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Theorem mulmarep1gsum1 19575
Description: The sum of element by element multiplications of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.)
Hypotheses
Ref Expression
marepvcl.a  |-  A  =  ( N Mat  R )
marepvcl.b  |-  B  =  ( Base `  A
)
marepvcl.v  |-  V  =  ( ( Base `  R
)  ^m  N )
ma1repvcl.1  |-  .1.  =  ( 1r `  A )
mulmarep1el.0  |-  .0.  =  ( 0g `  R )
mulmarep1el.e  |-  E  =  ( (  .1.  ( N matRepV  R ) C ) `
 K )
Assertion
Ref Expression
mulmarep1gsum1  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  ( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
`  R ) ( l E J ) ) ) )  =  ( I X J ) )
Distinct variable groups:    B, l    C, l    I, l    J, l    K, l    N, l    R, l    V, l    X, l    .0. , l
Allowed substitution hints:    A( l)    .1. ( l)    E( l)

Proof of Theorem mulmarep1gsum1
StepHypRef Expression
1 simp1 1005 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  R  e.  Ring )
21adantr 466 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/= 
K ) )  /\  l  e.  N )  ->  R  e.  Ring )
3 simp2 1006 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )
)
43adantr 466 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/= 
K ) )  /\  l  e.  N )  ->  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
) )
5 simp1 1005 . . . . . . 7  |-  ( ( I  e.  N  /\  J  e.  N  /\  J  =/=  K )  ->  I  e.  N )
653ad2ant3 1028 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  I  e.  N )
76adantr 466 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/= 
K ) )  /\  l  e.  N )  ->  I  e.  N )
8 simp2 1006 . . . . . . 7  |-  ( ( I  e.  N  /\  J  e.  N  /\  J  =/=  K )  ->  J  e.  N )
983ad2ant3 1028 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  J  e.  N )
109adantr 466 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/= 
K ) )  /\  l  e.  N )  ->  J  e.  N )
11 simpr 462 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/= 
K ) )  /\  l  e.  N )  ->  l  e.  N )
12 marepvcl.a . . . . . 6  |-  A  =  ( N Mat  R )
13 marepvcl.b . . . . . 6  |-  B  =  ( Base `  A
)
14 marepvcl.v . . . . . 6  |-  V  =  ( ( Base `  R
)  ^m  N )
15 ma1repvcl.1 . . . . . 6  |-  .1.  =  ( 1r `  A )
16 mulmarep1el.0 . . . . . 6  |-  .0.  =  ( 0g `  R )
17 mulmarep1el.e . . . . . 6  |-  E  =  ( (  .1.  ( N matRepV  R ) C ) `
 K )
1812, 13, 14, 15, 16, 17mulmarep1el 19574 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  l  e.  N
) )  ->  (
( I X l ) ( .r `  R ) ( l E J ) )  =  if ( J  =  K ,  ( ( I X l ) ( .r `  R ) ( C `
 l ) ) ,  if ( J  =  l ,  ( I X l ) ,  .0.  ) ) )
192, 4, 7, 10, 11, 18syl113anc 1276 . . . 4  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/= 
K ) )  /\  l  e.  N )  ->  ( ( I X l ) ( .r
`  R ) ( l E J ) )  =  if ( J  =  K , 
( ( I X l ) ( .r
`  R ) ( C `  l ) ) ,  if ( J  =  l ,  ( I X l ) ,  .0.  )
) )
2019mpteq2dva 4504 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  (
l  e.  N  |->  ( ( I X l ) ( .r `  R ) ( l E J ) ) )  =  ( l  e.  N  |->  if ( J  =  K , 
( ( I X l ) ( .r
`  R ) ( C `  l ) ) ,  if ( J  =  l ,  ( I X l ) ,  .0.  )
) ) )
2120oveq2d 6313 . 2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  ( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
