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Theorem frlmgsum 19096
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.) (Revised by AV, 23-Jun-2019.)
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 ) ) finSupp  .0.  )
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 19077 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
61, 2, 5syl2anc 659 . . 3  |-  ( ph  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
76oveq1d 6292 . 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 2402 . . 3  |-  ( Base `  ( (ringLMod `  R
)  ^s  I ) )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) )
9 eqid 2402 . . 3  |-  ( +g  `  ( (ringLMod `  R
)  ^s  I ) )  =  ( +g  `  (
(ringLMod `  R )  ^s  I
) )
10 eqid 2402 . . 3  |-  ( ( (ringLMod `  R )  ^s  I )s  B )  =  ( ( (ringLMod `  R
)  ^s  I )s  B )
11 ovex 6305 . . . 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 2402 . . . . . 6  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  I ) )
153, 4, 14frlmlss 19078 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  I ) ) )
161, 2, 15syl2anc 659 . . . 4  |-  ( ph  ->  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )
178, 14lssss 17901 . . . 4  |-  ( B  e.  ( LSubSp `  (
(ringLMod `  R )  ^s  I
) )  ->  B  C_  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
1816, 17syl 17 . . 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 2402 . . . 4  |-  ( y  e.  J  |->  ( x  e.  I  |->  U ) )  =  ( y  e.  J  |->  ( x  e.  I  |->  U ) )
2119, 20fmptd 6032 . . 3  |-  ( ph  ->  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) : J --> B )
22 rlmlmod 18169 . . . . . 6  |-  ( R  e.  Ring  ->  (ringLMod `  R
)  e.  LMod )
231, 22syl 17 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  LMod )
24 eqid 2402 . . . . . 6  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
2524pwslmod 17934 . . . . 5  |-  ( ( (ringLMod `  R )  e.  LMod  /\  I  e.  V )  ->  (
(ringLMod `  R )  ^s  I
)  e.  LMod )
2623, 2, 25syl2anc 659 . . . 4  |-  ( ph  ->  ( (ringLMod `  R
)  ^s  I )  e.  LMod )
27 eqid 2402 . . . . 5  |-  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) )  =  ( 0g `  (
(ringLMod `  R )  ^s  I
) )
2827, 14lss0cl 17911 . . . 4  |-  ( ( ( (ringLMod `  R
)  ^s  I )  e.  LMod  /\  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )  ->  ( 0g `  ( (ringLMod `  R )  ^s  I ) )  e.  B )
2926, 16, 28syl2anc 659 . . 3  |-  ( ph  ->  ( 0g `  (
(ringLMod `  R )  ^s  I
) )  e.  B
)
30 lmodcmn 17876 . . . . . . 7  |-  ( (ringLMod `  R )  e.  LMod  -> 
(ringLMod `  R )  e. CMnd
)
3123, 30syl 17 . . . . . 6  |-  ( ph  ->  (ringLMod `  R )  e. CMnd )
32 cmnmnd 17135 . . . . . 6  |-  ( (ringLMod `  R )  e. CMnd  ->  (ringLMod `  R )  e.  Mnd )
3331, 32syl 17 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  Mnd )
3424pwsmnd 16277 . . . . 5  |-  ( ( (ringLMod `  R )  e.  Mnd  /\  I  e.  V )  ->  (
(ringLMod `  R )  ^s  I
)  e.  Mnd )
3533, 2, 34syl2anc 659 . . . 4  |-  ( ph  ->  ( (ringLMod `  R
)  ^s  I )  e.  Mnd )
368, 9, 27mndlrid 16262 . . . 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 469 . . 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 16225 . 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 18159 . . . 4  |-  ( Base `  R )  =  (
Base `  (ringLMod `  R
) )
402adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  J )  ->  I  e.  V )
41 eqid 2402 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
423, 41, 4frlmbasf 19088 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( x  e.  I  |->  U )  e.  B
)  ->  ( x  e.  I  |->  U ) : I --> ( Base `  R ) )
4340, 19, 42syl2anc 659 . . . . . . . 8  |-  ( (
ph  /\  y  e.  J )  ->  (
x  e.  I  |->  U ) : I --> ( Base `  R ) )
44 eqid 2402 . . . . . . . . 9  |-  ( x  e.  I  |->  U )  =  ( x  e.  I  |->  U )
4544fmpt 6029 . . . . . . . 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 2772 . . . . . 6  |-  ( ( ( ph  /\  y  e.  J )  /\  x  e.  I )  ->  U  e.  ( Base `  R
) )
