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Theorem frlmgsumOLD 18290
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.) Obsolete version of frlmgsum 18291 as of 23-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
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 )
frlmgsumOLD.w  |-  ( ph  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) ) " ( _V 
\  {  .0.  }
) )  e.  Fin )
Assertion
Ref Expression
frlmgsumOLD  |-  ( 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 frlmgsumOLD
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 18270 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
61, 2, 5syl2anc 661 . . 3  |-  ( ph  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
76oveq1d 6191 . 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 2450 . . 3  |-  ( Base `  ( (ringLMod `  R
)  ^s  I ) )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) )
9 eqid 2450 . . 3  |-  ( +g  `  ( (ringLMod `  R
)  ^s  I ) )  =  ( +g  `  (
(ringLMod `  R )  ^s  I
) )
10 eqid 2450 . . 3  |-  ( ( (ringLMod `  R )  ^s  I )s  B )  =  ( ( (ringLMod `  R
)  ^s  I )s  B )
11 ovex 6201 . . . 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 2450 . . . . . 6  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  I ) )
153, 4, 14frlmlss 18271 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  I ) ) )
161, 2, 15syl2anc 661 . . . 4  |-  ( ph  ->  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )
178, 14lssss 17110 . . . 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 2450 . . . 4  |-  ( y  e.  J  |->  ( x  e.  I  |->  U ) )  =  ( y  e.  J  |->  ( x  e.  I  |->  U ) )
2119, 20fmptd 5952 . . 3  |-  ( ph  ->  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) : J --> B )
22 rlmlmod 17378 . . . . . 6  |-  ( R  e.  Ring  ->  (ringLMod `  R
)  e.  LMod )
231, 22syl 16 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  LMod )
24 eqid 2450 . . . . . 6  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
2524pwslmod 17143 . . . . 5  |-  ( ( (ringLMod `  R )  e.  LMod  /\  I  e.  V )  ->  (
(ringLMod `  R )  ^s  I
)  e.  LMod )
2623, 2, 25syl2anc 661 . . . 4  |-  ( ph  ->  ( (ringLMod `  R
)  ^s  I )  e.  LMod )
27 eqid 2450 . . . . 5  |-  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) )  =  ( 0g `  (
(ringLMod `  R )  ^s  I
) )
2827, 14lss0cl 17120 . . . 4  |-  ( ( ( (ringLMod `  R
)  ^s  I )  e.  LMod  /\  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )  ->  ( 0g `  ( (ringLMod `  R )  ^s  I ) )  e.  B )
2926, 16, 28syl2anc 661 . . 3  |-  ( ph  ->  ( 0g `  (
(ringLMod `  R )  ^s  I
) )  e.  B
)
30 lmodcmn 17085 . . . . . . 7  |-  ( (ringLMod `  R )  e.  LMod  -> 
(ringLMod `  R )  e. CMnd
)
3123, 30syl 16 . . . . . 6  |-  ( ph  ->  (ringLMod `  R )  e. CMnd )
32 cmnmnd 16382 . . . . . 6  |-  ( (ringLMod `  R )  e. CMnd  ->  (ringLMod `  R )  e.  Mnd )
3331, 32syl 16 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  Mnd )
3424pwsmnd 15544 . . . . 5  |-  ( ( (ringLMod `  R )  e.  Mnd  /\  I  e.  V )  ->  (
(ringLMod `  R )  ^s  I
)  e.  Mnd )
3533, 2, 34syl2anc 661 . . . 4  |-  ( ph  ->  ( (ringLMod `  R
)  ^s  I )  e.  Mnd )
368, 9, 27mndlrid 15528 . . . 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 471 . . 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 15595 . 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 17368 . . . 4  |-  ( Base `  R )  =  (
Base `  (ringLMod `  R
) )
402adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  J )  ->  I  e.  V )
41 eqid 2450 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
423, 41, 4frlmbasf 18283 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( x  e.  I  |->  U )  e.  B
)  ->  ( x  e.  I  |->  U ) : I --> ( Base `  R ) )
4340, 19, 42syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  y  e.  J )  ->  (
x  e.  I  |->  U ) : I --> ( Base `  R ) )
44 eqid 2450 . . . . . . . . 9  |-  ( x  e.  I  |->  U )  =  ( x  e.  I  |->  U )
4544fmpt 5949 . . . . . . . 8  |-  ( A. x  e.  I  U  e.  ( Base `  R
)  <->  ( x  e.  I  |->  U ) : I --> ( Base `  R
) )
4643, 45sylibr 212 . . . . . . 7  |-  ( (
ph  /\  y  e.  J )  ->  A. x  e.  I  U  e.  ( Base `  R )
)
4746r19.21bi 2896 . . . . . 6  |-  ( ( ( ph  /\  y  e.  J )  /\  x  e.  I )  ->  U  e.  ( Base `  R
) )
4847an32s 802 . . . . 5  |-  ( ( ( ph  /\  x  e.  I )  /\  y  e.  J )  ->  U  e.  ( Base `  R