`  R ) ( l E J ) ) ) )  =  ( R  gsumg  ( l  e.  N  |->  if ( J  =  K ,  ( ( I X l ) ( .r `  R
) ( C `  l ) ) ,  if ( J  =  l ,  ( I X l ) ,  .0.  ) ) ) ) )
22 df-ne 2618 . . . . . . . 8  |-  ( J  =/=  K  <->  -.  J  =  K )
2322biimpi 197 . . . . . . 7  |-  ( J  =/=  K  ->  -.  J  =  K )
24233ad2ant3 1028 . . . . . 6  |-  ( ( I  e.  N  /\  J  e.  N  /\  J  =/=  K )  ->  -.  J  =  K
)
25243ad2ant3 1028 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  -.  J  =  K )
2625iffalsed 3917 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  if ( J  =  K ,  ( ( I X l ) ( .r `  R ) ( C `  l
) ) ,  if ( J  =  l ,  ( I X l ) ,  .0.  ) )  =  if ( J  =  l ,  ( I X l ) ,  .0.  ) )
2726mpteq2dv 4505 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  (
l  e.  N  |->  if ( J  =  K ,  ( ( I X l ) ( .r `  R ) ( C `  l
) ) ,  if ( J  =  l ,  ( I X l ) ,  .0.  ) ) )  =  ( l  e.  N  |->  if ( J  =  l ,  ( I X l ) ,  .0.  ) ) )
2827oveq2d 6313 . 2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  ( R  gsumg  ( l  e.  N  |->  if ( J  =  K ,  ( ( I X l ) ( .r `  R
) ( C `  l ) ) ,  if ( J  =  l ,  ( I X l ) ,  .0.  ) ) ) )  =  ( R 
gsumg  ( l  e.  N  |->  if ( J  =  l ,  ( I X l ) ,  .0.  ) ) ) )
29 ringmnd 17767 . . . 4  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
30293ad2ant1 1026 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  R  e.  Mnd )
3112, 13matrcl 19414 . . . . . 6  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
3231simpld 460 . . . . 5  |-  ( X  e.  B  ->  N  e.  Fin )
33323ad2ant1 1026 . . . 4  |-  ( ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  N  e.  Fin )
34333ad2ant2 1027 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  N  e.  Fin )
35 eqcom 2429 . . . . 5  |-  ( J  =  l  <->  l  =  J )
36 ifbi 3927 . . . . . 6  |-  ( ( J  =  l  <->  l  =  J )  ->  if ( J  =  l ,  ( I X l ) ,  .0.  )  =  if (
l  =  J , 
( I X l ) ,  .0.  )
)
37 oveq2 6305 . . . . . . . 8  |-  ( l  =  J  ->  (
I X l )  =  ( I X J ) )
3837adantl 467 . . . . . . 7  |-  ( ( ( J  =  l  <-> 
l  =  J )  /\  l  =  J )  ->  ( I X l )  =  ( I X J ) )
3938ifeq1da 3936 . . . . . 6  |-  ( ( J  =  l  <->  l  =  J )  ->  if ( l  =  J ,  ( I X l ) ,  .0.  )  =  if (
l  =  J , 
( I X J ) ,  .0.  )
)
4036, 39eqtrd 2461 . . . . 5  |-  ( ( J  =  l  <->  l  =  J )  ->  if ( J  =  l ,  ( I X l ) ,  .0.  )  =  if (
l  =  J , 
( I X J ) ,  .0.  )
)
4135, 40ax-mp 5 . . . 4  |-  if ( J  =  l ,  ( I X l ) ,  .0.  )  =  if ( l  =  J ,  ( I X J ) ,  .0.  )
4241mpteq2i 4501 . . 3  |-  ( l  e.  N  |->  if ( J  =  l ,  ( I X l ) ,  .0.  )
)  =  ( l  e.  N  |->  if ( l  =  J , 
( I X J ) ,  .0.  )
)
4313eleq2i 2498 . . . . . . 7  |-  ( X  e.  B  <->  X  e.  ( Base `  A )
)
4443biimpi 197 . . . . . 6  |-  ( X  e.  B  ->  X  e.  ( Base `  A
) )
45443ad2ant1 1026 . . . . 5  |-  ( ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  X  e.  ( Base `  A ) )
46453ad2ant2 1027 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  X  e.  ( Base `  A
) )
47 eqid 2420 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
4812, 47matecl 19427 . . . 4  |-  ( ( I  e.  N  /\  J  e.  N  /\  X  e.  ( Base `  A ) )  -> 
