Theorem frlmgsum 20111

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 20094	. . . 4 |- ( ( R e. Ring /\ I e. V ) -> Y = ( ( (ringLMod ` R ) ^s I ) ↾s B ) )
6	1, 2, 5	syl2anc 693	. . 3 |- ( ph -> Y = ( ( (ringLMod ` R ) ^s I ) ↾s B ) )
7	6	oveq1d 6665	. 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 2622	. . 3 |- ( Base ` ( (ringLMod ` R ) ^s I ) ) = ( Base ` ( (ringLMod ` R ) ^s I ) )
9		eqid 2622	. . 3 |- ( +g ` ( (ringLMod ` R ) ^s I ) ) = ( +g ` ( (ringLMod ` R ) ^s I ) )
10		eqid 2622	. . 3 |- ( ( (ringLMod ` R ) ^s I ) ↾s B ) = ( ( (ringLMod ` R ) ^s I ) ↾s B )
11		ovexd 6680	. . 3 |- ( ph -> ( (ringLMod ` R ) ^s I ) e. _V )
12		frlmgsum.j	. . 3 |- ( ph -> J e. W )
13		eqid 2622	. . . . . 6 |- ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) = ( LSubSp ` ( (ringLMod ` R ) ^s I ) )
14	3, 4, 13	frlmlss 20095	. . . . 5 |- ( ( R e. Ring /\ I e. V ) -> B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) )
15	1, 2, 14	syl2anc 693	. . . 4 |- ( ph -> B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) )
16	8, 13	lssss 18937	. . . 4 |- ( B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) -> B C_ ( Base ` ( (ringLMod ` R ) ^s I ) ) )
17	15, 16	syl 17	. . 3 |- ( ph -> B C_ ( Base ` ( (ringLMod ` R ) ^s I ) ) )
18		frlmgsum.f	. . . 4 |- ( ( ph /\ y e. J ) -> ( x e. I |-> U ) e. B )
19		eqid 2622	. . . 4 |- ( y e. J |-> ( x e. I |-> U ) ) = ( y e. J |-> ( x e. I |-> U ) )
20	18, 19	fmptd 6385	. . 3 |- ( ph -> ( y e. J |-> ( x e. I |-> U ) ) : J --> B )
21		rlmlmod 19205	. . . . . 6 |- ( R e. Ring -> (ringLMod ` R ) e. LMod )
22	1, 21	syl 17	. . . . 5 |- ( ph -> (ringLMod ` R ) e. LMod )
23		eqid 2622	. . . . . 6 |- ( (ringLMod ` R ) ^s I ) = ( (ringLMod ` R ) ^s I )
24	23	pwslmod 18970	. . . . 5 |- ( ( (ringLMod ` R ) e. LMod /\ I e. V ) -> ( (ringLMod ` R ) ^s I ) e. LMod )
25	22, 2, 24	syl2anc 693	. . . 4 |- ( ph -> ( (ringLMod ` R ) ^s I ) e. LMod )
26		eqid 2622	. . . . 5 |- ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( (ringLMod ` R ) ^s I ) )
27	26, 13	lss0cl 18947	. . . 4 |- ( ( ( (ringLMod ` R ) ^s I ) e. LMod /\ B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) ) -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) e. B )
28	25, 15, 27	syl2anc 693	. . 3 |- ( ph -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) e. B )
29		lmodcmn 18911	. . . . . . 7 |- ( (ringLMod ` R ) e. LMod -> (ringLMod ` R ) e. CMnd )
30	22, 29	syl 17	. . . . . 6 |- ( ph -> (ringLMod ` R ) e. CMnd )
31		cmnmnd 18208	. . . . . 6 |- ( (ringLMod ` R ) e. CMnd -> (ringLMod ` R ) e. Mnd )
32	30, 31	syl 17	. . . . 5 |- ( ph -> (ringLMod ` R ) e. Mnd )
33	23	pwsmnd 17325	. . . . 5 |- ( ( (ringLMod ` R ) e. Mnd /\ I e. V ) -> ( (ringLMod ` R ) ^s I ) e. Mnd )
34	32, 2, 33	syl2anc 693	. . . 4 |- ( ph -> ( (ringLMod ` R ) ^s I ) e. Mnd )
35	8, 9, 26	mndlrid 17310	. . . 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 ) )
36	34, 35	sylan 488	. . 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 ) )
37	8, 9, 10, 11, 12, 17, 20, 28, 36	gsumress 17276	. 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 ) ) ) )
38		rlmbas 19195	. . . 4 |- ( Base ` R ) = ( Base ` (ringLMod ` R ) )
39	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ y e. J ) -> I e. V )
40		eqid 2622	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
