Theorem gsumpt 18361

Description: Sum of a family that is nonzero at at most one point. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
gsumpt.b	|- B = ( Base ` G )
gsumpt.z	|- .0. = ( 0g ` G )
gsumpt.g	|- ( ph -> G e. Mnd )
gsumpt.a	|- ( ph -> A e. V )
gsumpt.x	|- ( ph -> X e. A )
gsumpt.f	|- ( ph -> F : A --> B )
gsumpt.s	|- ( ph -> ( F supp .0. ) C_ { X } )

Assertion

Ref	Expression
gsumpt	|- ( ph -> ( G gsumg F ) = ( F ` X ) )

Proof of Theorem gsumpt

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumpt.f	. . . 4 |- ( ph -> F : A --> B )
2		gsumpt.x	. . . . 5 |- ( ph -> X e. A )
3	2	snssd 4340	. . . 4 |- ( ph -> { X } C_ A )
4	1, 3	feqresmpt 6250	. . 3 |- ( ph -> ( F |` { X } ) = ( a e. { X } |-> ( F ` a ) ) )
5	4	oveq2d 6666	. 2 |- ( ph -> ( G gsumg ( F |` { X } ) ) = ( G gsumg ( a e. { X } |-> ( F ` a ) ) ) )
6		gsumpt.b	. . 3 |- B = ( Base ` G )
7		gsumpt.z	. . 3 |- .0. = ( 0g ` G )
8		eqid 2622	. . 3 |- (Cntz ` G ) = (Cntz ` G )
9		gsumpt.g	. . 3 |- ( ph -> G e. Mnd )
10		gsumpt.a	. . 3 |- ( ph -> A e. V )
11	1, 2	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( F ` X ) e. B )
12		eqidd 2623	. . . . . . . 8 |- ( ph -> ( ( F ` X ) ( +g ` G ) ( F ` X ) ) = ( ( F ` X ) ( +g ` G ) ( F ` X ) ) )
13		eqid 2622	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
14	6, 13, 8	elcntzsn 17758	. . . . . . . . 9 |- ( ( F ` X ) e. B -> ( ( F ` X ) e. ( (Cntz ` G ) ` { ( F ` X ) } ) <-> ( ( F ` X ) e. B /\ ( ( F ` X ) ( +g ` G ) ( F ` X ) ) = ( ( F ` X ) ( +g ` G ) ( F ` X ) ) ) ) )
15	11, 14	syl 17	. . . . . . . 8 |- ( ph -> ( ( F ` X ) e. ( (Cntz ` G ) ` { ( F ` X ) } ) <-> ( ( F ` X ) e. B /\ ( ( F ` X ) ( +g ` G ) ( F ` X ) ) = ( ( F ` X ) ( +g ` G ) ( F ` X ) ) ) ) )
16	11, 12, 15	mpbir2and 957	. . . . . . 7 |- ( ph -> ( F ` X ) e. ( (Cntz ` G ) ` { ( F ` X ) } ) )
17	16	snssd 4340	. . . . . 6 |- ( ph -> { ( F ` X ) } C_ ( (Cntz ` G ) ` { ( F ` X ) } ) )
18		eqid 2622	. . . . . . 7 |- (mrCls ` (SubMnd ` G ) ) = (mrCls ` (SubMnd ` G ) )
19		eqid 2622	. . . . . . 7 |- ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) ) = ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
20	8, 18, 19	cntzspan 18247	. . . . . 6 |- ( ( G e. Mnd /\ { ( F ` X ) } C_ ( (Cntz ` G ) ` { ( F ` X ) } ) ) -> ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) ) e. CMnd )
21	9, 17, 20	syl2anc 693	. . . . 5 |- ( ph -> ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) ) e. CMnd )
22	6	submacs 17365	. . . . . . . 8 |- ( G e. Mnd -> (SubMnd ` G ) e. (ACS ` B ) )
23		acsmre 16313	. . . . . . . 8 |- ( (SubMnd ` G ) e. (ACS ` B ) -> (SubMnd ` G ) e. (Moore ` B ) )
24	9, 22, 23	3syl 18	. . . . . . 7 |- ( ph -> (SubMnd ` G ) e. (Moore ` B ) )
25	11	snssd 4340	. . . . . . 7 |- ( ph -> { ( F ` X ) } C_ B )
26	18	mrccl 16271	. . . . . . 7 |- ( ( (SubMnd ` G ) e. (Moore ` B ) /\ { ( F ` X ) } C_ B ) -> ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) e. (SubMnd ` G ) )
27	24, 25, 26	syl2anc 693	. . . . . 6 |- ( ph -> ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) e. (SubMnd ` G ) )
