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

Step	Hyp	Ref	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 )
5	3, 4	frlmpws 27086	. . . 4 |- ( ( R e. Ring /\ I e. V ) -> Y = ( ( (ringLMod ` R ) ^s I ) ↾s B ) )
6	1, 2, 5	syl2anc 643	. . 3 |- ( ph -> Y = ( ( (ringLMod ` R ) ^s I ) ↾s B ) )
7	6	oveq1d 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
12	11	a1i 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 ) )
15	3, 4, 14	frlmlss 27087	. . . . 5 |- ( ( R e. Ring /\ I e. V ) -> B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) )
16	1, 2, 15	syl2anc 643	. . . 4 |- ( ph -> B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) )
17	8, 14	lssss 15968	. . . 4 |- ( B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) -> B C_ ( Base ` ( (ringLMod ` R ) ^s I ) ) )
18	16, 17	syl 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 ) )
21	19, 20	fmptd 5852	. . 3 |- ( ph -> ( y e. J |-> ( x e. I |-> U ) ) : J --> B )
22		rlmlmod 16231	. . . . . 6 |- ( R e. Ring -> (ringLMod ` R ) e. LMod )
23	1, 22	syl 16	. . . . 5 |- ( ph -> (ringLMod ` R ) e. LMod )
24		eqid 2404	. . . . . 6 |- ( (ringLMod ` R ) ^s I ) = ( (ringLMod ` R ) ^s I )
25	24	pwslmod 16001	. . . . 5 |- ( ( (ringLMod ` R ) e. LMod /\ I e. V ) -> ( (ringLMod ` R ) ^s I ) e. LMod )
26	23, 2, 25	syl2anc 643	. . . 4 |- ( ph -> ( (ringLMod ` R ) ^s I ) e. LMod )
27		eqid 2404	. . . . 5 |- ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( (ringLMod ` R ) ^s I ) )
28	27, 14	lss0cl 15978	. . . 4 |- ( ( ( (ringLMod ` R ) ^s I ) e. LMod /\ B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) ) -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) e. B )
29	26, 16, 28	syl2anc 643	. . 3 |- ( ph -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) e. B )
30		lmodcmn 15947	. . . . . . 7 |- ( (ringLMod ` R ) e. LMod -> (ringLMod ` R ) e. CMnd )
31	23, 30	syl 16	. . . . . 6 |- ( ph -> (ringLMod ` R ) e. CMnd )
32		cmnmnd 15382	. . . . . 6 |- ( (ringLMod ` R ) e. CMnd -> (ringLMod ` R ) e. Mnd )
33	31, 32	syl 16	. . . . 5 |- ( ph -> (ringLMod ` R ) e. Mnd )
34	24	pwsmnd 14685	. . . . 5 |- ( ( (ringLMod ` R ) e. Mnd /\ I e. V ) -> ( (ringLMod ` R ) ^s I ) e. Mnd )
35	33, 2, 34	syl2anc 643	. . . 4 |- ( ph -> ( (ringLMod ` R ) ^s I ) e. Mnd )
36	8, 9, 27	mndlrid 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 ) )
37	35, 36	sylan 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 ) )
38	8, 9, 10, 12, 13, 18, 21, 29, 37	gsumress 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 ) )
40	2	adantr 452	. . . . . . . . 9 |- ( ( ph /\ y e. J ) -> I e. V )
41		eqid 2404	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
42	3, 41, 4	frlmbasf 27096	. . . . . . . . 9 |- ( ( I e. V /\ ( x e. I |-> U ) e. B ) -> ( x e. I |-> U ) : I --> ( Base ` R ) )
43	40, 19, 42	syl2anc 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 )