) )
4948anasss 647 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  J ) )  ->  U  e.  ( Base `  R ) )
50 frlmgsum.z . . . . . . . . 9  |-  .0.  =  ( 0g `  Y )
516fveq2d 5779 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  Y
)  =  ( 0g
`  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
5214lsssubg 17130 . . . . . . . . . . . 12  |-  ( ( ( (ringLMod `  R
)  ^s  I )  e.  LMod  /\  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )  ->  B  e.  (SubGrp `  ( (ringLMod `  R
)  ^s  I ) ) )
5326, 16, 52syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  (SubGrp `  ( (ringLMod `  R )  ^s  I ) ) )
5410, 27subg0 15775 . . . . . . . . . . 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 2493 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  Y
)  =  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) )
5750, 56syl5eq 2502 . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) )
5857sneqd 3973 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  ( (ringLMod `  R )  ^s  I ) ) } )
5958difeq2d 3558 . . . . . 6  |-  ( ph  ->  ( _V  \  {  .0.  } )  =  ( _V  \  { ( 0g `  ( (ringLMod `  R )  ^s  I ) ) } ) )
6059imaeq2d 5253 . . . . 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 frlmgsumOLD.w . . . . 5  |-  ( ph  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) ) " ( _V 
\  {  .0.  }
) )  e.  Fin )
6260, 61eqeltrrd 2537 . . . 4  |-  ( ph  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) ) " ( _V 
\  { ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) } ) )  e.  Fin )
6324, 39, 27, 2, 13, 31, 49, 62pwsgsumOLD 16565 . . 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 6032 . . . . . 6  |-  ( J  e.  W  ->  (
y  e.  J  |->  U )  e.  _V )
6513, 64syl 16 . . . . 5  |-  ( ph  ->  ( y  e.  J  |->  U )  e.  _V )
66 fvex 5785 . . . . . 6  |-  (ringLMod `  R
)  e.  _V
6766a1i 11 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  _V )
6839a1i 11 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (ringLMod `  R )
) )
69 rlmplusg 17369 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  (ringLMod `  R
) )
7069a1i 11 . . . . 5  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  (ringLMod `  R )
) )
7165, 1, 67, 68, 70gsumpropd 15592 . . . 4  |-  ( ph  ->  ( R  gsumg  ( y  e.  J  |->  U ) )  =  ( (ringLMod `  R
)  gsumg  ( y  e.  J  |->  U ) ) )
7271mpteq2dv 4463 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) )  =  ( x  e.  I  |->  ( (ringLMod `  R
)  gsumg  ( y  e.  J  |->  U ) ) ) )
7363, 72eqtr4d 2493 . 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 2496 1  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757   A.wral 2792   _Vcvv 3054    \ cdif 3409    C_ wss 3412   {csn 3961    |-> cmpt 4434   `'ccnv 4923   "cima 4927   -->wf 5498   ` cfv 5502  (class class class)co 6176   Fincfn 7396   Basecbs 14262   ↾s cress 14263   +g cplusg 14326   0gc0g 14466    gsumg cgsu 14467    ^s cpws 14473   Mndcmnd 15497  SubGrpcsubg 15763  CMndccmn 16367   Ringcrg 16737   LModclmod 17040   LSubSpclss 17105  ringLModcrglmod 17342   freeLMod cfrlm 18266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-se 4764  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-isom 5511  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-supp 6777  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-map 7302  df-ixp 7350  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-fsupp 7708  df-sup 7778  df-oi 7811  df-card 8196  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-3 10468  df-4 10469  df-5 10470  df-6 10471  df-7 10472  df-8 10473  df-9 10474  df-10 10475  df-n0 10667  df-z 10734  df-dec 10843  df-uz 10949  df-fz 11525  df-fzo 11636  df-seq 11894  df-hash 12191  df-struct 14264  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-ress 14269  df-plusg 14339  df-mulr 14340  df-sca 14342  df-vsca 14343  df-ip 14344  df-tset 14345  df-ple 14346  df-ds 14348  df-hom 14350  df-cco 14351  df-0g 14468  df-gsum 14469  df-prds 14474  df-pws 14476  df-mnd 15503  df-mhm 15552  df-grp 15633  df-minusg 15634  df-sbg 15635  df-subg 15766  df-cntz 15923  df-cmn 16369  df-abl 16370  df-mgp 16683  df-ur 16695  df-rng 16739  df-subrg 16955  df-lmod 17042  df-lss 17106  df-sra 17345  df-rgmod 17346  df-dsmm 18252  df-frlm 18267
This theorem is referenced by: (None)
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