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Theorem mdetunilem2 19638
Description: Lemma for mdetuni 19647. (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 17 . . 3  |-  ( ps 
->  N  e.  Fin )
7 mdetuni.r . . . 4  |-  ( ph  ->  R  e.  Ring )
81, 7syl 17 . . 3  |-  ( ps 
->  R  e.  Ring )
9 mdetunilem2.f . . . . 5  |-  ( ( ps  /\  b  e.  N )  ->  F  e.  K )
1093adant2 1027 . . . 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 3924 . . . 4  |-  ( ( ps  /\  a  e.  N  /\  b  e.  N )  ->  if ( a  =  G ,  F ,  H
)  e.  K )
1310, 12ifcld 3924 . . 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 19448 . 2  |-  ( ps 
->  ( a  e.  N ,  b  e.  N  |->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) ) )  e.  B )
15 eqidd 2452 . . . . 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 3887 . . . . . . 7  |-  ( a  =  E  ->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) )  =  F )
17 csbeq1a 3372 . . . . . . 7  |-  ( b  =  w  ->  F  =  [_ w  /  b ]_ F )
1816, 17sylan9eq 2505 . . . . . 6  |-  ( ( a  =  E  /\  b  =  w )  ->  if ( a  =  E ,  F ,  if ( a  =  G ,  F ,  H
) )  =  [_ w  /  b ]_ F
)
1918adantl 468 . . . . 5  |-  ( ( ( ps  /\  w  e.  N )  /\  (
a  =  E  /\  b  =  w )
)  ->  if (
a  =  E ,  F ,  if (
a  =  G ,  F ,  H )
)  =  [_ w  /  b ]_ F
)
20 eqidd 2452 . . . . 5  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  E )  ->  N  =  N )
21 mdetunilem2.eg . . . . . . 7  |-  ( ps 
->  ( E  e.  N  /\  G  e.  N  /\  E  =/=  G
) )
2221simp1d 1020 . . . . . 6  |-  ( ps 
->  E  e.  N
)
2322adantr 467 . . . . 5  |-  ( ( ps  /\  w  e.  N )  ->  E  e.  N )
24 simpr 463 . . . . 5  |-  ( ( ps  /\  w  e.  N )  ->  w  e.  N )
25 nfv 1761 . . . . . . 7  |-  F/ b ( ps  /\  w  e.  N )
26 nfcsb1v 3379 . . . . . . . 8  |-  F/_ b [_ w  /  b ]_ F
2726nfel1 2606 . . . . . . 7  |-  F/ b
[_ w  /  b ]_ F  e.  K
2825, 27nfim 2003 . . . . . 6  |-  F/ b ( ( ps  /\  w  e.  N )  ->  [_ w  /  b ]_ F  e.  K
)
29 eleq1 2517 . . . . . . . 8  |-  ( b  =  w  ->  (
b  e.  N  <->  w  e.  N ) )
3029anbi2d 710 . . . . . . 7  |-  ( b  =  w  ->  (
( ps  /\  b  e.  N )  <->  ( ps  /\  w  e.  N ) ) )
3117eleq1d 2513 . . . . . . 7  |-  ( b  =  w  ->  ( F  e.  K  <->  [_ w  / 
b ]_ F  e.  K
) )
3230, 31imbi12d 322 . . . . . 6  |-  ( b  =  w  ->  (
( ( ps  /\  b  e.  N )  ->  F  e.  K )  <-> 
( ( ps  /\  w  e.  N )  ->  [_ w  /  b ]_ F  e.  K
) ) )
3328, 32, 9chvar 2106 . . . . 5  |-  ( ( ps  /\  w  e.  N )  ->  [_ w  /  b ]_ F  e.  K )
34 nfv 1761 . . . . 5  |-  F/ a ( ps  /\  w  e.  N )
35 nfcv 2592 . . . . 5  |-  F/_ b E
36 nfcv 2592 . . . . 5  |-  F/_ a
w
37 nfcv 2592 . . . . 5  |-  F/_ a [_ w  /  b ]_ F
3815, 19, 20, 23, 24, 33, 34, 25, 35, 36, 37, 26ovmpt2dxf 6422 . . . 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 1022 . . . . . . . . . . . . 13  |-  ( ps 
->  E  =/=  G
)
4039adantr 467 . . . . . . . . . . . 12  |-  ( ( ps  /\  w  e.  N )  ->  E  =/=  G )
41 neeq2 2687 . . . . . . . . . . . 12  |-  ( a  =  G  ->  ( E  =/=  a  <->  E  =/=  G ) )
4240, 41syl5ibrcom 226 . . . . . . . . . . 11  |-  ( ( ps  /\  w  e.  N )  ->  (
a  =  G  ->  E  =/=  a ) )
4342imp 431 . . . . . . . . . 10  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  G )  ->  E  =/=  a )