45	44	fmpt 5849	. . . . . . . 8 |- ( A. x e. I U e. ( Base ` R ) <-> ( x e. I |-> U ) : I --> ( Base ` R ) )
46	43, 45	sylibr 204	. . . . . . 7 |- ( ( ph /\ y e. J ) -> A. x e. I U e. ( Base ` R ) )
47	46	r19.21bi 2764	. . . . . 6 |- ( ( ( ph /\ y e. J ) /\ x e. I ) -> U e. ( Base ` R ) )
48	47	an32s 780	. . . . 5 |- ( ( ( ph /\ x e. I ) /\ y e. J ) -> U e. ( Base ` R ) )
49	48	anasss 629	. . . 4 |- ( ( ph /\ ( x e. I /\ y e. J ) ) -> U e. ( Base ` R ) )
50		frlmgsum.z	. . . . . . . . 9 |- .0. = ( 0g ` Y )
51	6	fveq2d 5691	. . . . . . . . . 10 |- ( ph -> ( 0g ` Y ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
52	14	lsssubg 15988	. . . . . . . . . . . 12 |- ( ( ( (ringLMod ` R ) ^s I ) e. LMod /\ B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) ) -> B e. (SubGrp ` ( (ringLMod ` R ) ^s I ) ) )
53	26, 16, 52	syl2anc 643	. . . . . . . . . . 11 |- ( ph -> B e. (SubGrp ` ( (ringLMod ` R ) ^s I ) ) )
54	10, 27	subg0 14905	. . . . . . . . . . 11 |- ( B e. (SubGrp ` ( (ringLMod ` R ) ^s I ) ) -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
55	53, 54	syl 16	. . . . . . . . . 10 |- ( ph -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
56	51, 55	eqtr4d 2439	. . . . . . . . 9 |- ( ph -> ( 0g ` Y ) = ( 0g ` ( (ringLMod ` R ) ^s I ) ) )
57	50, 56	syl5eq 2448	. . . . . . . 8 |- ( ph -> .0. = ( 0g ` ( (ringLMod ` R ) ^s I ) ) )
58	57	sneqd 3787	. . . . . . 7 |- ( ph -> { .0. } = { ( 0g ` ( (ringLMod ` R ) ^s I ) ) } )
59	58	difeq2d 3425	. . . . . 6 |- ( ph -> ( _V \ { .0. } ) = ( _V \ { ( 0g ` ( (ringLMod ` R ) ^s I ) ) } ) )
60	59	imaeq2d 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 )
62	60, 61	eqeltrrd 2479	. . . 4 |- ( ph -> ( `' ( y e. J |-> ( x e. I |-> U ) ) " ( _V \ { ( 0g ` ( (ringLMod ` R ) ^s I ) ) } ) ) e. Fin )
63	24, 39, 27, 2, 13, 31, 49, 62	pwsgsum 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 )
65	13, 64	syl 16	. . . . 5 |- ( ph -> ( y e. J |-> U ) e. _V )
66		fvex 5701	. . . . . 6 |- (ringLMod ` R ) e. _V
67	66	a1i 11	. . . . 5 |- ( ph -> (ringLMod ` R ) e. _V )
68	39	a1i 11	. . . . 5 |- ( ph -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
69		rlmplusg 16223	. . . . . 6 |- ( +g ` R ) = ( +g ` (ringLMod ` R ) )
70	69	a1i 11	. . . . 5 |- ( ph -> ( +g ` R ) = ( +g ` (ringLMod ` R ) ) )
71	65, 1, 67, 68, 70	gsumpropd 14731	. . . 4 |- ( ph -> ( R gsumg ( y e. J |-> U ) ) = ( (ringLMod ` R ) gsumg ( y e. J |-> U ) ) )
72	71	mpteq2dv 4256	. . 3 |- ( ph -> ( x e. I |-> ( R gsumg ( y e. J |-> U ) ) ) = ( x e. I |-> ( (ringLMod ` R ) gsumg ( y e. J |-> U ) ) ) )
73	63, 72	eqtr4d 2439	. 2 |- ( ph -> ( ( (ringLMod ` R ) ^s I ) gsumg ( y e. J |-> ( x e. I |-> U ) ) ) = ( x e. I |-> ( R gsumg ( y e. J |-> U ) ) ) )
74	7, 38, 73	3eqtr2d 2442	1 |- ( ph -> ( Y gsumg ( y e. J |-> ( x e. I |-> U ) ) ) = ( x e. I |-> ( R gsumg ( y e. J |-> U ) ) ) )

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   gsum_g 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