Theorem tsmsadd 21153

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 21083	. . . . . . 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 21141	. . . . 5 |- ( ph -> ( G tsums F ) C_ B )
9		tsmsadd.x	. . . . 5 |- ( ph -> X e. ( G tsums F ) )
10	8, 9	sseldd 3466	. . . 4 |- ( ph -> X e. B )
11		tsmsadd.h	. . . . . 6 |- ( ph -> H : A --> B )
12	1, 2, 5, 6, 11	tsmscl 21141	. . . . 5 |- ( ph -> ( G tsums H ) C_ B )
13		tsmsadd.y	. . . . 5 |- ( ph -> Y e. ( G tsums H ) )
14	12, 13	sseldd 3466	. . . 4 |- ( ph -> Y e. B )
15		tsmsadd.p	. . . . 5 |- .+ = ( +g ` G )
16		eqid 2423	. . . . 5 |- ( +f ` G ) = ( +f ` G )
17	1, 15, 16	plusfval 16487	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( X ( +f ` G ) Y ) = ( X .+ Y ) )
18	10, 14, 17	syl2anc 666	. . 3 |- ( ph -> ( X ( +f ` G ) Y ) = ( X .+ Y ) )
19		eqid 2423	. . . . . 6 |- ( TopOpen ` G ) = ( TopOpen ` G )
20	1, 19	istps 19943	. . . . 5 |- ( G e. TopSp <-> ( TopOpen ` G ) e. (TopOn ` B ) )
21	5, 20	sylib 200	. . . 4 |- ( ph -> ( TopOpen ` G ) e. (TopOn ` B ) )
22		eqid 2423	. . . . . 6 |- ( ~P A i^i Fin ) = ( ~P A i^i Fin )
23		eqid 2423	. . . . . 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 2423	. . . . . 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 21134	. . . . 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 20885	. . . . 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 21135	. . . 4 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` z ) ) e. B )
29	1, 22, 2, 6, 11	tsmslem1 21135	. . . 4 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsumg ( H |` z ) ) e. B )
30	1, 19, 22, 24, 2, 6, 7	tsmsval 21137	. . . . 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 2513	. . . 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 21137	. . . . 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 2513	. . . 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 21090	. . . . . 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 4883	. . . . . . 7 |- ( ( X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) )
37	10, 14, 36	syl2anc 666	. . . . . 6 |- ( ph -> <. X , Y >. e. ( B X. B ) )
38		txtopon 20598	. . . . . . . 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 666	. . . . . . 7 |- ( ph -> ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) e. (TopOn ` ( B X. B ) ) )
40		toponuni 19934	. . . . . . 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 2513	. . . . 5 |- ( ph -> <. X , Y >. e. U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) )
43		eqid 2423	. . . . . 6 |- U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) = U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) )
44	43	cncnpi 20286	. . . . 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 666	. . . 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 21014	. . 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 2512	. 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 17438	. . . . . . 7 |- ( G e. CMnd -> G e. Mnd )
49	2, 48	syl 17	. . . . . 6 |- ( ph -> G e. Mnd )
50	1, 15	mndcl 16538	. . . . . . 7 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
51	50	3expb 1207	. . . . . 6 |- ( ( G e. Mnd /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B )
52	49, 51	sylan 474	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B )
53		inidm 3672	. . . . 5 |- ( A i^i A ) = A
54	52, 7, 11, 6, 6, 53	off 6558	. . . 4 |- ( ph -> ( F oF .+ H ) : A --> B )
55	1, 19, 22, 24, 2, 6, 54	tsmsval 21137	. . 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 2423	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
57	2	adantr 467	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> G e. CMnd )
58		elfpw 7880	. . . . . . . . 9 |- ( z e. ( ~P A i^i Fin ) <-> ( z C_ A /\ z e. Fin ) )
59	58	simprbi 466	. . . . . . . 8 |- ( z e. ( ~P A i^i Fin ) -> z e. Fin )
60	59	adantl 468	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> z e. Fin )
61	58	simplbi 462	. . . . . . . 8 |- ( z e. ( ~P A i^i Fin ) -> z C_ A )
62		fssres 5764	. . . . . . . 8 |- ( ( F : A --> B /\ z C_ A ) -> ( F |` z ) : z --> B )
63	7, 61, 62	syl2an 480	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( F |` z ) : z --> B )
64		fssres 5764	. . . . . . . 8 |- ( ( H : A --> B /\ z C_ A ) -> ( H |` z ) : z --> B )
65	11, 61, 64	syl2an 480	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( H |` z ) : z --> B )
66		fvex 5889	. . . . . . . . 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 7897	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( F |` z ) finSupp ( 0g ` G ) )
69	65, 60, 67	fdmfifsupp 7897	. . . . . . 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 17549	. . . . . 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 5889	. . . . . . . . . . . 12 |- ( Base ` G ) e. _V
72	1, 71	eqeltri 2507	. . . . . . . . . . 11 |- B e. _V
73	72	a1i 11	. . . . . . . . . 10 |- ( ph -> B e. _V )
74		fex2 6760	. . . . . . . . . 10 |- ( ( F : A --> B /\ A e. V /\ B e. _V ) -> F e. _V )
75	7, 6, 73, 74	syl3anc 1265	. . . . . . . . 9 |- ( ph -> F e. _V )
76		fex2 6760	. . . . . . . . . 10 |- ( ( H : A --> B /\ A e. V /\ B e. _V ) -> H e. _V )
77	11, 6, 73, 76	syl3anc 1265	. . . . . . . . 9 |- ( ph -> H e. _V )
78		offres 6800	. . . . . . . . 9 |- ( ( F e. _V /\ H e. _V ) -> ( ( F oF .+ H ) |` z ) = ( ( F |` z ) oF .+ ( H |` z ) ) )
79	75, 77, 78	syl2anc 666	. . . . . . . 8 |- ( ph -> ( ( F oF .+ H ) |` z ) = ( ( F |` z ) oF .+ ( H |` z ) ) )
80	79	adantr 467	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( ( F oF .+ H ) |` z ) = ( ( F |` z ) oF .+ ( H |` z ) ) )
81	80	oveq2d 6319	. . . . . 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 16487	. . . . . . 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 666	. . . . . 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 2474	. . . . 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 4508	. . . 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 5883	. . 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 2464	. 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 2514	1 |- ( ph -> ( X .+ Y ) e. ( G tsums ( F oF .+ H ) ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1438   e. wcel 1869   {crab 2780   _Vcvv 3082   i^i cin 3436   C_ wss 3437   ~Pcpw 3980   <.cop 4003   U.cuni 4217   |-> cmpt 4480   X. cxp 4849   ran crn 4852   |` cres 4853   -->wf 5595   ` cfv 5599  (class class class)co 6303   oFcof 6541   Fincfn 7575   Basecbs 15114   +g cplusg 15183   TopOpenctopn 15313   0gc0g 15331   gsum_g cgsu 15332   +fcplusf 16478   Mndcmnd 16528  CMndccmn 17423   fBascfbas 18951   filGencfg 18952  TopOnctopon 19910   TopSpctps 19911   Cn ccn 20232   CnP ccnp 20233   tX ctx 20567   Filcfil 20852   fLimf cflf 20942  TopMndctmd 21077   tsums ctsu 21132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-oi 8029  df-card 8376  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-fzo 11918  df-seq 12215  df-hash 12517  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-0g 15333  df-gsum 15334  df-topgen 15335  df-plusf 16480  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-cntz 16964  df-cmn 17425  df-fbas 18960  df-fg 18961  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-ntr 20027  df-nei 20106  df-cn 20235  df-cnp 20236  df-tx 20569  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-tmd 21079  df-tsms 21133

This theorem is referenced by:  tsmssub  21155  tsmssplit  21158  esumadd  28880  esumaddf  28884