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Theorem mamutpos 18355
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 2443 . . . 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 6794 . . 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 991 . . . . . . . . 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 7246 . . . . . . . . . 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 993 . . . . . . . . 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 6247 . . . . . . . 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 7246 . . . . . . . . . 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 992 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  P  /\  j  e.  M )  /\  k  e.  N )  ->  i  e.  P )
1614, 10, 15fovrnd 6247 . . . . . . . 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 2443 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
1917, 18crngcom 16671 . . . . . . . 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 1218 . . . . . . 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 6772 . . . . . . . 8  |-  ( itpos 
Y k )  =  ( k Y i )
22 ovtpos 6772 . . . . . . . 8  |-  ( ktpos 
X j )  =  ( j X k )
2321, 22oveq12i 6115 . . . . . . 7  |-  ( ( itpos  Y k ) ( .r `  R
) ( ktpos  X
j ) )  =  ( ( k Y i ) ( .r
`  R ) ( j X k ) )
2420, 23syl6eqr 2493 . . . . . 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 4390 . . . . 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 6119 . . . 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 6162 . . 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 2487 . 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 18296 . . 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 6760 . 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 18354 . . . 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 18354 . . . 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 18296 . 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 2485 1  |-  ( ph  -> tpos  ( X F Y )  =  (tpos  Y Gtpos  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   <.cotp 3897    e. cmpt 4362    X. cxp 4850   -->wf 5426   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105  tpos ctpos 6756    ^m cmap 7226   Fincfn 7322   Basecbs 14186   .rcmulr 14251    gsumg cgsu 14391   CRingccrg 16658   maMul cmmul 18291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-ot 3898  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-tpos 6757  df-recs 6844  df-rdg 6878  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-plusg 14263  df-cmn 16291  df-mgp 16604  df-cring 16660  df-mamu 18293
This theorem is referenced by:  mattposm  18356
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