . 6  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( 1 ... ( N  -  1 ) )  e.  Fin )
139 eqid 2422 . . . . . 6  |-  ( 1r
`  ( ( 1 ... ( N  - 
1 ) ) Mat  R
) )  =  ( 1r `  ( ( 1 ... ( N  -  1 ) ) Mat 
R ) )
140137, 26, 27, 138, 29, 19, 21, 139mat1ov 19460 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( i ( 1r
`  ( ( 1 ... ( N  - 
1 ) ) Mat  R
) ) j )  =  if ( i  =  j ,  ( 1r `  R ) ,  ( 0g `  R ) ) )
141136, 140eqtr4d 2466 . . . 4  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  ->  if ( if ( i  <  I ,  i ,  ( i  +  1 ) )  =  if ( j  < 
I ,  j ,  ( j  +  1 ) ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  =  ( i ( 1r
`  ( ( 1 ... ( N  - 
1 ) ) Mat  R
) ) j ) )
14225, 51, 1413eqtrd 2467 . . 3  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( i ( I (subMat1 `  .1.  ) I ) j )  =  ( i ( 1r
`  ( ( 1 ... ( N  - 
1 ) ) Mat  R
) ) j ) )
143142ralrimivva 2846 . 2  |-  ( ph  ->  A. i  e.  ( 1 ... ( N  -  1 ) ) A. j  e.  ( 1 ... ( N  -  1 ) ) ( i ( I (subMat1 `  .1.  ) I ) j )  =  ( i ( 1r
`  ( ( 1 ... ( N  - 
1 ) ) Mat  R
) ) j ) )
1441, 2, 2, 4, 4, 16smatrcl 28618 . . . 4  |-  ( ph  ->  ( I (subMat1 `  .1.  ) I )  e.  ( ( Base `  R
)  ^m  ( (
1 ... ( N  - 
1 ) )  X.  ( 1 ... ( N  -  1 ) ) ) ) )
145 elmapfn 7499 . . . 4  |-  ( ( I (subMat1 `  .1.  ) I )  e.  ( ( Base `  R
)  ^m  ( (
1 ... ( N  - 
1 ) )  X.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( I (subMat1 `  .1.  ) I )  Fn  ( ( 1 ... ( N  -  1 ) )  X.  (
1 ... ( N  - 
1 ) ) ) )
146144, 145syl 17 . . 3  |-  ( ph  ->  ( I (subMat1 `  .1.  ) I )  Fn  ( ( 1 ... ( N  -  1 ) )  X.  (
1 ... ( N  - 
1 ) ) ) )
147 fzfi 12185 . . . . . 6  |-  ( 1 ... ( N  - 
1 ) )  e. 
Fin
148 eqid 2422 . . . . . . 7  |-  ( Base `  ( ( 1 ... ( N  -  1 ) ) Mat  R ) )  =  ( Base `  ( ( 1 ... ( N  -  1 ) ) Mat  R ) )
149137, 148, 139mat1bas 19461 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
1 ... ( N  - 
1 ) )  e. 
