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Theorem mdetunilem2 18554
Description: Lemma for mdetuni 18563. (Contributed by SO, 15-Jul-2018.)
Hypotheses
Ref Expression
mdetuni.a  |-  A  =  ( N Mat  R )
mdetuni.b  |-  B  =  ( Base `  A
)
mdetuni.k  |-  K  =  ( Base `  R
)
mdetuni.0g  |-  .0.  =  ( 0g `  R )
mdetuni.1r  |-  .1.  =  ( 1r `  R )
mdetuni.pg  |-  .+  =  ( +g  `  R )
mdetuni.tg  |-  .x.  =  ( .r `  R )
mdetuni.n  |-  ( ph  ->  N  e.  Fin )
mdetuni.r  |-  ( ph  ->  R  e.  Ring )
mdetuni.ff  |-  ( ph  ->  D : B --> K )
mdetuni.al  |-  ( ph  ->  A. x  e.  B  A. y  e.  N  A. z  e.  N  ( ( y  =/=  z  /\  A. w  e.  N  ( y
x w )  =  ( z x w ) )  ->  ( D `  x )  =  .0.  ) )
mdetuni.li  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  A. z  e.  B  A. w  e.  N  ( ( ( x  |`  ( { w }  X.  N ) )  =  ( ( y  |`  ( { w }  X.  N ) )  oF  .+  ( z  |`  ( { w }  X.  N ) ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( y  |`  ( ( N  \  { w } )  X.  N ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  x )  =  ( ( D `  y
)  .+  ( D `  z ) ) ) )
mdetuni.sc  |-  ( ph  ->  A. x  e.  B  A. y  e.  K  A. z  e.  B  A. w  e.  N  ( ( ( x  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  x )  =  ( y  .x.  ( D `
 z ) ) ) )
mdetunilem2.ph  |-  ( ps 
->  ph )
mdetunilem2.eg  |-  ( ps 
->  ( E  e.  N  /\  G  e.  N  /\  E  =/=  G
) )
mdetunilem2.f  |-  ( ( ps  /\  b  e.  N )  ->  F  e.  K )
mdetunilem2.h  |-  ( ( ps  /\  a  e.  N  /\  b  e.  N )  ->  H  e.  K )
Assertion
Ref Expression
mdetunilem2  |-  ( ps 
->  ( D `  (
a  e.  N , 
b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) ) ) )  =  .0.  )
Distinct variable groups:    ph, x, y, z, w, a, b   
x, B, y, z, w, a, b    x, K, y, z, w, a, b    x, N, y, z, w, a, b   
x, D, y, z, w, a, b    x,  .x. , y, z, w    .+ , a,
b, x, y, z, w    .0. , a, b, x, y, z, w    .1. , a, b, x, y, z, w    x, R, y, z, w    A, a, b, x, y, z, w    x, E, y, z, w    x, F, y, z, w    x, G, y, z, w    x, H, y, z, w    ps, a, b, x, y, z, w    E, a, b    G, a, b    F, a
Allowed substitution hints:    R( a, b)    .x. ( a, b)    F( b)    H( a, b)

Proof of Theorem mdetunilem2
StepHypRef Expression
1 mdetunilem2.ph . 2  |-  ( ps 
->  ph )
2 mdetuni.a . . 3  |-  A  =  ( N Mat  R )
3 mdetuni.k . . 3  |-  K  =  ( Base `  R
)
4 mdetuni.b . . 3  |-  B  =  ( Base `  A
)
5 mdetuni.n . . . 4  |-  ( ph  ->  N  e.  Fin )
61, 5syl 16 . . 3  |-  ( ps 
->  N  e.  Fin )
7 mdetuni.r . . . 4  |-  ( ph  ->  R  e.  Ring )
81, 7syl 16 . . 3  |-  ( ps 
->  R  e.  Ring )
9 mdetunilem2.f . . . . 5  |-  ( ( ps  /\  b  e.  N )  ->  F  e.  K )
1093adant2 1007 . . . 4  |-  ( ( ps  /\  a  e.  N  /\  b  e.  N )  ->  F  e.  K )
11 mdetunilem2.h . . . . 5  |-  ( ( ps  /\  a  e.  N  /\  b  e.  N )  ->  H  e.  K )
1210, 11ifcld 3943 . . . 4  |-  ( ( ps  /\  a  e.  N  /\  b  e.  N )  ->  if ( a  =  G ,  F ,  H
)  e.  K )
1310, 12ifcld 3943 . . 3  |-  ( ( ps  /\  a  e.  N  /\  b  e.  N )  ->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) )  e.  K
)
142, 3, 4, 6, 8, 13matbas2d 18452 . 2  |-  ( ps 
->  ( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) ) )  e.  B )
15 eqidd 2455 . . . . 5  |-  ( ( ps  /\  w  e.  N )  ->  (
a  e.  N , 
b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) ) )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) ) ) )
16 iftrue 3908 . . . . . . 7  |-  ( a  =  E  ->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) )  =  F )
