Theorem tsmsadd 21950

Description: The sum of two infinite group sums. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)

Hypotheses

Ref	Expression
tsmsadd.b	|- B = ( Base ` G )
tsmsadd.p	|- .+ = ( +g ` G )
tsmsadd.1	|- ( ph -> G e. CMnd )
tsmsadd.2	|- ( ph -> G e. TopMnd )
tsmsadd.a	|- ( ph -> A e. V )
tsmsadd.f	|- ( ph -> F : A --> B )
tsmsadd.h	|- ( ph -> H : A --> B )
tsmsadd.x	|- ( ph -> X e. ( G tsums F ) )
tsmsadd.y	|- ( ph -> Y e. ( G tsums H ) )

Assertion

Ref	Expression
tsmsadd	|- ( ph -> ( X .+ Y ) e. ( G tsums ( F oF .+ H ) ) )

Proof of Theorem tsmsadd

Dummy variables y z x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tsmsadd.b	. . . . . 6 |- B = ( Base ` G )
2		tsmsadd.1	. . . . . 6 |- ( ph -> G e. CMnd )
3		tsmsadd.2	. . . . . . 7 |- ( ph -> G e. TopMnd )
4		tmdtps 21880	. . . . . . 7 |- ( G e. TopMnd -> G e. TopSp )
5	3, 4	syl 17	. . . . . 6 |- ( ph -> G e. TopSp )
6		tsmsadd.a	. . . . . 6 |- ( ph -> A e. V )
7		tsmsadd.f	. . . . . 6 |- ( ph -> F : A --> B )
8	1, 2, 5, 6, 7	tsmscl 21938	. . . . 5 |- ( ph -> ( G tsums F ) C_ B )
9		tsmsadd.x	. . . . 5 |- ( ph -> X e. ( G tsums F ) )
10	8, 9	sseldd 3604	. . . 4 |- ( ph -> X e. B )
11		tsmsadd.h	. . . . . 6 |- ( ph -> H : A --> B )
12	1, 2, 5, 6, 11	tsmscl 21938	. . . . 5 |- ( ph -> ( G tsums H ) C_ B )
13		tsmsadd.y	. . . . 5 |- ( ph -> Y e. ( G tsums H ) )
14	12, 13	sseldd 3604	. . . 4 |- ( ph -> Y e. B )
15		tsmsadd.p	. . . . 5 |- .+ = ( +g ` G )
16		eqid 2622	. . . . 5 |- ( +f ` G ) = ( +f ` G )
17	1, 15, 16	plusfval 17248	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( X ( +f ` G ) Y ) = ( X .+ Y ) )
18	10, 14, 17	syl2anc 693	. . 3 |- ( ph -> ( X ( +f ` G ) Y ) = ( X .+ Y ) )
19		eqid 2622	. . . . . 6 |- ( TopOpen ` G ) = ( TopOpen ` G )
20	1, 19	istps 20738	. . . . 5 |- ( G e. TopSp <-> ( TopOpen ` G ) e. (TopOn ` B ) )
21	5, 20	sylib 208	. . . 4 |- ( ph -> ( TopOpen ` G ) e. (TopOn ` B ) )
22		eqid 2622	. . . . . 6 |- ( ~P A i^i Fin ) = ( ~P A i^i Fin )
23		eqid 2622	. . . . . 6 |- ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) = ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } )
24		eqid 2622	. . . . . 6 |- ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) = ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } )
25	22, 23, 24, 6	tsmsfbas 21931	. . . . 5 |- ( ph -> ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) e. ( fBas ` ( ~P A i^i Fin ) ) )
26		fgcl 21682	. . . . 5 |- ( ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) e. ( fBas ` ( ~P A i^i Fin ) ) -> ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) e. ( Fil ` ( ~P A i^i Fin ) ) )
27	25, 26	syl 17	. . . 4 |- ( ph -> ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) e. ( Fil ` ( ~P A i^i Fin ) ) )
28	1, 22, 2, 6, 7	tsmslem1 21932	. . . 4 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` z ) ) e. B )
29	1, 22, 2, 6, 11	tsmslem1 21932	. . . 4 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsumg ( H |` z ) ) e. B )
30	1, 19, 22, 24, 2, 6, 7	tsmsval 21934	. . . . 5 |- ( ph -> ( G tsums F ) = ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` z ) ) ) ) )
31	9, 30	eleqtrd 2703	. . . 4 |- ( ph -> X e. ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` z ) ) ) ) )
32	1, 19, 22, 24, 2, 6, 11	tsmsval 21934	. . . . 5 |- ( ph -> ( G tsums H ) = ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( G gsumg ( H |` z ) ) ) ) )
33	13, 32	eleqtrd 2703	. . . 4 |- ( ph -> Y e. ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( G gsumg ( H |` z ) ) ) ) )
34	19, 16	tmdcn 21887	. . . . . 6 |- ( G e. TopMnd -> ( +f ` G ) e. ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) Cn ( TopOpen ` G ) ) )
35	3, 34	syl 17	. . . . 5 |- ( ph -> ( +f ` G ) e. ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) Cn ( TopOpen ` G ) ) )