4847an32s 805 . . . . 5  |-  ( ( ( ph  /\  x  e.  I )  /\  y  e.  J )  ->  U  e.  ( Base `  R
) )
4948anasss 645 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  J ) )  ->  U  e.  ( Base `  R ) )
50 frlmgsum.w . . . . 5  |-  ( ph  ->  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) finSupp  .0.  )
51 frlmgsum.z . . . . . 6  |-  .0.  =  ( 0g `  Y )
526fveq2d 5852 . . . . . . 7  |-  ( ph  ->  ( 0g `  Y
)  =  ( 0g
`  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
5314lsssubg 17921 . . . . . . . . 9  |-  ( ( ( (ringLMod `  R
)  ^s  I )  e.  LMod  /\  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )  ->  B  e.  (SubGrp `  ( (ringLMod `  R
)  ^s  I ) ) )
5426, 16, 53syl2anc 659 . . . . . . . 8  |-  ( ph  ->  B  e.  (SubGrp `  ( (ringLMod `  R )  ^s  I ) ) )
5510, 27subg0 16529 . . . . . . . 8  |-  ( B  e.  (SubGrp `  (
(ringLMod `  R )  ^s  I
) )  ->  ( 0g `  ( (ringLMod `  R
)  ^s  I ) )  =  ( 0g `  (
( (ringLMod `  R )  ^s  I )s  B ) ) )
5654, 55syl 17 . . . . . . 7  |-  ( ph  ->  ( 0g `  (
(ringLMod `  R )  ^s  I
) )  =  ( 0g `  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
5752, 56eqtr4d 2446 . . . . . 6  |-  ( ph  ->  ( 0g `  Y
)  =  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) )
5851, 57syl5eq 2455 . . . . 5  |-  ( ph  ->  .0.  =  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) )
5950, 58breqtrd 4418 . . . 4  |-  ( ph  ->  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) finSupp  ( 0g `  ( (ringLMod `  R
)  ^s  I ) ) )
6024, 39, 27, 2, 13, 31, 49, 59pwsgsum 17327 . . 3  |-  ( ph  ->  ( ( (ringLMod `  R
)  ^s  I )  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( (ringLMod `  R
)  gsumg  ( y  e.  J  |->  U ) ) ) )
61 mptexg 6122 . . . . . 6  |-  ( J  e.  W  ->  (
y  e.  J  |->  U )  e.  _V )
6213, 61syl 17 . . . . 5  |-  ( ph  ->  ( y  e.  J  |->  U )  e.  _V )
63 fvex 5858 . . . . . 6  |-  (ringLMod `  R
)  e.  _V
6463a1i 11 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  _V )
6539a1i 11 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (ringLMod `  R )
) )
66 rlmplusg 18160 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  (ringLMod `  R
) )
6766a1i 11 . . . . 5  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  (ringLMod `  R )
) )
6862, 1, 64, 65, 67gsumpropd 16221 . . . 4  |-  ( ph  ->  ( R  gsumg  ( y  e.  J  |->  U ) )  =  ( (ringLMod `  R
)  gsumg  ( y  e.  J  |->  U ) ) )
6968mpteq2dv 4481 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) )  =  ( x  e.  I  |->  ( (ringLMod `  R
)  gsumg  ( y  e.  J  |->  U ) ) ) )
7060, 69eqtr4d 2446 . 2  |-  ( ph  ->  ( ( (ringLMod `  R
)  ^s  I )  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
717, 38, 703eqtr2d 2449 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 367    = wceq 1405    e. wcel 1842   A.wral 2753   _Vcvv 3058    C_ wss 3413   class class class wbr 4394    |-> cmpt 4452   -->wf 5564   ` cfv 5568  (class class class)co 6277   finSupp cfsupp 7862   Basecbs 14839   ↾s cress 14840   +g cplusg 14907   0gc0g 15052    gsumg cgsu 15053    ^s cpws 15059   Mndcmnd 16241  SubGrpcsubg 16517  CMndccmn 17120   Ringcrg 17516   LModclmod 17830   LSubSpclss 17896  ringLModcrglmod 18133   freeLMod cfrlm 19073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-sup 7934  df-oi 7968  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-fz 11725  df-fzo 11853  df-seq 12150  df-hash 12451  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-hom 14931  df-cco 14932  df-0g 15054  df-gsum 15055  df-prds 15060  df-pws 15062  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-mhm 16288  df-grp 16379  df-minusg 16380  df-sbg 16381  df-subg 16520  df-cntz 16677  df-cmn 17122  df-abl 17123  df-mgp 17460  df-ur 17472  df-ring 17518  df-subrg 17745  df-lmod 17832  df-lss 17897  df-sra 18136  df-rgmod 18137  df-dsmm 19059  df-frlm 19074
This theorem is referenced by:  uvcresum  19118  matgsum  19229  aacllem  38841
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