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Theorem frlmgsum 18569
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 18548 . . . 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 6297 . 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 2467 . . 3  |-  ( Base `  ( (ringLMod `  R
)  ^s  I ) )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) )
9 eqid 2467 . . 3  |-  ( +g  `  ( (ringLMod `  R
)  ^s  I ) )  =  ( +g  `  (
(ringLMod `  R )  ^s  I
) )
10 eqid 2467 . . 3  |-  ( ( (ringLMod `  R )  ^s  I )s  B )  =  ( ( (ringLMod `  R
)  ^s  I )s  B )
11 ovex 6307 . . . 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 2467 . . . . . 6  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  I ) )
153, 4, 14frlmlss 18549 . . . . 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 17366 . . . 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 2467 . . . 4  |-  ( y  e.  J  |->  ( x  e.  I  |->  U ) )  =  ( y  e.  J  |->  ( x  e.  I  |->  U ) )
2119, 20fmptd 6043 . . 3  |-  ( ph  ->  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) : J --> B )
22 rlmlmod 17634 . . . . . 6  |-  ( R  e.  Ring  ->  (ringLMod `  R
)  e.  LMod )
231, 22syl 16 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  LMod )
24 eqid 2467 . . . . . 6  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
2524pwslmod 17399 . . . . 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 2467 . . . . 5  |-  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) )  =  ( 0g `  (
(ringLMod `  R )  ^s  I
) )
2827, 14lss0cl 17376 . . . 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 17341 . . . . . . 7  |-  ( (ringLMod `  R )  e.  LMod  -> 
(ringLMod `  R )  e. CMnd
)
3123, 30syl 16 . . . . . 6  |-  ( ph  ->  (ringLMod `  R )  e. CMnd )
32 cmnmnd 16609 . . . . . 6  |-  ( (ringLMod `  R )  e. CMnd  ->  (ringLMod `  R )  e.  Mnd )
3331, 32syl 16 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  Mnd )
3424pwsmnd 15769 . . . . 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 15753 . . . 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 15820 . 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 17624 . . . 4  |-  ( Base `  R )  =  (
Base `  (ringLMod `  R
) )
402adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  J )  ->  I  e.  V )
41 eqid 2467 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
423, 41, 4frlmbasf 18561 . . . . . . . . 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 2467 . . . . . . . . 9  |-  ( x  e.  I  |->  U )  =  ( x  e.  I  |->  U )
4544fmpt 6040 . . . . . . . 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 2833 . . . . . 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.w . . . . 5  |-  ( ph  ->  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) finSupp  .0.  )
51 frlmgsum.z . . . . . 6  |-  .0.  =  ( 0g `  Y )
526fveq2d 5868 . . . . . . 7  |-  ( ph  ->  ( 0g `  Y
)  =  ( 0g
`  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
5314lsssubg 17386 . . . . . . . . 9  |-  ( ( ( (ringLMod `  R
)  ^s  I )  e.  LMod  /\  B  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )  ->  B  e.  (SubGrp `  ( (ringLMod `  R
)  ^s  I ) ) )
5426, 16, 53syl2anc 661 . . . . . . . 8  |-  ( ph  ->  B  e.  (SubGrp `  ( (ringLMod `  R )  ^s  I ) ) )
5510, 27subg0 16002 . . . . . . . 8  |-  ( B  e.  (SubGrp `  (
(ringLMod `  R )  ^s  I
) )  ->  ( 0g `  ( (ringLMod `  R
)  ^s  I ) )  =  ( 0g `  (
( (ringLMod `  R )  ^s  I )s  B ) ) )
5654, 55syl 16 . . . . . . 7  |-  ( ph  ->  ( 0g `  (
(ringLMod `  R )  ^s  I
) )  =  ( 0g `  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
5752, 56eqtr4d 2511 . . . . . 6  |-  ( ph  ->  ( 0g `  Y
)  =  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) )
5851, 57syl5eq 2520 . . . . 5  |-  ( ph  ->  .0.  =  ( 0g
`  ( (ringLMod `  R
)  ^s  I ) ) )
5950, 58breqtrd 4471 . . . 4  |-  ( ph  ->  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) finSupp  ( 0g `  ( (ringLMod `  R
)  ^s  I ) ) )
6024, 39, 27, 2, 13, 31, 49, 59pwsgsum 16800 . . 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 6128 . . . . . 6  |-  ( J  e.  W  ->  (
y  e.  J  |->  U )  e.  _V )
6213, 61syl 16 . . . . 5  |-  ( ph  ->  ( y  e.  J  |->  U )  e.  _V )
63 fvex 5874 . . . . . 6  |-  (ringLMod `  R
)  e.  _V
6463a1i 11 . . . . 5  |-  ( ph  ->  (ringLMod `  R )  e.  _V )
6539a1i 11 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (ringLMod `  R )
) )
66 rlmplusg 17625 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  (ringLMod `  R
) )
6766a1i 11 . . . . 5  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  (ringLMod `  R )
) )
6862, 1, 64, 65, 67gsumpropd 15817 . . . 4  |-  ( ph  ->  ( R  gsumg  ( y  e.  J  |->  U ) )  =  ( (ringLMod `  R
)  gsumg  ( y  e.  J  |->  U ) ) )
6968mpteq2dv 4534 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) )  =  ( x  e.  I  |->  ( (ringLMod `  R
)  gsumg  ( y  e.  J  |->  U ) ) ) )
7060, 69eqtr4d 2511 . 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 2514 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 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   -->wf 5582   ` cfv 5586  (class class class)co 6282   finSupp cfsupp 7825   Basecbs 14486   ↾s cress 14487   +g cplusg 14551   0gc0g 14691    gsumg cgsu 14692    ^s cpws 14698   Mndcmnd 15722  SubGrpcsubg 15990  CMndccmn 16594   Ringcrg 16986   LModclmod 17295   LSubSpclss 17361  ringLModcrglmod 17598   freeLMod cfrlm 18544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-hom 14575  df-cco 14576  df-0g 14693  df-gsum 14694  df-prds 14699  df-pws 14701  df-mnd 15728  df-mhm 15777  df-grp 15858  df-minusg 15859  df-sbg 15860  df-subg 15993  df-cntz 16150  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-subrg 17210  df-lmod 17297  df-lss 17362  df-sra 17601  df-rgmod 17602  df-dsmm 18530  df-frlm 18545
This theorem is referenced by:  uvcresum  18591  matgsum  18706
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