( I X J )  e.  ( Base `  R ) )
496, 9, 46, 48syl3anc 1264 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  (
I X J )  e.  ( Base `  R
) )
5016, 30, 34, 9, 42, 49gsummptif1n0 17576 . 2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  ( R  gsumg  ( l  e.  N  |->  if ( J  =  l ,  ( I X l ) ,  .0.  ) ) )  =  ( I X J ) )
5121, 28, 503eqtrd 2465 1  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  ( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
`  R ) ( l E J ) ) ) )  =  ( I X J ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   _Vcvv 3078   ifcif 3906    |-> cmpt 4476   ` cfv 5593  (class class class)co 6297    ^m cmap 7472   Fincfn 7569   Basecbs 15099   .rcmulr 15169   0gc0g 15316    gsumg cgsu 15317   Mndcmnd 16513   1rcur 17713   Ringcrg 17758   Mat cmat 19409   matRepV cmatrepV 19559
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 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-inf2 8144  ax-cnex 9591  ax-resscn 9592  ax-1cn 9593  ax-icn 9594  ax-addcl 9595  ax-addrcl 9596  ax-mulcl 9597  ax-mulrcl 9598  ax-mulcom 9599  ax-addass 9600  ax-mulass 9601  ax-distr 9602  ax-i2m1 9603  ax-1ne0 9604  ax-1rid 9605  ax-rnegex 9606  ax-rrecex 9607  ax-cnre 9608  ax-pre-lttri 9609  ax-pre-lttrn 9610  ax-pre-ltadd 9611  ax-pre-mulgt0 9612
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  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-rmo 2781  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-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4477  df-mpt 4478  df-tr 4513  df-eprel 4757  df-id 4761  df-po 4767  df-so 4768  df-fr 4805  df-se 4806  df-we 4807  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-pred 5391  df-ord 5437  df-on 5438  df-lim 5439  df-suc 5440  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6259  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-of 6537  df-om 6699  df-1st 6799  df-2nd 6800  df-supp 6918  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 7882  df-sup 7954  df-oi 8023  df-card 8370  df-pnf 9673  df-mnf 9674  df-xr 9675  df-ltxr 9676  df-le 9677  df-sub 9858  df-neg 9859  df-nn 10606  df-2 10664  df-3 10665  df-4 10666  df-5 10667  df-6 10668  df-7 10669  df-8 10670  df-9 10671  df-10 10672  df-n0 10866  df-z 10934  df-dec 11048  df-uz 11156  df-fz 11779  df-fzo 11910  df-seq 12207  df-hash 12509  df-struct 15101  df-ndx 15102  df-slot 15103  df-base 15104  df-sets 15105  df-ress 15106  df-plusg 15181  df-mulr 15182  df-sca 15184  df-vsca 15185  df-ip 15186  df-tset 15187  df-ple 15188  df-ds 15190  df-hom 15192  df-cco 15193  df-0g 15318  df-gsum 15319  df-prds 15324  df-pws 15326  df-mre 15470  df-mrc 15471  df-acs 15473  df-mgm 16466  df-sgrp 16505  df-mnd 16515  df-mhm 16560  df-submnd 16561  df-grp 16651  df-minusg 16652  df-sbg 16653  df-mulg 16654  df-subg 16792  df-ghm 16859  df-cntz 16949  df-cmn 17410  df-abl 17411  df-mgp 17702  df-ur 17714  df-ring 17760  df-subrg 17984  df-lmod 18071  df-lss 18134  df-sra 18373  df-rgmod 18374  df-dsmm 19272  df-frlm 19287  df-mamu 19386  df-mat 19410  df-marepv 19561
This theorem is referenced by:  mulmarep1gsum2  19576
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