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Theorem 1smat1 28630
Description: The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 19608. (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 2451 . . . . 5  |-  ( I (subMat1 `  .1.  ) I )  =  ( I (subMat1 `  .1.  ) I )
2 1smat1.n . . . . . 6  |-  ( ph  ->  N  e.  NN )
32adantr 467 . . . . 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 467 . . . . 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 2451 . . . . . . . . 9  |-  ( ( 1 ... N ) Mat 
R )  =  ( ( 1 ... N
) Mat  R )
9 eqid 2451 . . . . . . . . 9  |-  ( Base `  ( ( 1 ... N ) Mat  R ) )  =  ( Base `  ( ( 1 ... N ) Mat  R ) )
10 1smat1.1 . . . . . . . . 9  |-  .1.  =  ( 1r `  ( ( 1 ... N ) Mat 
R ) )
118, 9, 10mat1bas 19474 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
1 ... N )  e. 
Fin )  ->  .1.  e.  ( Base `  (
( 1 ... N
) Mat  R ) ) )
126, 7, 11sylancl 668 . . . . . . 7  |-  ( ph  ->  .1.  e.  ( Base `  ( ( 1 ... N ) Mat  R ) ) )
13 eqid 2451 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
148, 13matbas2 19446 . . . . . . . 8  |-  ( ( ( 1 ... N
)  e.  Fin  /\  R  e.  Ring )  -> 
( ( Base `  R
)  ^m  ( (
1 ... N )  X.  ( 1 ... N
) ) )  =  ( Base `  (
( 1 ... N
) Mat  R ) ) )
157, 6, 14sylancr 669 . . . . . . 7  |-  ( ph  ->  ( ( Base `  R
)  ^m  ( (
1 ... N )  X.  ( 1 ... N
) ) )  =  ( Base `  (
( 1 ... N
) Mat  R ) ) )
1612, 15eleqtrrd 2532 . . . . . 6  |-  ( ph  ->  .1.  e.  ( (
Base `  R )  ^m  ( ( 1 ... N )  X.  (
1 ... N ) ) ) )
1716adantr 467 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  ->  .1.  e.  ( ( Base `  R )  ^m  (
( 1 ... N
)  X.  ( 1 ... N ) ) ) )
18 fz1ssnn 11830 . . . . . 6  |-  ( 1 ... ( N  - 
1 ) )  C_  NN
19 simprl 764 . . . . . 6  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  ( 1 ... ( N  - 
1 ) ) )
2018, 19sseldi 3430 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  NN )
21 simprr 766 . . . . . 6  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  ( 1 ... ( N  - 
1 ) ) )
2218, 21sseldi 3430 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  NN )
23 eqidd 2452 . . . . 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 2452 . . . . 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 28623 . . . 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 2451 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
27 eqid 2451 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
287a1i 11 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( 1 ... N
)  e.  Fin )
296adantr 467 . . . . 5  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  ->  R  e.  Ring )
30 nnuz 11194 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
3120, 30syl6eleq 2539 . . . . . . . 8  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  ( ZZ>= ` 
1 ) )
32 fznatpl1 11850 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  i  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( i  +  1 )  e.  ( 1 ... N ) )
333, 19, 32syl2anc 667 . . . . . . . 8  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( i  +  1 )  e.  ( 1 ... N ) )
34 peano2fzr 11812 . . . . . . . 8  |-  ( ( i  e.  ( ZZ>= ` 
1 )  /\  (
i  +  1 )  e.  ( 1 ... N ) )  -> 
i  e.  ( 1 ... N ) )
3531, 33, 34syl2anc 667 . . . . . . 7  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  ( 1 ... N ) )
3635, 33jca 535 . . . . . 6  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( i  e.  ( 1 ... N )  /\  ( i  +  1 )  e.  ( 1 ... N ) ) )
37 eleq1 2517 . . . . . . 7  |-  ( i  =  if ( i  <  I ,  i ,  ( i  +  1 ) )  -> 
( i  e.  ( 1 ... N )  <-> 
if ( i  < 
I ,  i ,  ( i  +  1 ) )  e.  ( 1 ... N ) ) )
38 eleq1 2517 . . . . . . 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 3917 . . . . . 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 2539 . . . . . . . 8  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  ( ZZ>= ` 
1 ) )
42 fznatpl1 11850 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  j  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( j  +  1 )  e.  ( 1 ... N ) )
433, 21, 42syl2anc 667 . . . . . . . 8  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( j  +  1 )  e.  ( 1 ... N ) )
44 peano2fzr 11812 . . . . . . . 8  |-  ( ( j  e.  ( ZZ>= ` 
1 )  /\  (
j  +  1 )  e.  ( 1 ... N ) )  -> 
j  e.  ( 1 ... N ) )
4541, 43, 44syl2anc 667 . . . . . . 7  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  ( 1 ... N ) )
4645, 43jca 535 . . . . . 6  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( j  e.  ( 1 ... N )  /\  ( j  +  1 )  e.  ( 1 ... N ) ) )
47 eleq1 2517 . . . . . . 7  |-  ( j  =  if ( j  <  I ,  j ,  ( j  +  1 ) )  -> 
( j  e.  ( 1 ... N )  <-> 
if ( j  < 
I ,  j ,  ( j  +  1 ) )  e.  ( 1 ... N ) ) )
48 eleq1 2517 . . . . . . 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 3917 . . . . . 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 19473 . . . 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 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  i  < 
I )  ->  i  <  I )
5352iftrued 3889 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  i  < 
I )  ->  if ( i  <  I ,  i ,  ( i  +  1 ) )  =  i )