28	19, 8	submcmn2 18244	. . . . . 6 |- ( ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) e. (SubMnd ` G ) -> ( ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) ) e. CMnd <-> ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) C_ ( (Cntz ` G ) ` ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) ) ) )
29	27, 28	syl 17	. . . . 5 |- ( ph -> ( ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) ) e. CMnd <-> ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) C_ ( (Cntz ` G ) ` ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) ) ) )
30	21, 29	mpbid 222	. . . 4 |- ( ph -> ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) C_ ( (Cntz ` G ) ` ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) ) )
31		ffn 6045	. . . . . . 7 |- ( F : A --> B -> F Fn A )
32	1, 31	syl 17	. . . . . 6 |- ( ph -> F Fn A )
33		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ a e. A ) /\ a = X ) -> a = X )
34	33	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ph /\ a e. A ) /\ a = X ) -> ( F ` a ) = ( F ` X ) )
35	24, 18, 25	mrcssidd 16285	. . . . . . . . . . 11 |- ( ph -> { ( F ` X ) } C_ ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
36		fvex 6201	. . . . . . . . . . . 12 |- ( F ` X ) e. _V
37	36	snss 4316	. . . . . . . . . . 11 |- ( ( F ` X ) e. ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) <-> { ( F ` X ) } C_ ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
38	35, 37	sylibr 224	. . . . . . . . . 10 |- ( ph -> ( F ` X ) e. ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
39	38	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ a e. A ) /\ a = X ) -> ( F ` X ) e. ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
40	34, 39	eqeltrd 2701	. . . . . . . 8 |- ( ( ( ph /\ a e. A ) /\ a = X ) -> ( F ` a ) e. ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
41		eldifsn 4317	. . . . . . . . . . 11 |- ( a e. ( A \ { X } ) <-> ( a e. A /\ a =/= X ) )
42		gsumpt.s	. . . . . . . . . . . 12 |- ( ph -> ( F supp .0. ) C_ { X } )
43		fvex 6201	. . . . . . . . . . . . . 14 |- ( 0g ` G ) e. _V
44	7, 43	eqeltri 2697	. . . . . . . . . . . . 13 |- .0. e. _V
45	44	a1i 11	. . . . . . . . . . . 12 |- ( ph -> .0. e. _V )
46	1, 42, 10, 45	suppssr 7326	. . . . . . . . . . 11 |- ( ( ph /\ a e. ( A \ { X } ) ) -> ( F ` a ) = .0. )
47	41, 46	sylan2br 493	. . . . . . . . . 10 |- ( ( ph /\ ( a e. A /\ a =/= X ) ) -> ( F ` a ) = .0. )
48	7	subm0cl 17352	. . . . . . . . . . . 12 |- ( ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) e. (SubMnd ` G ) -> .0. e. ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
49	27, 48	syl 17	. . . . . . . . . . 11 |- ( ph -> .0. e. ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
50	49	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( a e. A /\ a =/= X ) ) -> .0. e. ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
51	47, 50	eqeltrd 2701	. . . . . . . . 9 |- ( ( ph /\ ( a e. A /\ a =/= X ) ) -> ( F ` a ) e. ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