36		opelxpi 5148	. . . . . . 7 |- ( ( X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) )
37	10, 14, 36	syl2anc 693	. . . . . 6 |- ( ph -> <. X , Y >. e. ( B X. B ) )
38		txtopon 21394	. . . . . . . 8 |- ( ( ( TopOpen ` G ) e. (TopOn ` B ) /\ ( TopOpen ` G ) e. (TopOn ` B ) ) -> ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) e. (TopOn ` ( B X. B ) ) )
39	21, 21, 38	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) e. (TopOn ` ( B X. B ) ) )
40		toponuni 20719	. . . . . . 7 |- ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) e. (TopOn ` ( B X. B ) ) -> ( B X. B ) = U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) )
41	39, 40	syl 17	. . . . . 6 |- ( ph -> ( B X. B ) = U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) )
42	37, 41	eleqtrd 2703	. . . . 5 |- ( ph -> <. X , Y >. e. U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) )
43		eqid 2622	. . . . . 6 |- U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) = U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) )
44	43	cncnpi 21082	. . . . 5 |- ( ( ( +f ` G ) e. ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) Cn ( TopOpen ` G ) ) /\ <. X , Y >. e. U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) ) -> ( +f ` G ) e. ( ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) CnP ( TopOpen ` G ) ) ` <. X , Y >. ) )
45	35, 42, 44	syl2anc 693	. . . 4 |- ( ph -> ( +f ` G ) e. ( ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) CnP ( TopOpen ` G ) ) ` <. X , Y >. ) )
46	21, 21, 27, 28, 29, 31, 33, 45	flfcnp2 21811	. . 3 |- ( ph -> ( X ( +f ` G ) Y ) e. ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( ( G gsumg ( F |` z ) ) ( +f ` G ) ( G gsumg ( H |` z ) ) ) ) ) )
47	18, 46	eqeltrrd 2702	. 2 |- ( ph -> ( X .+ Y ) e. ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( ( G gsumg ( F |` z ) ) ( +f ` G ) ( G gsumg ( H |` z ) ) ) ) ) )
48		cmnmnd 18208	. . . . . . 7 |- ( G e. CMnd -> G e. Mnd )
49	2, 48	syl 17	. . . . . 6 |- ( ph -> G e. Mnd )
50	1, 15	mndcl 17301	. . . . . . 7 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
51	50	3expb 1266	. . . . . 6 |- ( ( G e. Mnd /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B )
52	49, 51	sylan 488	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B )
53		inidm 3822	. . . . 5 |- ( A i^i A ) = A
54	52, 7, 11, 6, 6, 53	off 6912	. . . 4 |- ( ph -> ( F oF .+ H ) : A --> B )
55	1, 19, 22, 24, 2, 6, 54	tsmsval 21934	. . 3 |- ( ph -> ( G tsums ( F oF .+ H ) ) = ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( G gsumg ( ( F oF .+ H ) |` z ) ) ) ) )
56		eqid 2622	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
57	2	adantr 481	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> G e. CMnd )
58		elfpw 8268	. . . . . . . . 9 |- ( z e. ( ~P A i^i Fin ) <-> ( z C_ A /\ z e. Fin ) )
59	58	simprbi 480	. . . . . . . 8 |- ( z e. ( ~P A i^i Fin ) -> z e. Fin )
60	59	adantl 482	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> z e. Fin )
61	58	simplbi 476	. . . . . . . 8 |- ( z e. ( ~P A i^i Fin ) -> z C_ A )
62		fssres 6070	. . . . . . . 8 |- ( ( F : A --> B /\ z C_ A ) -> ( F |` z ) : z --> B )
63	7, 61, 62	syl2an 494	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( F |` z ) : z --> B )
64		fssres 6070	. . . . . . . 8 |- ( ( H : A --> B /\ z C_ A ) -> ( H |` z ) : z --> B )
65	11, 61, 64	syl2an 494	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( H |` z ) : z --> B )
66		fvex 6201	. . . . . . . . 9 |- ( 0g ` G ) e. _V
67	66	a1i 11	. . . . . . . 8 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( 0g ` G ) e. _V )
68	63, 60, 67	fdmfifsupp 8285	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( F |` z ) finSupp ( 0g ` G ) )