5453eqeq1d 2453 . . . . . . . 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 463 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  j  <  I )  ->  j  <  I )
5655iftrued 3889 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  j  <  I )  ->  if ( j  <  I ,  j ,  ( j  +  1 ) )  =  j )
5756eqeq2d 2461 . . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  -.  j  <  I )
5958iffalsed 3892 . . . . . . . . . . 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 2461 . . . . . . . . . 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 10624 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  RR )
6261ad2antrr 732 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
i  e.  RR )
63 fz1ssnn 11830 . . . . . . . . . . . . . . . . 17  |-  ( 1 ... N )  C_  NN
6463, 4sseldi 3430 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  I  e.  NN )
6564nnred 10624 . . . . . . . . . . . . . . 15  |-  ( ph  ->  I  e.  RR )
6665ad3antrrr 736 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  I  e.  RR )
6722nnred 10624 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  RR )
6867ad2antrr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
j  e.  RR )
69 1red 9658 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
1  e.  RR )
7068, 69readdcld 9670 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
( j  +  1 )  e.  RR )
7152adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
i  <  I )
7264nnzd 11039 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  I  e.  ZZ )
7372ad3antrrr 736 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  I  e.  ZZ )
7422nnzd 11039 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  ZZ )
7574ad2antrr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
j  e.  ZZ )
7666, 68, 58nltled 9785 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  I  <_  j )
77 zleltp1 10987 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  ZZ  /\  j  e.  ZZ )  ->  ( I  <_  j  <->  I  <  ( j  +  1 ) ) )
7877biimpa 487 . . . . . . . . . . . . . . 15  |-  ( ( ( I  e.  ZZ  /\  j  e.  ZZ )  /\  I  <_  j
)  ->  I  <  ( j  +  1 ) )
7973, 75, 76, 78syl21anc 1267 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  I  <  ( j  +  1 ) )
8062, 66, 70, 71, 79lttrd 9796 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
i  <  ( j  +  1 ) )
8162, 80ltned 9771 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
i  =/=  ( j  +  1 ) )
8281neneqd 2629 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  -.  i  =  (
j  +  1 ) )
8362, 66, 68, 71, 76ltletrd 9795 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
i  <  j )
8462, 83ltned 9771 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
i  =/=  j )
8584neneqd 2629 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  ->  -.  i  =  j
)
8682, 852falsed 353 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  i  <  I )  /\  -.  j  <  I )  -> 
( i  =  ( j  +  1 )  <-> 
i  =  j ) )
8760, 86bitrd 257 . . . . . . . . 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 800 . . . . . . . 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 257 . . . . . . 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 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  -.  i  <  I )  ->  -.  i  <  I )
9190iffalsed 3892 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  e.  ( 1 ... ( N  - 
1 ) )  /\  j  e.  ( 1 ... ( N  - 
1 ) ) ) )  /\  -.  i  <  I )  ->  if ( i  <  I ,  i ,  ( i  +  1 ) )  =  ( i  +  1 ) )
9291eqeq1d 2453 . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
j  <  I )
9493iftrued 3889 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  if ( j  <  I ,  j ,  ( j  +  1 ) )  =  j )
9594eqeq2d 2461 . . . . . . . . . 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 732 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
j  e.  RR )
9765ad3antrrr 736 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  I  e.  RR )
9861ad2antrr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
i  e.  RR )
99 1red 9658 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
1  e.  RR )
10098, 99readdcld 9670 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
( i  +  1 )  e.  RR )
10172ad3antrrr 736 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  I  e.  ZZ )
10220nnzd 11039 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  ZZ )
103102ad2antrr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
i  e.  ZZ )
10490adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  -.  i  <  I )
10597, 98, 104nltled 9785 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  I  <_  i )
106 zleltp1 10987 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  ZZ  /\  i  e.  ZZ )  ->  ( I  <_  i  <->  I  <  ( i  +  1 ) ) )
107106biimpa 487 . . . . . . . . . . . . . . . 16  |-  ( ( ( I  e.  ZZ  /\  i  e.  ZZ )  /\  I  <_  i
)  ->  I  <  ( i  +  1 ) )
108101, 103, 105, 107syl21anc 1267 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  I  <  ( i  +  1 ) )
10996, 97, 100, 93, 108lttrd 9796 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
j  <  ( i  +  1 ) )
11096, 109ltned 9771 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
