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Theorem frlmgsum 27232
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 27218 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
61, 2, 5syl2anc 642 . . 3  |-  ( ph  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
76oveq1d 5873 . 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 2283 . . 3  |-  ( Base `  ( (ringLMod `  R
)  ^s  I ) )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) )
9 eqid 2283 . . 3  |-  ( +g  `  ( (ringLMod `  R
)  ^s  I ) )  =  ( +g  `  (
(ringLMod `  R )  ^s  I
) )
10 eqid 2283 . . 3  |-  ( ( (ringLMod `  R )  ^s  I )s  B )  =  ( ( (ringLMod `  R
)  ^s  I )s  B )
11 ovex 5883 . . . 4  |-  ( (ringLMod `  R )  ^s  I )  e.  _V
1211a1i 10 . . 3  |-  ( ph  ->  ( (ringLMod `  R
)  ^s  I )  e.  _V )
13 frlmgsum.j . . 3  |-  ( ph  ->  J  e.  W )
14 eqid 2283 . . . . . 6  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  I ) )
153, 4, 14frlmlss 27219 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  I ) ) )
161, 2, 15syl2anc 642 . . . 4  |-  ( ph  ->  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )
178, 14lssss 15694 . . . 4  |-  ( B  e.  ( LSubSp `  (
(ringLMod `  R )  ^s  I
) )  ->  B  C_  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
1816, 17syl 15 . . 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 2283 . . . 4  |-  ( y  e.  J  |->  ( x  e.  I  |->  U ) )  =  ( y  e.  J  |->  ( x  e.  I  |->  U ) )
2119, 20fmptd 5684 . . 3  |-  ( ph  ->  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) : J --> B )
22 rlmlmod 15957 . . . . . 6  |-  ( R  e.  Ring  ->  (ringLMod `  R
)  e.  LMod )
231, 22syl 15 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  LMod )
24 eqid 2283 . . . . . 6  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
2524pwslmod 15727 . . . . 5  |-  ( ( (ringLMod `  R )  e.  LMod  /\  I  e.  V )  ->  (
(ringLMod `  R )  ^s  I
)  e.  LMod )
2623, 2, 25syl2anc 642 . . . 4  |-  ( ph  ->  ( (ringLMod `  R
)  ^s  I )  e.  LMod )
27 eqid 2283 . . . . 5  |-  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) )  =  ( 0g `  (
(ringLMod `  R )  ^s  I
) )
2827, 14lss0cl 15704 . . . 4  |-  ( ( ( (ringLMod `  R
)  ^s  I )  e.  LMod  /\  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )  ->  ( 0g `  ( (ringLMod `  R )  ^s  I ) )  e.  B )
2926, 16, 28syl2anc 642 . . 3  |-  ( ph  ->  ( 0g `  (
(ringLMod `  R )  ^s  I
) )  e.  B
)
30 lmodcmn 15673 . . . . . . 7  |-  ( (ringLMod `  R )  e.  LMod  -> 
(ringLMod `  R )  e. CMnd
)
3123, 30syl 15 . . . . . 6  |-  ( ph  ->  (ringLMod `  R )  e. CMnd )
32 cmnmnd 15104 . . . . . 6  |-  ( (ringLMod `  R )  e. CMnd  ->  (ringLMod `  R )  e.  Mnd )
3331, 32syl 15 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  Mnd )
3424pwsmnd 14407 . . . . 5  |-  ( ( (ringLMod `  R )  e.  Mnd  /\  I  e.  V )  ->  (
(ringLMod `  R )  ^s  I
)  e.  Mnd )
3533, 2, 34syl2anc 642 . . . 4  |-  ( ph  ->  ( (ringLMod `  R
)  ^s  I )  e.  Mnd )
368, 9, 27mndlrid 14392 . . . 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 457 . . 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 14454 . 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 15948 . . . 4  |-  ( Base `  R )  =  (
Base `  (ringLMod `  R
) )
402adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  J )  ->  I  e.  V )
41 eqid 2283 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
423, 41, 4frlmbasf 27228 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( x  e.  I  |->  U )  e.  B
)  ->  ( x  e.  I  |->  U ) : I --> ( Base `  R ) )
4340, 19, 42syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  y  e.  J )  ->  (
x  e.  I  |->  U ) : I --> ( Base `  R ) )
44 eqid 2283 . . . . . . . . 9  |-  ( x  e.  I  |->  U )  =  ( x  e.  I  |->  U )