17 csbeq1a 3407 . . . . . . 7  |-  ( b  =  w  ->  F  =  [_ w  /  b ]_ F )
1816, 17sylan9eq 2515 . . . . . 6  |-  ( ( a  =  E  /\  b  =  w )  ->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) )  =  [_ w  /  b ]_ F
)
1918adantl 466 . . . . 5  |-  ( ( ( ps  /\  w  e.  N )  /\  (
a  =  E  /\  b  =  w )
)  ->  if (
a  =  E ,  F ,  if (
a  =  G ,  F ,  H )
)  =  [_ w  /  b ]_ F
)
20 eqidd 2455 . . . . 5  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  E )  ->  N  =  N )
21 mdetunilem2.eg . . . . . . 7  |-  ( ps 
->  ( E  e.  N  /\  G  e.  N  /\  E  =/=  G
) )
2221simp1d 1000 . . . . . 6  |-  ( ps 
->  E  e.  N
)
2322adantr 465 . . . . 5  |-  ( ( ps  /\  w  e.  N )  ->  E  e.  N )
24 simpr 461 . . . . 5  |-  ( ( ps  /\  w  e.  N )  ->  w  e.  N )
25 nfv 1674 . . . . . . 7  |-  F/ b ( ps  /\  w  e.  N )
26 nfcsb1v 3414 . . . . . . . 8  |-  F/_ b [_ w  /  b ]_ F
2726nfel1 2632 . . . . . . 7  |-  F/ b
[_ w  /  b ]_ F  e.  K
2825, 27nfim 1858 . . . . . 6  |-  F/ b ( ( ps  /\  w  e.  N )  ->  [_ w  /  b ]_ F  e.  K
)
29 eleq1 2526 . . . . . . . 8  |-  ( b  =  w  ->  (
b  e.  N  <->  w  e.  N ) )
3029anbi2d 703 . . . . . . 7  |-  ( b  =  w  ->  (
( ps  /\  b  e.  N )  <->  ( ps  /\  w  e.  N ) ) )
3117eleq1d 2523 . . . . . . 7  |-  ( b  =  w  ->  ( F  e.  K  <->  [_ w  / 
b ]_ F  e.  K
) )
3230, 31imbi12d 320 . . . . . 6  |-  ( b  =  w  ->  (
( ( ps  /\  b  e.  N )  ->  F  e.  K )  <-> 
( ( ps  /\  w  e.  N )  ->  [_ w  /  b ]_ F  e.  K
) ) )
3328, 32, 9chvar 1969 . . . . 5  |-  ( ( ps  /\  w  e.  N )  ->  [_ w  /  b ]_ F  e.  K )
34 nfv 1674 . . . . 5  |-  F/ a ( ps  /\  w  e.  N )
35 nfcv 2616 . . . . 5  |-  F/_ b E
36 nfcv 2616 . . . . 5  |-  F/_ a
w
37 nfcv 2616 . . . . 5  |-  F/_ a [_ w  /  b ]_ F
3815, 19, 20, 23, 24, 33, 34, 25, 35, 36, 37, 26ovmpt2dxf 6329 . . . 4  |-  ( ( ps  /\  w  e.  N )  ->  ( E ( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H ) ) ) w )  =  [_ w  /  b ]_ F
)
3921simp3d 1002 . . . . . . . . . . . . 13  |-  ( ps 
->  E  =/=  G
)
4039adantr 465 . . . . . . . . . . . 12  |-  ( ( ps  /\  w  e.  N )  ->  E  =/=  G )
41 neeq2 2735 . . . . . . . . . . . 12  |-  ( a  =  G  ->  ( E  =/=  a  <->  E  =/=  G ) )
4240, 41syl5ibrcom 222 . . . . . . . . . . 11  |-  ( ( ps  /\  w  e.  N )  ->  (
a  =  G  ->  E  =/=  a ) )
4342imp 429 . . . . . . . . . 10  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  G )  ->  E  =/=  a )
4443necomd 2723 . . . . . . . . 9  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  G )  ->  a  =/=  E )
4544neneqd 2655 . . . . . . . 8  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  G )  ->  -.  a  =  E )
4645adantrr 716 . . . . . . 7  |-  ( ( ( ps  /\  w  e.  N )  /\  (
a  =  G  /\  b  =  w )
)  ->  -.  a  =  E )
47 iffalse 3910 . . . . . . 7  |-  ( -.  a  =  E  ->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) )  =  if ( a  =  G ,  F ,  H
) )
4846, 47syl 16 . . . . . 6  |-  ( ( ( ps  /\  w  e.  N )  /\  (
a  =  G  /\  b  =  w )
)  ->  if (
a  =  E ,  F ,  if (
a  =  G ,  F ,  H )
)  =  if ( a  =  G ,  F ,  H )
)
49 iftrue 3908 . . . . . . . 8  |-  ( a  =  G  ->  if ( a  =  G ,  F ,  H
)  =  F )
5049, 17sylan9eq 2515 . . . . . . 7  |-  ( ( a  =  G  /\  b  =  w )  ->  if ( a  =  G ,  F ,  H )  =  [_ w  /  b ]_ F
)
5150adantl 466 . . . . . 6  |-  ( ( ( ps  /\  w  e.  N )  /\  (
a  =  G  /\  b  =  w )
)  ->  if (
a  =  G ,  F ,  H )  =  [_ w  /  b ]_ F )
5248, 51eqtrd 2495 . . . . 5  |-  ( ( ( ps  /\  w  e.  N )  /\  (