j  =/=  ( i  +  1 ) )
111110necomd 2679 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
( i  +  1 )  =/=  j )
112111neneqd 2629 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  -.  ( i  +  1 )  =  j )
11396, 97, 98, 93, 105ltletrd 9795 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
j  <  i )
11496, 113ltned 9771 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
j  =/=  i )
115114necomd 2679 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
i  =/=  j )
116115neneqd 2629 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  ->  -.  i  =  j
)
117112, 1162falsed 353 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  j  <  I )  -> 
( ( i  +  1 )  =  j  <-> 
i  =  j ) )
11895, 117bitrd 257 . . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  ->  -.  j  <  I )
120119iffalsed 3892 . . . . . . . . . . 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 2461 . . . . . . . . . 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 10625 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
i  e.  CC )
123122ad3antrrr 736 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  /\  ( i  +  1 )  =  ( j  +  1 ) )  ->  i  e.  CC )
12422nncnd 10625 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
j  e.  CC )
125124ad3antrrr 736 . . . . . . . . . . . 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 9659 . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 9839 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  /\  ( i  +  1 )  =  ( j  +  1 ) )  ->  i  =  j )
129 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  /\  i  =  j )  ->  i  =  j )
130129oveq1d 6305 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  /\  i  =  j )  ->  ( i  +  1 )  =  ( j  +  1 ) )
131128, 130impbida 843 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  e.  ( 1 ... ( N  -  1 ) )  /\  j  e.  ( 1 ... ( N  -  1 ) ) ) )  /\  -.  i  <  I )  /\  -.  j  <  I )  ->  ( ( i  +  1 )  =  ( j  +  1 )  <->  i  =  j ) )
132121, 131bitrd 257 . . . . . . . . 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 800 . . . . . . . 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 257 . . . . . . 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 800 . . . . . 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 3903 . . . . 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 2451 . . . . . 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 2451 . . . . . 6  |-  ( 1r
`  ( ( 1 ... ( N  - 
1 ) ) Mat  R
) )  =  ( 1r `  ( ( 1 ... ( N  -  1 ) ) Mat 
R ) )
140137, 26, 27, 138, 29, 19, 21, 139mat1ov 19473 . . . . 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 2488 . . . 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 2489 . . 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 2809 . 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 28622 . . . 4  |-  ( ph  ->  ( I (subMat1 `  .1.  ) I )  e.  ( ( Base `  R
)  ^m  ( (
1 ... ( N  - 
1 ) )  X.  ( 1 ... ( N  -  1 ) ) ) ) )
145 elmapfn 7494 . . . 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 2451 . . . . . . 7  |-  ( Base `  ( ( 1 ... ( N  -  1 ) ) Mat  R ) )  =  ( Base `  ( ( 1 ... ( N  -  1 ) ) Mat  R ) )
149137, 148, 139mat1bas 19474 . . . . . 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 668 . . . . 5  |-  ( ph  ->  ( 1r `  (
( 1 ... ( N  -  1 ) ) Mat  R ) )  e.  ( Base `  (
( 1 ... ( N  -  1 ) ) Mat  R ) ) )
151137, 13matbas2 19446 . . . . . 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 669 . . . . 5  |-  ( ph  ->  ( ( Base `  R
)  ^m  ( (
1 ... ( N  - 
1 ) )  X.  ( 1 ... ( N  -  1 ) ) ) )  =  ( Base `  (
( 1 ... ( N  -  1 ) ) Mat  R ) ) )
153150, 152eleqtrrd 2532 . . . 4  |-  ( ph  ->  ( 1r `  (
( 1 ... ( N  -  1 ) ) Mat  R ) )  e.  ( ( Base `  R )  ^m  (
( 1 ... ( N  -  1 ) )  X.  ( 1 ... ( N  - 
1 ) ) ) ) )
154 elmapfn 7494 . . . 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 6403 . . 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 667 . 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 236 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 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   ifcif 3881   class class class wbr 4402    X. cxp 4832    Fn wfn 5577   ` cfv 5582  (class class class)co 6290    ^m cmap 7472   Fincfn 7569   CCcc 9537   RRcr 9538   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676    - cmin 9860   NNcn 10609   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784   Basecbs 15121   0gc0g 15338   1rcur 17735   Ringcrg 17780   Mat cmat 19432  subMat1csmat 28619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-ot 3977  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-hom 15214  df-cco 15215  df-0g 15340  df-gsum 15341  df-prds 15346  df-pws 15348  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mulg 16676  df-subg 16814  df-ghm 16881  df-cntz 16971  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-subrg 18006  df-lmod 18093  df-lss 18156  df-sra 18395  df-rgmod 18396  df-dsmm 19295  df-frlm 19310  df-mamu 19409  df-mat 19433  df-smat 28620
This theorem is referenced by: (None)
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