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Theorem frlmgsum 27100
Description: Finite commutative sums in a free module are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 5-Jul-2015.)
Hypotheses
Ref Expression
frlmgsum.y  |-  Y  =  ( R freeLMod  I )
frlmgsum.b  |-  B  =  ( Base `  Y
)
frlmgsum.z  |-  .0.  =  ( 0g `  Y )
frlmgsum.i  |-  ( ph  ->  I  e.  V )
frlmgsum.j  |-  ( ph  ->  J  e.  W )
frlmgsum.r  |-  ( ph  ->  R  e.  Ring )
frlmgsum.f  |-  ( (
ph  /\  y  e.  J )  ->  (
x  e.  I  |->  U )  e.  B )
frlmgsum.w  |-  ( ph  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) ) " ( _V 
\  {  .0.  }
) )  e.  Fin )
Assertion
Ref Expression
frlmgsum  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
Distinct variable groups:    x, y, B    x, I, y    ph, x, y    x,  .0. , y    x, J, y    x, R, y   
x, Y, y
Allowed substitution hints:    U( x, y)    V( x, y)    W( x, y)

Proof of Theorem frlmgsum
StepHypRef Expression
1 frlmgsum.r . . . 4  |-  ( ph  ->  R  e.  Ring )
2 frlmgsum.i . . . 4  |-  ( ph  ->  I  e.  V )
3 frlmgsum.y . . . . 5  |-  Y  =  ( R freeLMod  I )
4 frlmgsum.b . . . . 5  |-  B  =  ( Base `  Y
)
53, 4frlmpws 27086 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
61, 2, 5syl2anc 643 . . 3  |-  ( ph  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
76oveq1d 6055 . 2  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( ( ( (ringLMod `  R )  ^s  I )s  B )  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) ) )
8 eqid 2404 . . 3  |-  ( Base `  ( (ringLMod `  R
)  ^s  I ) )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) )
9 eqid 2404 . . 3  |-  ( +g  `  ( (ringLMod `  R
)  ^s  I ) )  =  ( +g  `  (
(ringLMod `  R )  ^s  I
) )
10 eqid 2404 . . 3  |-  ( ( (ringLMod `  R )  ^s  I )s  B )  =  ( ( (ringLMod `  R
)  ^s  I )s  B )
11 ovex 6065 . . . 4  |-  ( (ringLMod `  R )  ^s  I )  e.  _V
1211a1i 11 . . 3  |-  ( ph  ->  ( (ringLMod `  R
)  ^s  I )  e.  _V )
13 frlmgsum.j . . 3  |-  ( ph  ->  J  e.  W )
14 eqid 2404 . . . . . 6  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  I ) )
153, 4, 14frlmlss 27087 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  I ) ) )
161, 2, 15syl2anc 643 . . . 4  |-  ( ph  ->  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )
178, 14lssss 15968 . . . 4  |-  ( B  e.  ( LSubSp `  (
(ringLMod `  R )  ^s  I
) )  ->  B  C_  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
1816, 17syl 16 . . 3  |-  ( ph  ->  B  C_  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
19 frlmgsum.f . . . 4  |-  ( (
ph  /\  y  e.  J )  ->  (
x  e.  I  |->  U )  e.  B )
20 eqid 2404 . . . 4  |-  ( y  e.  J  |->  ( x  e.  I  |->  U ) )  =  ( y  e.  J  |->  ( x  e.  I  |->  U ) )
2119, 20fmptd 5852 . . 3  |-  ( ph  ->  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) : J --> B )
22 rlmlmod 16231 . . . . . 6  |-  ( R  e.  Ring  ->  (ringLMod `  R
)  e.  LMod )
231, 22syl 16 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  LMod )
24 eqid 2404 . . . . . 6  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
2524pwslmod 16001 . . . . 5  |-  ( ( (ringLMod `  R )  e.  LMod  /\  I  e.  V )  ->  (
(ringLMod `  R )  ^s  I
)  e.  LMod )
2623, 2, 25syl2anc 643 . . . 4  |-  ( ph  ->  ( (ringLMod `  R
)  ^s  I )  e.  LMod )
27 eqid 2404 . . . . 5  |-  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) )  =  ( 0g `  (
(ringLMod `  R )  ^s  I
) )