Fin )  ->  ( 1r `  ( ( 1 ... ( N  - 
1 ) ) Mat  R
) )  e.  (
Base `  ( (
1 ... ( N  - 
1 ) ) Mat  R
) ) )
1506, 147, 149sylancl 666 . . . . 5  |-  ( ph  ->  ( 1r `  (
( 1 ... ( N  -  1 ) ) Mat  R ) )  e.  ( Base `  (
( 1 ... ( N  -  1 ) ) Mat  R ) ) )
151137, 13matbas2 19433 . . . . . 6  |-  ( ( ( 1 ... ( N  -  1 ) )  e.  Fin  /\  R  e.  Ring )  -> 
( ( Base `  R
)  ^m  ( (
1 ... ( N  - 
1 ) )  X.  ( 1 ... ( N  -  1 ) ) ) )  =  ( Base `  (
( 1 ... ( N  -  1 ) ) Mat  R ) ) )
152147, 6, 151sylancr 667 . . . . 5  |-  ( ph  ->  ( ( Base `  R
)  ^m  ( (
1 ... ( N  - 
1 ) )  X.  ( 1 ... ( N  -  1 ) ) ) )  =  ( Base `  (
( 1 ... ( N  -  1 ) ) Mat  R ) ) )
153150, 152eleqtrrd 2513 . . . 4  |-  ( ph  ->  ( 1r `  (
( 1 ... ( N  -  1 ) ) Mat  R ) )  e.  ( ( Base `  R )  ^m  (
( 1 ... ( N  -  1 ) )  X.  ( 1 ... ( N  - 
1 ) ) ) ) )
154 elmapfn 7499 . . . 4  |-  ( ( 1r `  ( ( 1 ... ( N  -  1 ) ) Mat 
R ) )  e.  ( ( Base `  R
)  ^m  ( (
1 ... ( N  - 
1 ) )  X.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( 1r `  (
( 1 ... ( N  -  1 ) ) Mat  R ) )  Fn  ( ( 1 ... ( N  - 
1 ) )  X.  ( 1 ... ( N  -  1 ) ) ) )
155153, 154syl 17 . . 3  |-  ( ph  ->  ( 1r `  (
( 1 ... ( N  -  1 ) ) Mat  R ) )  Fn  ( ( 1 ... ( N  - 
1 ) )  X.  ( 1 ... ( N  -  1 ) ) ) )
156 eqfnov2 6414 . . 3  |-  ( ( ( I (subMat1 `  .1.  ) I )  Fn  ( ( 1 ... ( N  -  1 ) )  X.  (
1 ... ( N  - 
1 ) ) )  /\  ( 1r `  ( ( 1 ... ( N  -  1 ) ) Mat  R ) )  Fn  ( ( 1 ... ( N  -  1 ) )  X.  ( 1 ... ( N  -  1 ) ) ) )  ->  ( ( I (subMat1 `  .1.  ) I )  =  ( 1r
`  ( ( 1 ... ( N  - 
1 ) ) Mat  R
) )  <->  A. i  e.  ( 1 ... ( N  -  1 ) ) A. j  e.  ( 1 ... ( N  -  1 ) ) ( i ( I (subMat1 `  .1.  ) I ) j )  =  ( i ( 1r `  (
( 1 ... ( N  -  1 ) ) Mat  R ) ) j ) ) )
157146, 155, 156syl2anc 665 . 2  |-  ( ph  ->  ( ( I (subMat1 `  .1.  ) I )  =  ( 1r `  ( ( 1 ... ( N  -  1 ) ) Mat  R ) )  <->  A. i  e.  ( 1 ... ( N  -  1 ) ) A. j  e.  ( 1 ... ( N  -  1 ) ) ( i ( I (subMat1 `  .1.  ) I ) j )  =  ( i ( 1r
`  ( ( 1 ... ( N  - 
1 ) ) Mat  R
) ) j ) ) )
158143, 157mpbird 235 1  |-  ( ph  ->  ( I (subMat1 `  .1.  ) I )  =  ( 1r `  (
( 1 ... ( N  -  1 ) ) Mat  R ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   ifcif 3909   class class class wbr 4420    X. cxp 4848    Fn wfn 5593   ` cfv 5598  (class class class)co 6302    ^m cmap 7477   Fincfn 7574   CCcc 9538   RRcr 9539   1c1 9541    + caddc 9543    < clt 9676    <_ cle 9677    - cmin 9861   NNcn 10610   ZZcz 10938   ZZ>=cuz 11160   ...cfz 11785   Basecbs 15109   0gc0g 15326   1rcur 17723   Ringcrg 17768   Mat cmat 19419  subMat1csmat 28615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-ot 4005  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-sup 7959  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-fz 11786  df-fzo 11917  df-seq 12214  df-hash 12516  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-hom 15202  df-cco 15203  df-0g 15328  df-gsum 15329  df-prds 15334  df-pws 15336  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-mhm 16570  df-submnd 16571  df-grp 16661  df-minusg 16662  df-sbg 16663  df-mulg 16664  df-subg 16802  df-ghm 16869  df-cntz 16959  df-cmn 17420  df-abl 17421  df-mgp 17712  df-ur 17724  df-ring 17770  df-subrg 17994  df-lmod 18081  df-lss 18144  df-sra 18383  df-rgmod 18384  df-dsmm 19282  df-frlm 19297  df-mamu 19396  df-mat 19420  df-smat 28616
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator