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

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 18548	. . . 4 |- ( ( R e. Ring /\ I e. V ) -> Y = ( ( (ringLMod ` R ) ^s I ) ↾s B ) )
6	1, 2, 5	syl2anc 661	. . 3 |- ( ph -> Y = ( ( (ringLMod ` R ) ^s I ) ↾s B ) )
7	6	oveq1d 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
12	11	a1i 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 ) )
15	3, 4, 14	frlmlss 18549	. . . . 5 |- ( ( R e. Ring /\ I e. V ) -> B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) )
16	1, 2, 15	syl2anc 661	. . . 4 |- ( ph -> B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) )
17	8, 14	lssss 17366	. . . 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 2467	. . . 4 |- ( y e. J |-> ( x e. I |-> U ) ) = ( y e. J |-> ( x e. I |-> U ) )
21	19, 20	fmptd 6043	. . 3 |- ( ph -> ( y e. J |-> ( x e. I |-> U ) ) : J --> B )
22		rlmlmod 17634	. . . . . 6 |- ( R e. Ring -> (ringLMod ` R ) e. LMod )
23	1, 22	syl 16	. . . . 5 |- ( ph -> (ringLMod ` R ) e. LMod )
24		eqid 2467	. . . . . 6 |- ( (ringLMod ` R ) ^s I ) = ( (ringLMod ` R ) ^s I )
25	24	pwslmod 17399	. . . . 5 |- ( ( (ringLMod ` R ) e. LMod /\ I e. V ) -> ( (ringLMod ` R ) ^s I ) e. LMod )
26	23, 2, 25	syl2anc 661	. . . 4 |- ( ph -> ( (ringLMod ` R ) ^s I ) e. LMod )
27		eqid 2467	. . . . 5 |- ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( (ringLMod ` R ) ^s I ) )
28	27, 14	lss0cl 17376	. . . 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 661	. . 3 |- ( ph -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) e. B )
30		lmodcmn 17341	. . . . . . 7 |- ( (ringLMod ` R ) e. LMod -> (ringLMod ` R ) e. CMnd )
31	23, 30	syl 16	. . . . . 6 |- ( ph -> (ringLMod ` R ) e. CMnd )
32		cmnmnd 16609	. . . . . 6 |- ( (ringLMod ` R ) e. CMnd -> (ringLMod ` R ) e. Mnd )
33	31, 32	syl 16	. . . . 5 |- ( ph -> (ringLMod ` R ) e. Mnd )
34	24	pwsmnd 15769	. . . . 5 |- ( ( (ringLMod ` R ) e. Mnd /\ I e. V ) -> ( (ringLMod ` R ) ^s I ) e. Mnd )
35	33, 2, 34	syl2anc 661	. . . 4 |- ( ph -> ( (ringLMod ` R ) ^s I ) e. Mnd )
36	8, 9, 27	mndlrid 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 ) )
37	35, 36	sylan 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 ) )
38	8, 9, 10, 12, 13, 18, 21, 29, 37	gsumress 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 ) )
40	2	adantr 465	. . . . . . . . 9 |- ( ( ph /\ y e. J ) -> I e. V )
41		eqid 2467	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
42	3, 41, 4	frlmbasf 18561	. . . . . . . . 9 |- ( ( I e. V /\ ( x e. I |-> U ) e. B ) -> ( x e. I |-> U ) : I --> ( Base ` R ) )
43	40, 19, 42	syl2anc 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 )
45	44	fmpt 6040	. . . . . . . 8 |- ( A. x e. I U e. ( Base ` R ) <-> ( x e. I |-> U ) : I --> ( Base ` R ) )
46	43, 45	sylibr 212	. . . . . . 7 |- ( ( ph /\ y e. J ) -> A. x e. I U e. ( Base ` R ) )
47	46	r19.21bi 2833	. . . . . 6 |- ( ( ( ph /\ y e. J ) /\ x e. I ) -> U e. ( Base ` R ) )
48	47	an32s 802	. . . . 5 |- ( ( ( ph /\ x e. I ) /\ y e. J ) -> U e. ( Base ` R ) )
49	48	anasss 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 )
52	6	fveq2d 5868	. . . . . . 7 |- ( ph -> ( 0g ` Y ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
53	14	lsssubg 17386	. . . . . . . . 9 |- ( ( ( (ringLMod ` R ) ^s I ) e. LMod /\ B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) ) -> B e. (SubGrp ` ( (ringLMod ` R ) ^s I ) ) )
54	26, 16, 53	syl2anc 661	. . . . . . . 8 |- ( ph -> B e. (SubGrp ` ( (ringLMod ` R ) ^s I ) ) )
55	10, 27	subg0 16002	. . . . . . . 8 |- ( B e. (SubGrp ` ( (ringLMod ` R ) ^s I ) ) -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
56	54, 55	syl 16	. . . . . . 7 |- ( ph -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
57	52, 56	eqtr4d 2511	. . . . . 6 |- ( ph -> ( 0g ` Y ) = ( 0g ` ( (ringLMod ` R ) ^s I ) ) )
58	51, 57	syl5eq 2520	. . . . 5 |- ( ph -> .0. = ( 0g ` ( (ringLMod ` R ) ^s I ) ) )
59	50, 58	breqtrd 4471	. . . 4 |- ( ph -> ( y e. J |-> ( x e. I |-> U ) ) finSupp ( 0g ` ( (ringLMod ` R ) ^s I ) ) )
60	24, 39, 27, 2, 13, 31, 49, 59	pwsgsum 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 )
62	13, 61	syl 16	. . . . 5 |- ( ph -> ( y e. J |-> U ) e. _V )
63		fvex 5874	. . . . . 6 |- (ringLMod ` R ) e. _V
64	63	a1i 11	. . . . 5 |- ( ph -> (ringLMod ` R ) e. _V )
65	39	a1i 11	. . . . 5 |- ( ph -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
66		rlmplusg 17625	. . . . . 6 |- ( +g ` R ) = ( +g ` (ringLMod ` R ) )
67	66	a1i 11	. . . . 5 |- ( ph -> ( +g ` R ) = ( +g ` (ringLMod ` R ) ) )
68	62, 1, 64, 65, 67	gsumpropd 15817	. . . 4 |- ( ph -> ( R gsumg ( y e. J |-> U ) ) = ( (ringLMod ` R ) gsumg ( y e. J |-> U ) ) )
69	68	mpteq2dv 4534	. . 3 |- ( ph -> ( x e. I |-> ( R gsumg ( y e. J |-> U ) ) ) = ( x e. I |-> ( (ringLMod ` R ) gsumg ( y e. J |-> U ) ) ) )
70	60, 69	eqtr4d 2511	. 2 |- ( ph -> ( ( (ringLMod ` R ) ^s I ) gsumg ( y e. J |-> ( x e. I |-> U ) ) ) = ( x e. I |-> ( R gsumg ( y e. J |-> U ) ) ) )
71	7, 38, 70	3eqtr2d 2514	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 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   gsum_g 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