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Theorem mamutpos 19407
Description: Behavior of transposes in matrix products, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
Hypotheses
Ref Expression
mamutpos.f  |-  F  =  ( R maMul  <. M ,  N ,  P >. )
mamutpos.g  |-  G  =  ( R maMul  <. P ,  N ,  M >. )
mamutpos.b  |-  B  =  ( Base `  R
)
mamutpos.r  |-  ( ph  ->  R  e.  CRing )
mamutpos.m  |-  ( ph  ->  M  e.  Fin )
mamutpos.n  |-  ( ph  ->  N  e.  Fin )
mamutpos.p  |-  ( ph  ->  P  e.  Fin )
mamutpos.x  |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )
mamutpos.y  |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )
Assertion
Ref Expression
mamutpos  |-  ( ph  -> tpos  ( X F Y )  =  (tpos  Y Gtpos  X ) )

Proof of Theorem mamutpos
Dummy variables  i 
j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2420 . . . 4  |-  ( j  e.  M ,  i  e.  P  |->  ( R 
gsumg  ( k  e.  N  |->  ( ( j X k ) ( .r
`  R ) ( k Y i ) ) ) ) )  =  ( j  e.  M ,  i  e.  P  |->  ( R  gsumg  ( k  e.  N  |->  ( ( j X k ) ( .r `  R
) ( k Y i ) ) ) ) )
21tposmpt2 7009 . . 3  |- tpos  ( j  e.  M ,  i  e.  P  |->  ( R 
gsumg  ( k  e.  N  |->  ( ( j X k ) ( .r
`  R ) ( k Y i ) ) ) ) )  =  ( i  e.  P ,  j  e.  M  |->  ( R  gsumg  ( k  e.  N  |->  ( ( j X k ) ( .r `  R
) ( k Y i ) ) ) ) )
3 simpl1 1008 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  ph )
4 mamutpos.r . . . . . . . . 9  |-  ( ph  ->  R  e.  CRing )
53, 4syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  R  e.  CRing )
6 mamutpos.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )
7 elmapi 7492 . . . . . . . . . 10  |-  ( X  e.  ( B  ^m  ( M  X.  N
) )  ->  X : ( M  X.  N ) --> B )
83, 6, 73syl 18 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  X : ( M  X.  N ) --> B )
9 simpl3 1010 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  j  e.  M )
10 simpr 462 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  k  e.  N )
118, 9, 10fovrnd 6446 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  (
j X k )  e.  B )
12 mamutpos.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )
13 elmapi 7492 . . . . . . . . . 10  |-  ( Y  e.  ( B  ^m  ( N  X.  P
) )  ->  Y : ( N  X.  P ) --> B )
143, 12, 133syl 18 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  Y : ( N  X.  P ) --> B )
15 simpl2 1009 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  i  e.  P )
1614, 10, 15fovrnd 6446 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  (
k Y i )  e.  B )
17 mamutpos.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
18 eqid 2420 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
1917, 18crngcom 17723 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  (
j X k )  e.  B  /\  (
k Y i )  e.  B )  -> 
( ( j X k ) ( .r
`  R ) ( k Y i ) )  =  ( ( k Y i ) ( .r `  R
) ( j X k ) ) )
205, 11, 16, 19syl3anc 1264 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  (
( j X k ) ( .r `  R ) ( k Y i ) )  =  ( ( k Y i ) ( .r `  R ) ( j X k ) ) )
21 ovtpos 6987 . . . . . . . 8  |-  ( itpos 
Y k )  =  ( k Y i )
22 ovtpos 6987 . . . . . . . 8  |-  ( ktpos 
X j )  =  ( j X k )
2321, 22oveq12i 6308 . . . . . . 7  |-  ( ( itpos  Y k ) ( .r `  R
) ( ktpos  X
j ) )  =  ( ( k Y i ) ( .r
