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Theorem mamutpos 18767
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 2467 . . . 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 6993 . . 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 999 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  ph )
4 mamutpos.r . . . . . . . . 9  |-  ( ph  ->  R  e.  CRing )
53, 4syl 16 . . . . . . . 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 7441 . . . . . . . . . 10  |-  ( X  e.  ( B  ^m  ( M  X.  N
) )  ->  X : ( M  X.  N ) --> B )
83, 6, 73syl 20 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  X : ( M  X.  N ) --> B )
9 simpl3 1001 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  j  e.  M )
10 simpr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  k  e.  N )
118, 9, 10fovrnd 6432 . . . . . . . 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 7441 . . . . . . . . . 10  |-  ( Y  e.  ( B  ^m  ( N  X.  P
) )  ->  Y : ( N  X.  P ) --> B )
143, 12, 133syl 20 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  Y : ( N  X.  P ) --> B )
15 simpl2 1000 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  i  e.  P )
1614, 10, 15fovrnd 6432 . . . . . . . 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 2467 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
1917, 18crngcom 17026 . . . . . . . 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 1228 . . . . . . 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 6971 . . . . . . . 8  |-  ( itpos 
Y k )  =  ( k Y i )
22 ovtpos 6971 . . . . . . . 8  |-  ( ktpos 
X j )  =  ( j X k )
2321, 22oveq12i 6297 . . . . . . 7  |-  ( ( itpos  Y k ) ( .r `  R
) ( ktpos  X
j ) )  =  ( ( k Y i ) ( .r
`  R ) ( j X k ) )
2420, 23syl6eqr 2526 . . . . . 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 4533 . . . . 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 6301 . . . 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 6346 . . 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 2520 . 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 18695 . . 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 6959 . 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 18766 . . . 4  |-  ( Y  e.  ( B  ^m  ( N  X.  P
) )  -> tpos  Y  e.  ( B  ^m  ( P  X.  N ) ) )
3712, 36syl 16 . . 3  |-  ( ph  -> tpos  Y  e.  ( B  ^m  ( P  X.  N ) ) )
38 tposmap 18766 . . . 4  |-  ( X  e.  ( B  ^m  ( M  X.  N
) )  -> tpos  X  e.  ( B  ^m  ( N  X.  M ) ) )
396, 38syl 16 . . 3  |-  ( ph  -> tpos  X  e.  ( B  ^m  ( N  X.  M ) ) )
4035, 17, 18, 4, 32, 31, 30, 37, 39mamuval 18695 . 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 2518 1  |-  ( ph  -> tpos  ( X F Y )  =  (tpos  Y Gtpos  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   <.cotp 4035    |-> cmpt 4505    X. cxp 4997   -->wf 5584   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287  tpos ctpos 6955    ^m cmap 7421   Fincfn 7517   Basecbs 14493   .rcmulr 14559    gsumg cgsu 14699   CRingccrg 17013   maMul cmmul 18692
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6956  df-recs 7043  df-rdg 7077  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-plusg 14571  df-cmn 16615  df-mgp 16956  df-cring 17015  df-mamu 18693
This theorem is referenced by:  mattposm  18768
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