2827, 14lss0cl 15978 . . . 4  |-  ( ( ( (ringLMod `  R
)  ^s  I )  e.  LMod  /\  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )  ->  ( 0g `  ( (ringLMod `  R )  ^s  I ) )  e.  B )
2926, 16, 28syl2anc 643 . . 3  |-  ( ph  ->  ( 0g `  (
(ringLMod `  R )  ^s  I
) )  e.  B
)
30 lmodcmn 15947 . . . . . . 7  |-  ( (ringLMod `  R )  e.  LMod  -> 
(ringLMod `  R )  e. CMnd
)
3123, 30syl 16 . . . . . 6  |-  ( ph  ->  (ringLMod `  R )  e. CMnd )
32 cmnmnd 15382 . . . . . 6  |-  ( (ringLMod `  R )  e. CMnd  ->  (ringLMod `  R )  e.  Mnd )
3331, 32syl 16 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  Mnd )
3424pwsmnd 14685 . . . . 5  |-  ( ( (ringLMod `  R )  e.  Mnd  /\  I  e.  V )  ->  (
(ringLMod `  R )  ^s  I
)  e.  Mnd )
3533, 2, 34syl2anc 643 . . . 4  |-  ( ph  ->  ( (ringLMod `  R
)  ^s  I )  e.  Mnd )
368, 9, 27mndlrid 14670 . . . 4  |-  ( ( ( (ringLMod `  R
)  ^s  I )  e.  Mnd  /\  x  e.  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )  ->  ( ( ( 0g `  ( (ringLMod `  R )  ^s  I ) ) ( +g  `  (
(ringLMod `  R )  ^s  I
) ) x )  =  x  /\  (
x ( +g  `  (
(ringLMod `  R )  ^s  I
) ) ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) )  =  x ) )
3735, 36sylan 458 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  ( (ringLMod `  R )  ^s  I ) ) )  ->  (
( ( 0g `  ( (ringLMod `  R )  ^s  I ) ) ( +g  `  ( (ringLMod `  R )  ^s  I ) ) x )  =  x  /\  ( x ( +g  `  (
(ringLMod `  R )  ^s  I
) ) ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) )  =  x ) )
388, 9, 10, 12, 13, 18, 21, 29, 37gsumress 14732 . 2  |-  ( ph  ->  ( ( (ringLMod `  R
)  ^s  I )  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( ( ( (ringLMod `  R )  ^s  I )s  B )  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) ) )
39 rlmbas 16222 . . . 4  |-  ( Base `  R )  =  (
Base `  (ringLMod `  R
) )
402adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  J )  ->  I  e.  V )
41 eqid 2404 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
423, 41, 4frlmbasf 27096 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( x  e.  I  |->  U )  e.  B
)  ->  ( x  e.  I  |->  U ) : I --> ( Base `  R ) )
4340, 19, 42syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  y  e.  J )  ->  (
x  e.  I  |->  U ) : I --> ( Base `  R ) )
44 eqid 2404 . . . . . . . . 9  |-  ( x  e.  I  |->  U )  =  ( x  e.  I  |->  U )
4544fmpt 5849 . . . . . . . 8  |-  ( A. x  e.  I  U  e.  ( Base `  R
)  <->  ( x  e.  I  |->  U ) : I --> ( Base `  R
) )
4643, 45sylibr 204 . . . . . . 7  |-  ( (
ph  /\  y  e.  J )  ->  A. x  e.  I  U  e.  ( Base `  R )
)
4746r19.21bi 2764 . . . . . 6  |-  ( ( ( ph  /\  y  e.  J )  /\  x  e.  I )  ->  U  e.  ( Base `  R
) )
4847an32s 780 . . . . 5  |-  ( ( ( ph  /\  x  e.  I )  /\  y  e.  J )  ->  U  e.  ( Base `  R
) )
4948anasss 629 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  J ) )  ->  U  e.  ( Base `  R ) )
50 frlmgsum.z . . . . . . . . 9  |-  .0.  =  ( 0g `  Y )
516fveq2d 5691 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  Y
)  =  ( 0g
`  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
5214lsssubg 15988 . . . . . . . . . . . 12  |-  ( ( ( (ringLMod `  R
)  ^s  I )  e.  LMod  /\  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )  ->  B  e.  (SubGrp `  ( (ringLMod `  R
)  ^s  I ) ) )
5326, 16, 52syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  (SubGrp `  ( (ringLMod `  R )  ^s  I ) ) )
