Theorem gsummgp0 18608

Description: If one factor in a finite group sum of the multiplicative group of a commutative ring is 0, the whole "sum" (i.e. product) is 0. (Contributed by AV, 3-Jan-2019.)

Hypotheses

Ref	Expression
gsummgp0.g	|- G = (mulGrp ` R )
gsummgp0.0	|- .0. = ( 0g ` R )
gsummgp0.r	|- ( ph -> R e. CRing )
gsummgp0.n	|- ( ph -> N e. Fin )
gsummgp0.a	|- ( ( ph /\ n e. N ) -> A e. ( Base ` R ) )
gsummgp0.e	|- ( ( ph /\ n = i ) -> A = B )
gsummgp0.b	|- ( ph -> E. i e. N B = .0. )

Assertion

Ref	Expression
gsummgp0	|- ( ph -> ( G gsumg ( n e. N |-> A ) ) = .0. )

Distinct variable groups:   A, i   B, n   i, n, G   i, N, n   R, n   ph, i, n   .0. , i, n

Allowed substitution hints:   A( n)   B( i)   R( i)

Proof of Theorem gsummgp0

Step	Hyp	Ref	Expression
1		gsummgp0.b	. 2 |- ( ph -> E. i e. N B = .0. )
2		difsnid 4341	. . . . . . 7 |- ( i e. N -> ( ( N \ { i } ) u. { i } ) = N )
3	2	eqcomd 2628	. . . . . 6 |- ( i e. N -> N = ( ( N \ { i } ) u. { i } ) )
4	3	ad2antrl 764	. . . . 5 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> N = ( ( N \ { i } ) u. { i } ) )
5	4	mpteq1d 4738	. . . 4 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> ( n e. N |-> A ) = ( n e. ( ( N \ { i } ) u. { i } ) |-> A ) )
6	5	oveq2d 6666	. . 3 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> ( G gsumg ( n e. N |-> A ) ) = ( G gsumg ( n e. ( ( N \ { i } ) u. { i } ) |-> A ) ) )
7		gsummgp0.g	. . . . 5 |- G = (mulGrp ` R )
8		eqid 2622	. . . . 5 |- ( Base ` R ) = ( Base ` R )
9	7, 8	mgpbas 18495	. . . 4 |- ( Base ` R ) = ( Base ` G )
10		eqid 2622	. . . . 5 |- ( .r ` R ) = ( .r ` R )
11	7, 10	mgpplusg 18493	. . . 4 |- ( .r ` R ) = ( +g ` G )
12		gsummgp0.r	. . . . . 6 |- ( ph -> R e. CRing )
13	7	crngmgp 18555	. . . . . 6 |- ( R e. CRing -> G e. CMnd )
14	12, 13	syl 17	. . . . 5 |- ( ph -> G e. CMnd )
15	14	adantr 481	. . . 4 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> G e. CMnd )
16		gsummgp0.n	. . . . . 6 |- ( ph -> N e. Fin )
17		diffi 8192	. . . . . 6 |- ( N e. Fin -> ( N \ { i } ) e. Fin )
18	16, 17	syl 17	. . . . 5 |- ( ph -> ( N \ { i } ) e. Fin )
19	18	adantr 481	. . . 4 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> ( N \ { i } ) e. Fin )
20		simpl 473	. . . . 5 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> ph )
21		eldifi 3732	. . . . 5 |- ( n e. ( N \ { i } ) -> n e. N )
22		gsummgp0.a	. . . . 5 |- ( ( ph /\ n e. N ) -> A e. ( Base ` R ) )
23	20, 21, 22	syl2an 494	. . . 4 |- ( ( ( ph /\ ( i e. N /\ B = .0. ) ) /\ n e. ( N \ { i } ) ) -> A e. ( Base ` R ) )
24		simprl 794	. . . 4 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> i e. N )
25		neldifsnd 4322	. . . 4 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> -. i e. ( N \ { i } ) )
26		crngring 18558	. . . . . . . 8 |- ( R e. CRing -> R e. Ring )
27	12, 26	syl 17	. . . . . . 7 |- ( ph -> R e. Ring )
28		ringmnd 18556	. . . . . . 7 |- ( R e. Ring -> R e. Mnd )
29		gsummgp0.0	. . . . . . . 8 |- .0. = ( 0g ` R )
30	8, 29	mndidcl 17308	. . . . . . 7 |- ( R e. Mnd -> .0. e. ( Base ` R ) )
31	27, 28, 30	3syl 18	. . . . . 6 |- ( ph -> .0. e. ( Base ` R ) )
32	31	adantr 481	. . . . 5 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> .0. e. ( Base ` R ) )
33		eleq1 2689	. . . . . 6 |- ( B = .0. -> ( B e. ( Base ` R ) <-> .0. e. ( Base ` R ) ) )
34	33	ad2antll 765	. . . . 5 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> ( B e. ( Base ` R ) <-> .0. e. ( Base ` R ) ) )
35	32, 34	mpbird 247	. . . 4 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> B e. ( Base ` R ) )
36		gsummgp0.e	. . . . 5 |- ( ( ph /\ n = i ) -> A = B )
37	36	adantlr 751	. . . 4 |- ( ( ( ph /\ ( i e. N /\ B = .0. ) ) /\ n = i ) -> A = B )
38	9, 11, 15, 19, 23, 24, 25, 35, 37	gsumunsnd 18357	. . 3 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> ( G gsumg ( n e. ( ( N \ { i } ) u. { i } ) |-> A ) ) = ( ( G gsumg ( n e. ( N \ { i } ) |-> A ) ) ( .r ` R ) B ) )
39		oveq2 6658	. . . . 5 |- ( B = .0. -> ( ( G gsumg ( n e. ( N \ { i } ) |-> A ) ) ( .r ` R ) B ) = ( ( G gsumg ( n e. ( N \ { i } ) |-> A ) ) ( .r ` R ) .0. ) )
40	39	ad2antll 765	. . . 4 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> ( ( G gsumg ( n e. ( N \ { i } ) |-> A ) ) ( .r ` R ) B ) = ( ( G gsumg ( n e. ( N \ { i } ) |-> A ) ) ( .r ` R ) .0. ) )
41	27	adantr 481	. . . . 5 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> R e. Ring )
42	21, 22	sylan2 491	. . . . . . . 8 |- ( ( ph /\ n e. ( N \ { i } ) ) -> A e. ( Base ` R ) )
43	42	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. n e. ( N \ { i } ) A e. ( Base ` R ) )
44	9, 14, 18, 43	gsummptcl 18366	. . . . . 6 |- ( ph -> ( G gsumg ( n e. ( N \ { i } ) |-> A ) ) e. ( Base ` R ) )
45	44	adantr 481	. . . . 5 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> ( G gsumg ( n e. ( N \ { i } ) |-> A ) ) e. ( Base ` R ) )
46	8, 10, 29	ringrz 18588	. . . . 5 |- ( ( R e. Ring /\ ( G gsumg ( n e. ( N \ { i } ) |-> A ) ) e. ( Base ` R ) ) -> ( ( G gsumg ( n e. ( N \ { i } ) |-> A ) ) ( .r ` R ) .0. ) = .0. )
47	41, 45, 46	syl2anc 693	. . . 4 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> ( ( G gsumg ( n e. ( N \ { i } ) |-> A ) ) ( .r ` R ) .0. ) = .0. )
48	40, 47	eqtrd 2656	. . 3 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> ( ( G gsumg ( n e. ( N \ { i } ) |-> A ) ) ( .r ` R ) B ) = .0. )
49	6, 38, 48	3eqtrd 2660	. 2 |- ( ( ph /\ ( i e. N /\ B = .0. ) ) -> ( G gsumg ( n e. N |-> A ) ) = .0. )
50	1, 49	rexlimddv 3035	1 |- ( ph -> ( G gsumg ( n e. N |-> A ) ) = .0. )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   u. cun 3572   {csn 4177   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   .rcmulr 15942   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  CMndccmn 18193  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548

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-inf2 8538  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-iin 4523  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-of 6897  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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-mgp 18490  df-ring 18549  df-cring 18550

This theorem is referenced by:  smadiadetlem0  20467