52	51	anassrs 680	. . . . . . . 8 |- ( ( ( ph /\ a e. A ) /\ a =/= X ) -> ( F ` a ) e. ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
53	40, 52	pm2.61dane 2881	. . . . . . 7 |- ( ( ph /\ a e. A ) -> ( F ` a ) e. ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
54	53	ralrimiva 2966	. . . . . 6 |- ( ph -> A. a e. A ( F ` a ) e. ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
55		ffnfv 6388	. . . . . 6 |- ( F : A --> ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) <-> ( F Fn A /\ A. a e. A ( F ` a ) e. ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) ) )
56	32, 54, 55	sylanbrc 698	. . . . 5 |- ( ph -> F : A --> ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
57		frn 6053	. . . . 5 |- ( F : A --> ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) -> ran F C_ ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
58	56, 57	syl 17	. . . 4 |- ( ph -> ran F C_ ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) )
59	8	cntzidss 17770	. . . 4 |- ( ( ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) C_ ( (Cntz ` G ) ` ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) ) /\ ran F C_ ( (mrCls ` (SubMnd ` G ) ) ` { ( F ` X ) } ) ) -> ran F C_ ( (Cntz ` G ) ` ran F ) )
60	30, 58, 59	syl2anc 693	. . 3 |- ( ph -> ran F C_ ( (Cntz ` G ) ` ran F ) )
61		ffun 6048	. . . . 5 |- ( F : A --> B -> Fun F )
62	1, 61	syl 17	. . . 4 |- ( ph -> Fun F )
63		snfi 8038	. . . . 5 |- { X } e. Fin
64		ssfi 8180	. . . . 5 |- ( ( { X } e. Fin /\ ( F supp .0. ) C_ { X } ) -> ( F supp .0. ) e. Fin )
65	63, 42, 64	sylancr 695	. . . 4 |- ( ph -> ( F supp .0. ) e. Fin )
66		fex 6490	. . . . . 6 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
67	1, 10, 66	syl2anc 693	. . . . 5 |- ( ph -> F e. _V )
68		isfsupp 8279	. . . . 5 |- ( ( F e. _V /\ .0. e. _V ) -> ( F finSupp .0. <-> ( Fun F /\ ( F supp .0. ) e. Fin ) ) )
69	67, 45, 68	syl2anc 693	. . . 4 |- ( ph -> ( F finSupp .0. <-> ( Fun F /\ ( F supp .0. ) e. Fin ) ) )
70	62, 65, 69	mpbir2and 957	. . 3 |- ( ph -> F finSupp .0. )
71	6, 7, 8, 9, 10, 1, 60, 42, 70	gsumzres 18310	. 2 |- ( ph -> ( G gsumg ( F |` { X } ) ) = ( G gsumg F ) )
72		fveq2 6191	. . . 4 |- ( a = X -> ( F ` a ) = ( F ` X ) )
73	6, 72	gsumsn 18354	. . 3 |- ( ( G e. Mnd /\ X e. A /\ ( F ` X ) e. B ) -> ( G gsumg ( a e. { X } |-> ( F ` a ) ) ) = ( F ` X ) )
74	9, 2, 11, 73	syl3anc 1326	. 2 |- ( ph -> ( G gsumg ( a e. { X } |-> ( F ` a ) ) ) = ( F ` X ) )
75	5, 71, 74	3eqtr3d 2664	1 |- ( ph -> ( G gsumg F ) = ( F ` X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   |` cres 5116   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   Mndcmnd 17294  SubMndcsubmnd 17334  Cntzccntz 17748  CMndccmn 18193

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-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-mulg 17541  df-cntz 17750  df-cmn 18195

This theorem is referenced by:  gsummpt1n0  18364  dprdfid  18416  evlslem3  19514  evlslem1  19515  coe1tmmul2  19646  coe1tmmul  19647  uvcresum  20132  frlmup2  20138  mamulid  20247  mamurid  20248  coe1mul3  23859  tayl0  24116  jensen  24715  linc1  42214