Theorem mamutpos 20264

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.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 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 ) ) ) ) )
2	1	tposmpt2 7389	. . 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 1064	. . . . . . . . 9 |- ( ( ( ph /\ i e. P /\ j e. M ) /\ k e. N ) -> ph )
4		mamutpos.r	. . . . . . . . 9 |- ( ph -> R e. CRing )
5	3, 4	syl 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 7879	. . . . . . . . . 10 |- ( X e. ( B ^m ( M X. N ) ) -> X : ( M X. N ) --> B )
8	3, 6, 7	3syl 18	. . . . . . . . 9 |- ( ( ( ph /\ i e. P /\ j e. M ) /\ k e. N ) -> X : ( M X. N ) --> B )
9		simpl3 1066	. . . . . . . . 9 |- ( ( ( ph /\ i e. P /\ j e. M ) /\ k e. N ) -> j e. M )
10		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ i e. P /\ j e. M ) /\ k e. N ) -> k e. N )
11	8, 9, 10	fovrnd 6806	. . . . . . . 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 7879	. . . . . . . . . 10 |- ( Y e. ( B ^m ( N X. P ) ) -> Y : ( N X. P ) --> B )
14	3, 12, 13	3syl 18	. . . . . . . . 9 |- ( ( ( ph /\ i e. P /\ j e. M ) /\ k e. N ) -> Y : ( N X. P ) --> B )
15		simpl2 1065	. . . . . . . . 9 |- ( ( ( ph /\ i e. P /\ j e. M ) /\ k e. N ) -> i e. P )
16	14, 10, 15	fovrnd 6806	. . . . . . . 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 2622	. . . . . . . . 9 |- ( .r ` R ) = ( .r ` R )
19	17, 18	crngcom 18562	. . . . . . . 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 ) ) )
20	5, 11, 16, 19	syl3anc 1326	. . . . . . 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 7367	. . . . . . . 8 |- ( itpos Y k ) = ( k Y i )
22		ovtpos 7367	. . . . . . . 8 |- ( ktpos X j ) = ( j X k )
23	21, 22	oveq12i 6662	. . . . . . 7 |- ( ( itpos Y k ) ( .r ` R ) ( ktpos X j ) ) = ( ( k Y i ) ( .r ` R ) ( j X k ) )
24	20, 23	syl6eqr 2674	. . . . . 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 ) ) )
25	24	mpteq2dva 4744	. . . . 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 ) ) ) )
26	25	oveq2d 6666	. . . 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 ) ) ) ) )
27	26	mpt2eq3dva 6719	. . 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 ) ) ) ) ) )
28	2, 27	syl5eq 2668	. 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 )
33	29, 17, 18, 4, 30, 31, 32, 6, 12	mamuval 20192	. . 3 |- ( ph -> ( X F Y ) = ( j e. M , i e. P |-> ( R gsumg ( k e. N |-> ( ( j X k ) ( .r ` R ) ( k Y i ) ) ) ) ) )
34	33	tposeqd 7355	. 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 20263	. . . 4 |- ( Y e. ( B ^m ( N X. P ) ) -> tpos Y e. ( B ^m ( P X. N ) ) )
37	12, 36	syl 17	. . 3 |- ( ph -> tpos Y e. ( B ^m ( P X. N ) ) )
38		tposmap 20263	. . . 4 |- ( X e. ( B ^m ( M X. N ) ) -> tpos X e. ( B ^m ( N X. M ) ) )
39	6, 38	syl 17	. . 3 |- ( ph -> tpos X e. ( B ^m ( N X. M ) ) )
40	35, 17, 18, 4, 32, 31, 30, 37, 39	mamuval 20192	. 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 ) ) ) ) ) )
41	28, 34, 40	3eqtr4d 2666	1 |- ( ph -> tpos ( X F Y ) = (tpos Y Gtpos X ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cotp 4185   |-> cmpt 4729   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652  tpos ctpos 7351   ^m cmap 7857   Fincfn 7955   Basecbs 15857   .rcmulr 15942   gsum_g cgsu 16101   CRingccrg 18548   maMul cmmul 20189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-cmn 18195  df-mgp 18490  df-cring 18550  df-mamu 20190

This theorem is referenced by:  mattposm  20265