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Theorem mdetunilem2 18984
Description: Lemma for mdetuni 18993. (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 1015 . . . 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 3988 . . . 4  |-  ( ( ps  /\  a  e.  N  /\  b  e.  N )  ->  if ( a  =  G ,  F ,  H
)  e.  K )
1310, 12ifcld 3988 . . 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 18794 . 2  |-  ( ps 
->  ( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) ) )  e.  B )
15 eqidd 2468 . . . . 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 3951 . . . . . . 7  |-  ( a  =  E  ->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) )  =  F )
17 csbeq1a 3449 . . . . . . 7  |-  ( b  =  w  ->  F  =  [_ w  /  b ]_ F )
1816, 17sylan9eq 2528 . . . . . 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 2468 . . . . 5  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  E )  ->  N  =  N )
21 mdetunilem2.eg . . . . . . 7  |-  ( ps 
->  ( E  e.  N  /\  G  e.  N  /\  E  =/=  G
) )
2221simp1d 1008 . . . . . 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 1683 . . . . . . 7  |-  F/ b ( ps  /\  w  e.  N )
26 nfcsb1v 3456 . . . . . . . 8  |-  F/_ b [_ w  /  b ]_ F
2726nfel1 2645 . . . . . . 7  |-  F/ b
[_ w  /  b ]_ F  e.  K
2825, 27nfim 1867 . . . . . 6  |-  F/ b ( ( ps  /\  w  e.  N )  ->  [_ w  /  b ]_ F  e.  K
)
29 eleq1 2539 . . . . . . . 8  |-  ( b  =  w  ->  (
b  e.  N  <->  w  e.  N ) )
3029anbi2d 703 . . . . . . 7  |-  ( b  =  w  ->  (
( ps  /\  b  e.  N )  <->  ( ps  /\  w  e.  N ) ) )
3117eleq1d 2536 . . . . . . 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 1982 . . . . 5  |-  ( ( ps  /\  w  e.  N )  ->  [_ w  /  b ]_ F  e.  K )
34 nfv 1683 . . . . 5  |-  F/ a ( ps  /\  w  e.  N )
35 nfcv 2629 . . . . 5  |-  F/_ b E
36 nfcv 2629 . . . . 5  |-  F/_ a
w
37 nfcv 2629 . . . . 5  |-  F/_ a [_ w  /  b ]_ F
3815, 19, 20, 23, 24, 33, 34, 25, 35, 36, 37, 26ovmpt2dxf 6423 . . . 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 1010 . . . . . . . . . . . . 13  |-  ( ps 
->  E  =/=  G
)
4039adantr 465 . . . . . . . . . . . 12  |-  ( ( ps  /\  w  e.  N )  ->  E  =/=  G )
41 neeq2 2750 . . . . . . . . . . . 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 2738 . . . . . . . . 9  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  G )  ->  a  =/=  E )
4544neneqd 2669 . . . . . . . 8  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  G )  ->  -.  a  =  E )
4645adantrr 716 . . . . . . 7  |-  ( ( ( ps  /\  w  e.  N )  /\  (
a  =  G  /\  b  =  w )
)  ->  -.  a  =  E )
47 iffalse 3954 . . . . . . 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 3951 . . . . . . . 8  |-  ( a  =  G  ->  if ( a  =  G ,  F ,  H
)  =  F )
5049, 17sylan9eq 2528 . . . . . . 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 2508 . . . . 5  |-  ( ( ( ps  /\  w  e.  N )  /\  (
a  =  G  /\  b  =  w )
)  ->  if (
a  =  E ,  F ,  if (
a  =  G ,  F ,  H )
)  =  [_ w  /  b ]_ F
)
53 eqidd 2468 . . . . 5  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  G )  ->  N  =  N )
5421simp2d 1009 . . . . . 6  |-  ( ps 
->  G  e.  N
)
5554adantr 465 . . . . 5  |-  ( ( ps  /\  w  e.  N )  ->  G  e.  N )
56 nfcv 2629 . . . . 5  |-  F/_ b G
5715, 52, 53, 55, 24, 33, 34, 25, 56, 36, 37, 26ovmpt2dxf 6423 . . . 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 2511 . . 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 2881 . 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 18983 . 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 1231 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   [_csb 3440    \ cdif 3478   ifcif 3945   {csn 4033    X. cxp 5003    |` cres 5007   -->wf 5590   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297    oFcof 6533   Fincfn 7528   Basecbs 14507   +g cplusg 14572   .rcmulr 14573   0gc0g 14712   1rcur 17025   Ringcrg 17070   Mat cmat 18778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-0g 14714  df-prds 14720  df-pws 14722  df-sra 17689  df-rgmod 17690  df-dsmm 18632  df-frlm 18647  df-mat 18779
This theorem is referenced by:  mdetunilem6  18988  mdetunilem8  18990
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