41	3, 40, 4	frlmbasf 20104	. . . . . . . . 9 |- ( ( I e. V /\ ( x e. I |-> U ) e. B ) -> ( x e. I |-> U ) : I --> ( Base ` R ) )
42	39, 18, 41	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ y e. J ) -> ( x e. I |-> U ) : I --> ( Base ` R ) )
43		eqid 2622	. . . . . . . . 9 |- ( x e. I |-> U ) = ( x e. I |-> U )
44	43	fmpt 6381	. . . . . . . 8 |- ( A. x e. I U e. ( Base ` R ) <-> ( x e. I |-> U ) : I --> ( Base ` R ) )
45	42, 44	sylibr 224	. . . . . . 7 |- ( ( ph /\ y e. J ) -> A. x e. I U e. ( Base ` R ) )
46	45	r19.21bi 2932	. . . . . 6 |- ( ( ( ph /\ y e. J ) /\ x e. I ) -> U e. ( Base ` R ) )
47	46	an32s 846	. . . . 5 |- ( ( ( ph /\ x e. I ) /\ y e. J ) -> U e. ( Base ` R ) )
48	47	anasss 679	. . . 4 |- ( ( ph /\ ( x e. I /\ y e. J ) ) -> U e. ( Base ` R ) )
49		frlmgsum.w	. . . . 5 |- ( ph -> ( y e. J |-> ( x e. I |-> U ) ) finSupp .0. )
50		frlmgsum.z	. . . . . 6 |- .0. = ( 0g ` Y )
51	6	fveq2d 6195	. . . . . . 7 |- ( ph -> ( 0g ` Y ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
52	13	lsssubg 18957	. . . . . . . . 9 |- ( ( ( (ringLMod ` R ) ^s I ) e. LMod /\ B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) ) -> B e. (SubGrp ` ( (ringLMod ` R ) ^s I ) ) )
53	25, 15, 52	syl2anc 693	. . . . . . . 8 |- ( ph -> B e. (SubGrp ` ( (ringLMod ` R ) ^s I ) ) )
54	10, 26	subg0 17600	. . . . . . . 8 |- ( B e. (SubGrp ` ( (ringLMod ` R ) ^s I ) ) -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
55	53, 54	syl 17	. . . . . . 7 |- ( ph -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
56	51, 55	eqtr4d 2659	. . . . . 6 |- ( ph -> ( 0g ` Y ) = ( 0g ` ( (ringLMod ` R ) ^s I ) ) )
57	50, 56	syl5eq 2668	. . . . 5 |- ( ph -> .0. = ( 0g ` ( (ringLMod ` R ) ^s I ) ) )
58	49, 57	breqtrd 4679	. . . 4 |- ( ph -> ( y e. J |-> ( x e. I |-> U ) ) finSupp ( 0g ` ( (ringLMod ` R ) ^s I ) ) )
59	23, 38, 26, 2, 12, 30, 48, 58	pwsgsum 18378	. . 3 |- ( ph -> ( ( (ringLMod ` R ) ^s I ) gsumg ( y e. J |-> ( x e. I |-> U ) ) ) = ( x e. I |-> ( (ringLMod ` R ) gsumg ( y e. J |-> U ) ) ) )
60		mptexg 6484	. . . . . 6 |- ( J e. W -> ( y e. J |-> U ) e. _V )
61	12, 60	syl 17	. . . . 5 |- ( ph -> ( y e. J |-> U ) e. _V )
62		fvexd 6203	. . . . 5 |- ( ph -> (ringLMod ` R ) e. _V )
63	38	a1i 11	. . . . 5 |- ( ph -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
64		rlmplusg 19196	. . . . . 6 |- ( +g ` R ) = ( +g ` (ringLMod ` R ) )
65	64	a1i 11	. . . . 5 |- ( ph -> ( +g ` R ) = ( +g ` (ringLMod ` R ) ) )
66	61, 1, 62, 63, 65	gsumpropd 17272	. . . 4 |- ( ph -> ( R gsumg ( y e. J |-> U ) ) = ( (ringLMod ` R ) gsumg ( y e. J |-> U ) ) )
67	66	mpteq2dv 4745	. . 3 |- ( ph -> ( x e. I |-> ( R gsumg ( y e. J |-> U ) ) ) = ( x e. I |-> ( (ringLMod ` R ) gsumg ( y e. J |-> U ) ) ) )
68	59, 67	eqtr4d 2659	. 2 |- ( ph -> ( ( (ringLMod ` R ) ^s I ) gsumg ( y e. J |-> ( x e. I |-> U ) ) ) = ( x e. I |-> ( R gsumg ( y e. J |-> U ) ) ) )
69	7, 37, 68	3eqtr2d 2662	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 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   finSupp cfsupp 8275   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   ^_s cpws 16107   Mndcmnd 17294  SubGrpcsubg 17588  CMndccmn 18193   Ringcrg 18547   LModclmod 18863   LSubSpclss 18932  ringLModcrglmod 19169   freeLMod cfrlm 20090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091

This theorem is referenced by:  uvcresum  20132  matgsum  20243  matunitlindflem1  33405  matunitlindflem2  33406  aacllem  42547