Theorem frlmgsum 19323

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 19306	. . . 4 |- ( ( R e. Ring /\ I e. V ) -> Y = ( ( (ringLMod ` R ) ^s I ) ↾s B ) )
6	1, 2, 5	syl2anc 666	. . 3 |- ( ph -> Y = ( ( (ringLMod ` R ) ^s I ) ↾s B ) )
7	6	oveq1d 6303	. 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 2450	. . 3 |- ( Base ` ( (ringLMod ` R ) ^s I ) ) = ( Base ` ( (ringLMod ` R ) ^s I ) )
9		eqid 2450	. . 3 |- ( +g ` ( (ringLMod ` R ) ^s I ) ) = ( +g ` ( (ringLMod ` R ) ^s I ) )
10		eqid 2450	. . 3 |- ( ( (ringLMod ` R ) ^s I ) ↾s B ) = ( ( (ringLMod ` R ) ^s I ) ↾s B )
11		ovex 6316	. . . 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 2450	. . . . . 6 |- ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) = ( LSubSp ` ( (ringLMod ` R ) ^s I ) )
15	3, 4, 14	frlmlss 19307	. . . . 5 |- ( ( R e. Ring /\ I e. V ) -> B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) )
16	1, 2, 15	syl2anc 666	. . . 4 |- ( ph -> B e. ( LSubSp ` ( (ringLMod ` R ) ^s I ) ) )
17	8, 14	lssss 18153	. . . 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 2450	. . . 4 |- ( y e. J |-> ( x e. I |-> U ) ) = ( y e. J |-> ( x e. I |-> U ) )
21	19, 20	fmptd 6044	. . 3 |- ( ph -> ( y e. J |-> ( x e. I |-> U ) ) : J --> B )
22		rlmlmod 18421	. . . . . 6 |- ( R e. Ring -> (ringLMod ` R ) e. LMod )
23	1, 22	syl 17	. . . . 5 |- ( ph -> (ringLMod ` R ) e. LMod )
24		eqid 2450	. . . . . 6 |- ( (ringLMod ` R ) ^s I ) = ( (ringLMod ` R ) ^s I )
25	24	pwslmod 18186	. . . . 5 |- ( ( (ringLMod ` R ) e. LMod /\ I e. V ) -> ( (ringLMod ` R ) ^s I ) e. LMod )
26	23, 2, 25	syl2anc 666	. . . 4 |- ( ph -> ( (ringLMod ` R ) ^s I ) e. LMod )
27		eqid 2450	. . . . 5 |- ( 0g ` ( (ringLMod ` R ) ^s I ) ) = ( 0g ` ( (ringLMod ` R ) ^s I ) )
28	27, 14	lss0cl 18163	. . . 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 666	. . 3 |- ( ph -> ( 0g ` ( (ringLMod ` R ) ^s I ) ) e. B )
30		lmodcmn 18129	. . . . . . 7 |- ( (ringLMod ` R ) e. LMod -> (ringLMod ` R ) e. CMnd )
31	23, 30	syl 17	. . . . . 6 |- ( ph -> (ringLMod ` R ) e. CMnd )
32		cmnmnd 17438	. . . . . 6 |- ( (ringLMod ` R ) e. CMnd -> (ringLMod ` R ) e. Mnd )
33	31, 32	syl 17	. . . . 5 |- ( ph -> (ringLMod ` R ) e. Mnd )
34	24	pwsmnd 16564	. . . . 5 |- ( ( (ringLMod ` R ) e. Mnd /\ I e. V ) -> ( (ringLMod ` R ) ^s I ) e. Mnd )
35	33, 2, 34	syl2anc 666	. . . 4 |- ( ph -> ( (ringLMod ` R ) ^s I ) e. Mnd )
36	8, 9, 27	mndlrid 16549	. . . 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 474	. . 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 16512	. 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 18411	. . . 4 |- ( Base ` R ) = ( Base ` (ringLMod ` R ) )
40	2	adantr 467	. . . . . . . . 9 |- ( ( ph /\ y e. J ) -> I e. V )
41		eqid 2450	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
42	3, 41, 4	frlmbasf 19316	. . . . . . . . 9 |- ( ( I e. V /\ ( x e. I |-> U ) e. B ) -> ( x e. I |-> U ) : I --> ( Base ` R ) )