4443necomd 2679 . . . . . . . . 9  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  G )  ->  a  =/=  E )
4544neneqd 2629 . . . . . . . 8  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  G )  ->  -.  a  =  E )
4645adantrr 723 . . . . . . 7  |-  ( ( ( ps  /\  w  e.  N )  /\  (
a  =  G  /\  b  =  w )
)  ->  -.  a  =  E )
4746iffalsed 3892 . . . . . 6  |-  ( ( ( ps  /\  w  e.  N )  /\  (
a  =  G  /\  b  =  w )
)  ->  if (
a  =  E ,  F ,  if (
a  =  G ,  F ,  H )
)  =  if ( a  =  G ,  F ,  H )
)
48 iftrue 3887 . . . . . . . 8  |-  ( a  =  G  ->  if ( a  =  G ,  F ,  H
)  =  F )
4948, 17sylan9eq 2505 . . . . . . 7  |-  ( ( a  =  G  /\  b  =  w )  ->  if ( a  =  G ,  F ,  H )  =  [_ w  /  b ]_ F
)
5049adantl 468 . . . . . 6  |-  ( ( ( ps  /\  w  e.  N )  /\  (
a  =  G  /\  b  =  w )
)  ->  if (
a  =  G ,  F ,  H )  =  [_ w  /  b ]_ F )
5147, 50eqtrd 2485 . . . . 5  |-  ( ( ( ps  /\  w  e.  N )  /\  (
a  =  G  /\  b  =  w )
)  ->  if (
a  =  E ,  F ,  if (
a  =  G ,  F ,  H )
)  =  [_ w  /  b ]_ F
)
52 eqidd 2452 . . . . 5  |-  ( ( ( ps  /\  w  e.  N )  /\  a  =  G )  ->  N  =  N )
5321simp2d 1021 . . . . . 6  |-  ( ps 
->  G  e.  N
)
5453adantr 467 . . . . 5  |-  ( ( ps  /\  w  e.  N )  ->  G  e.  N )
55 nfcv 2592 . . . . 5  |-  F/_ b G
5615, 51, 52, 54, 24, 33, 34, 25, 55, 36, 37, 26ovmpt2dxf 6422 . . . 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
)
5738, 56eqtr4d 2488 . . 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 ) )
5857ralrimiva 2802 . 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 ) )
59 mdetuni.0g . . 3  |-  .0.  =  ( 0g `  R )
60 mdetuni.1r . . 3  |-  .1.  =  ( 1r `  R )
61 mdetuni.pg . . 3  |-  .+  =  ( +g  `  R )
62 mdetuni.tg . . 3  |-  .x.  =  ( .r `  R )
63 mdetuni.ff . . 3  |-  ( ph  ->  D : B --> K )
64 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.  ) )
65 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 ) ) ) )
66 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 ) ) ) )
672, 4, 3, 59, 60, 61, 62, 5, 7, 63, 64, 65, 66mdetunilem1 19637 . 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.  )
681, 14, 58, 21, 67syl31anc 1271 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 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   [_csb 3363    \ cdif 3401   ifcif 3881   {csn 3968    X. cxp 4832    |` cres 4836   -->wf 5578   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292    oFcof 6529   Fincfn 7569   Basecbs 15121   +g cplusg 15190   .rcmulr 15191   0gc0g 15338   1rcur 17735   Ringcrg 17780   Mat cmat 19432
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-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-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-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-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-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  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-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-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-prds 15346  df-pws 15348  df-sra 18395  df-rgmod 18396  df-dsmm 19295  df-frlm 19310  df-mat 19433
This theorem is referenced by:  mdetunilem6  19642  mdetunilem8  19644
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