a  =  G  /\  b  =  w )
)  ->  if (
a  =  E ,  F ,  if (
a  =  G ,  F ,  H )
)  =  [_ w  /  b ]_ F
)
53 eqidd 2455 . . . . 5  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  G )  ->  N  =  N )
5421simp2d 1001 . . . . . 6  |-  ( ps 
->  G  e.  N
)
5554adantr 465 . . . . 5  |-  ( ( ps  /\  w  e.  N )  ->  G  e.  N )
56 nfcv 2616 . . . . 5  |-  F/_ b G
5715, 52, 53, 55, 24, 33, 34, 25, 56, 36, 37, 26ovmpt2dxf 6329 . . . 4  |-  ( ( ps  /\  w  e.  N )  ->  ( G ( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H ) ) ) w )  =  [_ w  /  b ]_ F
)
5838, 57eqtr4d 2498 . . 3  |-  ( ( ps  /\  w  e.  N )  ->  ( E ( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H ) ) ) w )  =  ( G ( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H ) ) ) w ) )
5958ralrimiva 2830 . 2  |-  ( ps 
->  A. w  e.  N  ( E ( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H ) ) ) w )  =  ( G ( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H ) ) ) w ) )
60 mdetuni.0g . . 3  |-  .0.  =  ( 0g `  R )
61 mdetuni.1r . . 3  |-  .1.  =  ( 1r `  R )
62 mdetuni.pg . . 3  |-  .+  =  ( +g  `  R )
63 mdetuni.tg . . 3  |-  .x.  =  ( .r `  R )
64 mdetuni.ff . . 3  |-  ( ph  ->  D : B --> K )
65 mdetuni.al . . 3  |-  ( ph  ->  A. x  e.  B  A. y  e.  N  A. z  e.  N  ( ( y  =/=  z  /\  A. w  e.  N  ( y
x w )  =  ( z x w ) )  ->  ( D `  x )  =  .0.  ) )
66 mdetuni.li . . 3  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  A. z  e.  B  A. w  e.  N  ( ( ( x  |`  ( { w }  X.  N ) )  =  ( ( y  |`  ( { w }  X.  N ) )  oF  .+  ( z  |`  ( { w }  X.  N ) ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( y  |`  ( ( N  \  { w } )  X.  N ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  x )  =  ( ( D `  y
)  .+  ( D `  z ) ) ) )
67 mdetuni.sc . . 3  |-  ( ph  ->  A. x  e.  B  A. y  e.  K  A. z  e.  B  A. w  e.  N  ( ( ( x  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  x )  =  ( y  .x.  ( D `
 z ) ) ) )
682, 4, 3, 60, 61, 62, 63, 5, 7, 64, 65, 66, 67mdetunilem1 18553 . 2  |-  ( ( ( ph  /\  (
a  e.  N , 
b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) ) )  e.  B  /\  A. w  e.  N  ( E
( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) ) ) w )  =  ( G ( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) ) ) w ) )  /\  ( E  e.  N  /\  G  e.  N  /\  E  =/=  G ) )  ->  ( D `  ( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) ) ) )  =  .0.  )
691, 14, 59, 21, 68syl31anc 1222 1  |-  ( ps 
->  ( D `  (
a  e.  N , 
b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) ) ) )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   [_csb 3398    \ cdif 3436   ifcif 3902   {csn 3988    X. cxp 4949    |` cres 4953   -->wf 5525   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205    oFcof 6431   Fincfn 7423   Basecbs 14295   +g cplusg 14360   .rcmulr 14361   0gc0g 14500   1rcur 16728   Ringcrg 16771   Mat cmat 18408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-ot 3997  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-fz 11558  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-sca 14376  df-vsca 14377  df-ip 14378  df-tset 14379  df-ple 14380  df-ds 14382  df-hom 14384  df-cco 14385  df-0g 14502  df-prds 14508  df-pws 14510  df-sra 17379  df-rgmod 17380  df-dsmm 18285  df-frlm 18300  df-mat 18410
This theorem is referenced by:  mdetunilem6  18558  mdetunilem8  18560
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