43	40, 19, 42	syl2anc 666	. . . . . . . 8 |- ( ( ph /\ y e. J ) -> ( x e. I |-> U ) : I --> ( Base ` R ) )
44		eqid 2450	. . . . . . . . 9 |- ( x e. I |-> U ) = ( x e. I |-> U )
45	44	fmpt 6041	. . . . . . . 8 |- ( A. x e. I U e. ( Base ` R ) <-> ( x e. I |-> U ) : I --> ( Base ` R ) )
46	43, 45	sylibr 216	. . . . . . 7 |- ( ( ph /\ y e. J ) -> A. x e. I U e. ( Base ` R ) )
47	46	r19.21bi 2756	. . . . . 6 |- ( ( ( ph /\ y e. J ) /\ x e. I ) -> U e. ( Base ` R ) )
48	47	an32s 812	. . . . 5 |- ( ( ( ph /\ x e. I ) /\ y e. J ) -> U e. ( Base ` R ) )
49	48	anasss 652	. . . 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 5867	. . . . . . 7 |- ( ph -> ( 0g ` Y ) = ( 0g ` ( ( (ringLMod ` R ) ^s I ) ↾s B ) ) )
53	14	lsssubg 18173	. . . . . . . . 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 666	. . . . . . . 8 |- ( ph -> B e. (SubGrp ` ( (ringLMod ` R ) ^s I ) ) )
55	10, 27	subg0 16816	. . . . . . . 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 2487	. . . . . 6 |- ( ph -> ( 0g ` Y ) = ( 0g ` ( (ringLMod ` R ) ^s I ) ) )
58	51, 57	syl5eq 2496	. . . . 5 |- ( ph -> .0. = ( 0g ` ( (ringLMod ` R ) ^s I ) ) )
59	50, 58	breqtrd 4426	. . . 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 17604	. . 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 6133	. . . . . 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 5873	. . . . . 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 18412	. . . . . 6 |- ( +g ` R ) = ( +g ` (ringLMod ` R ) )
67	66	a1i 11	. . . . 5 |- ( ph -> ( +g ` R ) = ( +g ` (ringLMod ` R ) ) )
68	62, 1, 64, 65, 67	gsumpropd 16508	. . . 4 |- ( ph -> ( R gsumg ( y e. J |-> U ) ) = ( (ringLMod ` R ) gsumg ( y e. J |-> U ) ) )
69	68	mpteq2dv 4489	. . 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 2487	. 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 2490	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 371   = wceq 1443   e. wcel 1886   A.wral 2736   _Vcvv 3044   C_ wss 3403   class class class wbr 4401   |-> cmpt 4460   -->wf 5577   ` cfv 5581  (class class class)co 6288   finSupp cfsupp 7880   Basecbs 15114   ↾_s cress 15115   +g cplusg 15183   0gc0g 15331   gsum_g cgsu 15332   ^_s cpws 15338   Mndcmnd 16528  SubGrpcsubg 16804  CMndccmn 17423   Ringcrg 17773   LModclmod 18084   LSubSpclss 18148  ringLModcrglmod 18385   freeLMod cfrlm 19302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-sup 7953  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-fz 11782  df-fzo 11913  df-seq 12211  df-hash 12513  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-hom 15207  df-cco 15208  df-0g 15333  df-gsum 15334  df-prds 15339  df-pws 15341  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-grp 16666  df-minusg 16667  df-sbg 16668  df-subg 16807  df-cntz 16964  df-cmn 17425  df-abl 17426  df-mgp 17717  df-ur 17729  df-ring 17775  df-subrg 17999  df-lmod 18086  df-lss 18149  df-sra 18388  df-rgmod 18389  df-dsmm 19288  df-frlm 19303

This theorem is referenced by:  uvcresum  19344  matgsum  19455  aacllem  40527