69	65, 60, 67	fdmfifsupp 8285	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( H |` z ) finSupp ( 0g ` G ) )
70	1, 56, 15, 57, 60, 63, 65, 68, 69	gsumadd 18323	. . . . . 6 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsumg ( ( F |` z ) oF .+ ( H |` z ) ) ) = ( ( G gsumg ( F |` z ) ) .+ ( G gsumg ( H |` z ) ) ) )
71		fvex 6201	. . . . . . . . . . . 12 |- ( Base ` G ) e. _V
72	1, 71	eqeltri 2697	. . . . . . . . . . 11 |- B e. _V
73	72	a1i 11	. . . . . . . . . 10 |- ( ph -> B e. _V )
74		fex2 7121	. . . . . . . . . 10 |- ( ( F : A --> B /\ A e. V /\ B e. _V ) -> F e. _V )
75	7, 6, 73, 74	syl3anc 1326	. . . . . . . . 9 |- ( ph -> F e. _V )
76		fex2 7121	. . . . . . . . . 10 |- ( ( H : A --> B /\ A e. V /\ B e. _V ) -> H e. _V )
77	11, 6, 73, 76	syl3anc 1326	. . . . . . . . 9 |- ( ph -> H e. _V )
78		offres 7163	. . . . . . . . 9 |- ( ( F e. _V /\ H e. _V ) -> ( ( F oF .+ H ) |` z ) = ( ( F |` z ) oF .+ ( H |` z ) ) )
79	75, 77, 78	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( F oF .+ H ) |` z ) = ( ( F |` z ) oF .+ ( H |` z ) ) )
80	79	adantr 481	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( ( F oF .+ H ) |` z ) = ( ( F |` z ) oF .+ ( H |` z ) ) )
81	80	oveq2d 6666	. . . . . 6 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsumg ( ( F oF .+ H ) |` z ) ) = ( G gsumg ( ( F |` z ) oF .+ ( H |` z ) ) ) )
82	1, 15, 16	plusfval 17248	. . . . . . 7 |- ( ( ( G gsumg ( F |` z ) ) e. B /\ ( G gsumg ( H |` z ) ) e. B ) -> ( ( G gsumg ( F |` z ) ) ( +f ` G ) ( G gsumg ( H |` z ) ) ) = ( ( G gsumg ( F |` z ) ) .+ ( G gsumg ( H |` z ) ) ) )
83	28, 29, 82	syl2anc 693	. . . . . 6 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( ( G gsumg ( F |` z ) ) ( +f ` G ) ( G gsumg ( H |` z ) ) ) = ( ( G gsumg ( F |` z ) ) .+ ( G gsumg ( H |` z ) ) ) )
84	70, 81, 83	3eqtr4d 2666	. . . . 5 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsumg ( ( F oF .+ H ) |` z ) ) = ( ( G gsumg ( F |` z ) ) ( +f ` G ) ( G gsumg ( H |` z ) ) ) )
85	84	mpteq2dva 4744	. . . 4 |- ( ph -> ( z e. ( ~P A i^i Fin ) |-> ( G gsumg ( ( F oF .+ H ) |` z ) ) ) = ( z e. ( ~P A i^i Fin ) |-> ( ( G gsumg ( F |` z ) ) ( +f ` G ) ( G gsumg ( H |` z ) ) ) ) )
86	85	fveq2d 6195	. . 3 |- ( ph -> ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( G gsumg ( ( F oF .+ H ) |` z ) ) ) ) = ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( ( G gsumg ( F |` z ) ) ( +f ` G ) ( G gsumg ( H |` z ) ) ) ) ) )
87	55, 86	eqtrd 2656	. 2 |- ( ph -> ( G tsums ( F oF .+ H ) ) = ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( ( G gsumg ( F |` z ) ) ( +f ` G ) ( G gsumg ( H |` z ) ) ) ) ) )
88	47, 87	eleqtrrd 2704	1 |- ( ph -> ( X .+ Y ) e. ( G tsums ( F oF .+ H ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   <.cop 4183   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   Basecbs 15857   +g cplusg 15941   TopOpenctopn 16082   0gc0g 16100   gsum_g cgsu 16101   +fcplusf 17239   Mndcmnd 17294  CMndccmn 18193   fBascfbas 19734   filGencfg 19735  TopOnctopon 20715   TopSpctps 20736   Cn ccn 21028   CnP ccnp 21029   tX ctx 21363   Filcfil 21649   fLimf cflf 21739  TopMndctmd 21874   tsums ctsu 21929

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-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-map 7859  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-topgen 16104  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-cntz 17750  df-cmn 18195  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-ntr 20824  df-nei 20902  df-cn 21031  df-cnp 21032  df-tx 21365  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tmd 21876  df-tsms 21930

This theorem is referenced by:  tsmssub  21952  tsmssplit  21955  esumadd  30119  esumaddf  30123