5410, 27subg0 14905 . . . . . . . . . . 11  |-  ( B  e.  (SubGrp `  (
(ringLMod `  R )  ^s  I
) )  ->  ( 0g `  ( (ringLMod `  R
)  ^s  I ) )  =  ( 0g `  (
( (ringLMod `  R )  ^s  I )s  B ) ) )
5553, 54syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  (
(ringLMod `  R )  ^s  I
) )  =  ( 0g `  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
5651, 55eqtr4d 2439 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  Y
)  =  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) )
5750, 56syl5eq 2448 . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) )
5857sneqd 3787 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  ( (ringLMod `  R )  ^s  I ) ) } )
5958difeq2d 3425 . . . . . 6  |-  ( ph  ->  ( _V  \  {  .0.  } )  =  ( _V  \  { ( 0g `  ( (ringLMod `  R )  ^s  I ) ) } ) )
6059imaeq2d 5162 . . . . 5  |-  ( ph  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) ) " ( _V 
\  {  .0.  }
) )  =  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) )
" ( _V  \  { ( 0g `  ( (ringLMod `  R )  ^s  I ) ) } ) ) )
61 frlmgsum.w . . . . 5  |-  ( ph  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) ) " ( _V 
\  {  .0.  }
) )  e.  Fin )
6260, 61eqeltrrd 2479 . . . 4  |-  ( ph  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) ) " ( _V 
\  { ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) } ) )  e.  Fin )
6324, 39, 27, 2, 13, 31, 49, 62pwsgsum 15508 . . 3  |-  ( ph  ->  ( ( (ringLMod `  R
)  ^s  I )  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( (ringLMod `  R
)  gsumg  ( y  e.  J  |->  U ) ) ) )
64 mptexg 5924 . . . . . 6  |-  ( J  e.  W  ->  (
y  e.  J  |->  U )  e.  _V )
6513, 64syl 16 . . . . 5  |-  ( ph  ->  ( y  e.  J  |->  U )  e.  _V )
66 fvex 5701 . . . . . 6  |-  (ringLMod `  R
)  e.  _V
6766a1i 11 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  _V )
6839a1i 11 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (ringLMod `  R )
) )
69 rlmplusg 16223 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  (ringLMod `  R
) )
7069a1i 11 . . . . 5  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  (ringLMod `  R )
) )
7165, 1, 67, 68, 70gsumpropd 14731 . . . 4  |-  ( ph  ->  ( R  gsumg  ( y  e.  J  |->  U ) )  =  ( (ringLMod `  R
)  gsumg  ( y  e.  J  |->  U ) ) )
7271mpteq2dv 4256 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) )  =  ( x  e.  I  |->  ( (ringLMod `  R
)  gsumg  ( y  e.  J  |->  U ) ) ) )
7363, 72eqtr4d 2439 . 2  |-  ( ph  ->  ( ( (ringLMod `  R
)  ^s  I )  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
747, 38, 733eqtr2d 2442 1  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {csn 3774    e. cmpt 4226   `'ccnv 4836   "cima 4840   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   Basecbs 13424   ↾s cress 13425   +g cplusg 13484    ^s cpws 13625   0gc0g 13678    gsumg cgsu 13679   Mndcmnd 14639  SubGrpcsubg 14893  CMndccmn 15367   Ringcrg 15615   LModclmod 15905   LSubSpclss 15963  ringLModcrglmod 16196   freeLMod cfrlm 27080
This theorem is referenced by:  uvcresum  27110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-hom 13508  df-cco 13509  df-prds 13626  df-pws 13628  df-0g 13682  df-gsum 13683  df-mnd 14645  df-mhm 14693  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-subrg 15821  df-lmod 15907  df-lss 15964  df-sra 16199  df-rgmod 16200  df-dsmm 27066  df-frlm 27082
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