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

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 19077	. . . 4 |- ( ( R e. Ring /\ I e. V ) -> Y = ( ( (ringLMod ` R ) ^s I ) ↾s B ) )
6	1, 2, 5	syl2anc 659	. . 3 |- ( ph -> Y = ( ( (ringLMod ` R ) ^s I ) ↾s B ) )
7	6	oveq1d 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
12	11	a1i 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 ) )
15	3, 4, 14	frlmlss 19078	. . . . 5 |- ( ( R e. Ring /\ I e. V ) -> B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) )
16	1, 2, 15	syl2anc 659	. . . 4 |- ( ph -> B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) )
17	8, 14	lssss 17901	. . . 4 |- ( B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) -> B C_ ( Base ` ( (ringLMod ` R ) ^s I ) ) )
18	16, 17	syl 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 ) )
21	19, 20	fmptd 6032	. . 3 |- ( ph -> ( y e. J |-> ( x e. I |-> U ) ) : J --> B )
22		rlmlmod 18169	. . . . . 6 |- ( R e. Ring -> (ringLMod ` R ) e. LMod )
23	1, 22	syl 17	. . . . 5 |- ( ph -> (ringLMod ` R ) e. LMod )
24		eqid 2402	. . . . . 6 |- ( (ringLMod ` R ) ^s I ) = ( (ringLMod ` R ) ^s I )
25	24	pwslmod 17934	. . . . 5 |- ( ( (ringLMod ` R ) e. LMod /\ I e. V ) -> ( (ringLMod ` R ) ^s I ) e. LMod )
26	23, 2, 25	syl2anc 659	. . . 4 |- ( ph -> ( (ringLMod ` R ) ^s I ) e. LMod )
27		eqid 2402	. . . . 5 |- ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( (ringLMod ` R ) ^s I ) )
28	27, 14	lss0cl 17911	. . . 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 659	. . 3 |- ( ph -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) e. B )
30		lmodcmn 17876	. . . . . . 7 |- ( (ringLMod ` R ) e. LMod -> (ringLMod ` R ) e. CMnd )
31	23, 30	syl 17	. . . . . 6 |- ( ph -> (ringLMod ` R ) e. CMnd )
32		cmnmnd 17135	. . . . . 6 |- ( (ringLMod ` R ) e. CMnd -> (ringLMod ` R ) e. Mnd )
33	31, 32	syl 17	. . . . 5 |- ( ph -> (ringLMod ` R ) e. Mnd )
34	24	pwsmnd 16277	. . . . 5 |- ( ( (ringLMod ` R ) e. Mnd /\ I e. V ) -> ( (ringLMod ` R ) ^s I ) e. Mnd )
35	33, 2, 34	syl2anc 659	. . . 4 |- ( ph -> ( (ringLMod ` R ) ^s I ) e. Mnd )
36	8, 9, 27	mndlrid 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 ) )
37	35, 36	sylan 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 ) )
38	8, 9, 10, 12, 13, 18, 21, 29, 37	gsumress 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 ) )
40	2	adantr 463	. . . . . . . . 9 |- ( ( ph /\ y e. J ) -> I e. V )
41		eqid 2402	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
42	3, 41, 4	frlmbasf 19088	. . . . . . . . 9 |- ( ( I e. V /\ ( x e. I |-> U ) e. B ) -> ( x e. I |-> U ) : I --> ( Base ` R ) )
43	40, 19, 42	syl2anc 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 )
45	44	fmpt 6029	. . . . . . . 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 2772	. . . . . 6 |- ( ( ( ph /\ y e. J ) /\ x e. I ) -> U e. ( Base ` R ) )
48	47	an32s 805	. . . . 5 |- ( ( ( ph /\ x e. I ) /\ y e. J ) -> U e. ( Base ` R ) )
49	48	anasss 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 )
52	6	fveq2d 5852	. . . . . . 7 |- ( ph -> ( 0g ` Y ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
53	14	lsssubg 17921	. . . . . . . . 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 659	. . . . . . . 8 |- ( ph -> B e. (SubGrp ` ( (ringLMod ` R ) ^s I ) ) )
55	10, 27	subg0 16529	. . . . . . . 8 |- ( B e. (SubGrp ` ( (ringLMod ` R ) ^s I ) ) -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
56	54, 55	syl 17	. . . . . . 7 |- ( ph -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
57	52, 56	eqtr4d 2446	. . . . . 6 |- ( ph -> ( 0g ` Y ) = ( 0g ` ( (ringLMod ` R ) ^s I ) ) )
58	51, 57	syl5eq 2455	. . . . 5 |- ( ph -> .0. = ( 0g ` ( (ringLMod ` R ) ^s I ) ) )
59	50, 58	breqtrd 4418	. . . 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 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 )
62	13, 61	syl 17	. . . . 5 |- ( ph -> ( y e. J |-> U ) e. _V )
63		fvex 5858	. . . . . 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 18160	. . . . . 6 |- ( +g ` R ) = ( +g ` (ringLMod ` R ) )
67	66	a1i 11	. . . . 5 |- ( ph -> ( +g ` R ) = ( +g ` (ringLMod ` R ) ) )
68	62, 1, 64, 65, 67	gsumpropd 16221	. . . 4 |- ( ph -> ( R gsumg ( y e. J |-> U ) ) = ( (ringLMod ` R ) gsumg ( y e. J |-> U ) ) )
69	68	mpteq2dv 4481	. . 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 2446	. 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 2449	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 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   gsum_g 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