4544fmpt 5681 . . . . . . . 8  |-  ( A. x  e.  I  U  e.  ( Base `  R
)  <->  ( x  e.  I  |->  U ) : I --> ( Base `  R
) )
4643, 45sylibr 203 . . . . . . 7  |-  ( (
ph  /\  y  e.  J )  ->  A. x  e.  I  U  e.  ( Base `  R )
)
4746r19.21bi 2641 . . . . . 6  |-  ( ( ( ph  /\  y  e.  J )  /\  x  e.  I )  ->  U  e.  ( Base `  R
) )
4847an32s 779 . . . . 5  |-  ( ( ( ph  /\  x  e.  I )  /\  y  e.  J )  ->  U  e.  ( Base `  R
) )
4948anasss 628 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  J ) )  ->  U  e.  ( Base `  R ) )
50 frlmgsum.z . . . . . . . . 9  |-  .0.  =  ( 0g `  Y )
516fveq2d 5529 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  Y
)  =  ( 0g
`  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
5214lsssubg 15714 . . . . . . . . . . . 12  |-  ( ( ( (ringLMod `  R
)  ^s  I )  e.  LMod  /\  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )  ->  B  e.  (SubGrp `  ( (ringLMod `  R
)  ^s  I ) ) )
5326, 16, 52syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  (SubGrp `  ( (ringLMod `  R )  ^s  I ) ) )
5410, 27subg0 14627 . . . . . . . . . . 11  |-  ( B  e.  (SubGrp `  (
(ringLMod `  R )  ^s  I
) )  ->  ( 0g `  ( (ringLMod `  R
)  ^s  I ) )  =  ( 0g `  (
( (ringLMod `  R )  ^s  I )s  B ) ) )
5553, 54syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  (
(ringLMod `  R )  ^s  I
) )  =  ( 0g `  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
5651, 55eqtr4d 2318 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  Y
)  =  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) )
5750, 56syl5eq 2327 . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) )
5857sneqd 3653 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  ( (ringLMod `  R )  ^s  I ) ) } )
5958difeq2d 3294 . . . . . 6  |-  ( ph  ->  ( _V  \  {  .0.  } )  =  ( _V  \  { ( 0g `  ( (ringLMod `  R )  ^s  I ) ) } ) )
6059imaeq2d 5012 . . . . 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 2358 . . . 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 15230 . . 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 5745 . . . . . 6  |-  ( J  e.  W  ->  (
y  e.  J  |->  U )  e.  _V )
6513, 64syl 15 . . . . 5  |-  ( ph  ->  ( y  e.  J  |->  U )  e.  _V )
66 fvex 5539 . . . . . 6  |-  (ringLMod `  R
)  e.  _V
6766a1i 10 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  _V )
6839a1i 10 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (ringLMod `  R )
) )
69 rlmplusg 15949 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  (ringLMod `  R
) )
7069a1i 10 . . . . 5  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  (ringLMod `  R )
) )
7165, 1, 67, 68, 70gsumpropd 14453 . . . 4  |-  ( ph  ->  ( R  gsumg  ( y  e.  J  |->  U ) )  =  ( (ringLMod `  R
)  gsumg  ( y  e.  J  |->  U ) ) )
7271mpteq2dv 4107 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) )  =  ( x  e.  I  |->  ( (ringLMod `  R
)  gsumg  ( y  e.  J  |->  U ) ) ) )
7363, 72eqtr4d 2318 . 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 2321 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640    e. cmpt 4077   `'ccnv 4688   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   ↾s cress 13149   +g cplusg 13208    ^s cpws 13347   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361  SubGrpcsubg 14615  CMndccmn 15089   Ringcrg 15337   LModclmod 15627   LSubSpclss 15689  ringLModcrglmod 15922   freeLMod cfrlm 27212
This theorem is referenced by:  uvcresum  27242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-gsum 13405  df-mnd 14367  df-mhm 14415  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-sra 15925  df-rgmod 15926  df-dsmm 27198  df-frlm 27214
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