`  R ) ( j X k ) )
2420, 23syl6eqr 2479 . . . . . 6  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  (
( j X k ) ( .r `  R ) ( k Y i ) )  =  ( ( itpos 
Y k ) ( .r `  R ) ( ktpos  X j ) ) )
2524mpteq2dva 4503 . . . . 5  |-  ( (
ph  /\  i  e.  P  /\  j  e.  M
)  ->  ( k  e.  N  |->  ( ( j X k ) ( .r `  R
) ( k Y i ) ) )  =  ( k  e.  N  |->  ( ( itpos 
Y k ) ( .r `  R ) ( ktpos  X j ) ) ) )
2625oveq2d 6312 . . . 4  |-  ( (
ph  /\  i  e.  P  /\  j  e.  M
)  ->  ( R  gsumg  ( k  e.  N  |->  ( ( j X k ) ( .r `  R ) ( k Y i ) ) ) )  =  ( R  gsumg  ( k  e.  N  |->  ( ( itpos  Y
k ) ( .r
`  R ) ( ktpos  X j ) ) ) ) )
2726mpt2eq3dva 6360 . . 3  |-  ( ph  ->  ( i  e.  P ,  j  e.  M  |->  ( R  gsumg  ( k  e.  N  |->  ( ( j X k ) ( .r
`  R ) ( k Y i ) ) ) ) )  =  ( i  e.  P ,  j  e.  M  |->  ( R  gsumg  ( k  e.  N  |->  ( ( itpos  Y k ) ( .r `  R
) ( ktpos  X
j ) ) ) ) ) )
282, 27syl5eq 2473 . 2  |-  ( ph  -> tpos  ( j  e.  M ,  i  e.  P  |->  ( R  gsumg  ( k  e.  N  |->  ( ( j X k ) ( .r
`  R ) ( k Y i ) ) ) ) )  =  ( i  e.  P ,  j  e.  M  |->  ( R  gsumg  ( k  e.  N  |->  ( ( itpos  Y k ) ( .r `  R
) ( ktpos  X
j ) ) ) ) ) )
29 mamutpos.f . . . 4  |-  F  =  ( R maMul  <. M ,  N ,  P >. )
30 mamutpos.m . . . 4  |-  ( ph  ->  M  e.  Fin )
31 mamutpos.n . . . 4  |-  ( ph  ->  N  e.  Fin )
32 mamutpos.p . . . 4  |-  ( ph  ->  P  e.  Fin )
3329, 17, 18, 4, 30, 31, 32, 6, 12mamuval 19335 . . 3  |-  ( ph  ->  ( X F Y )  =  ( j  e.  M ,  i  e.  P  |->  ( R 
gsumg  ( k  e.  N  |->  ( ( j X k ) ( .r
`  R ) ( k Y i ) ) ) ) ) )
3433tposeqd 6975 . 2  |-  ( ph  -> tpos  ( X F Y )  = tpos  ( j  e.  M ,  i  e.  P  |->  ( R 
gsumg  ( k  e.  N  |->  ( ( j X k ) ( .r
`  R ) ( k Y i ) ) ) ) ) )
35 mamutpos.g . . 3  |-  G  =  ( R maMul  <. P ,  N ,  M >. )
36 tposmap 19406 . . . 4  |-  ( Y  e.  ( B  ^m  ( N  X.  P
) )  -> tpos  Y  e.  ( B  ^m  ( P  X.  N ) ) )
3712, 36syl 17 . . 3  |-  ( ph  -> tpos  Y  e.  ( B  ^m  ( P  X.  N ) ) )
38 tposmap 19406 . . . 4  |-  ( X  e.  ( B  ^m  ( M  X.  N
) )  -> tpos  X  e.  ( B  ^m  ( N  X.  M ) ) )
396, 38syl 17 . . 3  |-  ( ph  -> tpos  X  e.  ( B  ^m  ( N  X.  M ) ) )
4035, 17, 18, 4, 32, 31, 30, 37, 39mamuval 19335 . 2  |-  ( ph  ->  (tpos  Y Gtpos  X
)  =  ( i  e.  P ,  j  e.  M  |->  ( R 
gsumg  ( k  e.  N  |->  ( ( itpos  Y
k ) ( .r
`  R ) ( ktpos  X j ) ) ) ) ) )
4128, 34, 403eqtr4d 2471 1  |-  ( ph  -> tpos  ( X F Y )  =  (tpos  Y Gtpos  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   <.cotp 4001    |-> cmpt 4475    X. cxp 4843   -->wf 5588   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298  tpos ctpos 6971    ^m cmap 7471   Fincfn 7568   Basecbs 15073   .rcmulr 15143    gsumg cgsu 15291   CRingccrg 17709   maMul cmmul 19332
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 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
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 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-ot 4002  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-tpos 6972  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-plusg 15155  df-cmn 17360  df-mgp 17652  df-cring 17711  df-mamu 19333
This theorem is referenced by:  mattposm  19408
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