Theorem tsmsadd 20377

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 20303	. . . . . . 7 |- ( G e. TopMnd -> G e. TopSp )
5	3, 4	syl 16	. . . . . 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 20361	. . . . 5 |- ( ph -> ( G tsums F ) C_ B )
9		tsmsadd.x	. . . . 5 |- ( ph -> X e. ( G tsums F ) )
10	8, 9	sseldd 3498	. . . 4 |- ( ph -> X e. B )
11		tsmsadd.h	. . . . . 6 |- ( ph -> H : A --> B )
12	1, 2, 5, 6, 11	tsmscl 20361	. . . . 5 |- ( ph -> ( G tsums H ) C_ B )
13		tsmsadd.y	. . . . 5 |- ( ph -> Y e. ( G tsums H ) )
14	12, 13	sseldd 3498	. . . 4 |- ( ph -> Y e. B )
15		tsmsadd.p	. . . . 5 |- .+ = ( +g ` G )
16		eqid 2460	. . . . 5 |- ( +f ` G ) = ( +f ` G )
17	1, 15, 16	plusfval 15734	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( X ( +f ` G ) Y ) = ( X .+ Y ) )
18	10, 14, 17	syl2anc 661	. . 3 |- ( ph -> ( X ( +f ` G ) Y ) = ( X .+ Y ) )
19		eqid 2460	. . . . . 6 |- ( TopOpen ` G ) = ( TopOpen ` G )
20	1, 19	istps 19197	. . . . 5 |- ( G e. TopSp <-> ( TopOpen ` G ) e. (TopOn ` B ) )
21	5, 20	sylib 196	. . . 4 |- ( ph -> ( TopOpen ` G ) e. (TopOn ` B ) )
22		eqid 2460	. . . . . 6 |- ( ~P A i^i Fin ) = ( ~P A i^i Fin )
23		eqid 2460	. . . . . 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 2460	. . . . . 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 20354	. . . . 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 20107	. . . . 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 16	. . . 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 20355	. . . 4 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` z ) ) e. B )
29	1, 22, 2, 6, 11	tsmslem1 20355	. . . 4 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsumg ( H |` z ) ) e. B )
30	1, 19, 22, 24, 2, 6, 7	tsmsval 20357	. . . . 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 2550	. . . 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 20357	. . . . 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 2550	. . . 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 20310	. . . . . 6 |- ( G e. TopMnd -> ( +f ` G ) e. ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) Cn ( TopOpen ` G ) ) )
35	3, 34	syl 16	. . . . 5 |- ( ph -> ( +f ` G ) e. ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) Cn ( TopOpen ` G ) ) )
36		opelxpi 5023	. . . . . . 7 |- ( ( X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) )
37	10, 14, 36	syl2anc 661	. . . . . 6 |- ( ph -> <. X , Y >. e. ( B X. B ) )
38		txtopon 19820	. . . . . . . 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 661	. . . . . . 7 |- ( ph -> ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) e. (TopOn ` ( B X. B ) ) )
40		toponuni 19188	. . . . . . 7 |- ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) e. (TopOn ` ( B X. B ) ) -> ( B X. B ) = U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) )
41	39, 40	syl 16	. . . . . 6 |- ( ph -> ( B X. B ) = U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) )
42	37, 41	eleqtrd 2550	. . . . 5 |- ( ph -> <. X , Y >. e. U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) )
43		eqid 2460	. . . . . 6 |- U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) = U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) )
44	43	cncnpi 19538	. . . . 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 661	. . . 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 20236	. . 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 2549	. 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 16602	. . . . . . 7 |- ( G e. CMnd -> G e. Mnd )
49	2, 48	syl 16	. . . . . 6 |- ( ph -> G e. Mnd )
50	1, 15	mndcl 15726	. . . . . . 7 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
51	50	3expb 1192	. . . . . 6 |- ( ( G e. Mnd /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B )
52	49, 51	sylan 471	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B )
53		inidm 3700	. . . . 5 |- ( A i^i A ) = A
54	52, 7, 11, 6, 6, 53	off 6529	. . . 4 |- ( ph -> ( F oF .+ H ) : A --> B )
55	1, 19, 22, 24, 2, 6, 54	tsmsval 20357	. . 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 2460	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
57	2	adantr 465	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> G e. CMnd )
58		elfpw 7811	. . . . . . . . 9 |- ( z e. ( ~P A i^i Fin ) <-> ( z C_ A /\ z e. Fin ) )
59	58	simprbi 464	. . . . . . . 8 |- ( z e. ( ~P A i^i Fin ) -> z e. Fin )
60	59	adantl 466	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> z e. Fin )
61	58	simplbi 460	. . . . . . . 8 |- ( z e. ( ~P A i^i Fin ) -> z C_ A )
62		fssres 5742	. . . . . . . 8 |- ( ( F : A --> B /\ z C_ A ) -> ( F |` z ) : z --> B )
63	7, 61, 62	syl2an 477	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( F |` z ) : z --> B )
64		fssres 5742	. . . . . . . 8 |- ( ( H : A --> B /\ z C_ A ) -> ( H |` z ) : z --> B )
65	11, 61, 64	syl2an 477	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( H |` z ) : z --> B )
66		fvex 5867	. . . . . . . . 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 7828	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( F |` z ) finSupp ( 0g ` G ) )
69	65, 60, 67	fdmfifsupp 7828	. . . . . . 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 16722	. . . . . 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 5867	. . . . . . . . . . . 12 |- ( Base ` G ) e. _V
72	1, 71	eqeltri 2544	. . . . . . . . . . 11 |- B e. _V
73	72	a1i 11	. . . . . . . . . 10 |- ( ph -> B e. _V )
74		fex2 6729	. . . . . . . . . 10 |- ( ( F : A --> B /\ A e. V /\ B e. _V ) -> F e. _V )
75	7, 6, 73, 74	syl3anc 1223	. . . . . . . . 9 |- ( ph -> F e. _V )
76		fex2 6729	. . . . . . . . . 10 |- ( ( H : A --> B /\ A e. V /\ B e. _V ) -> H e. _V )
77	11, 6, 73, 76	syl3anc 1223	. . . . . . . . 9 |- ( ph -> H e. _V )
78		offres 6769	. . . . . . . . 9 |- ( ( F e. _V /\ H e. _V ) -> ( ( F oF .+ H ) |` z ) = ( ( F |` z ) oF .+ ( H |` z ) ) )
79	75, 77, 78	syl2anc 661	. . . . . . . 8 |- ( ph -> ( ( F oF .+ H ) |` z ) = ( ( F |` z ) oF .+ ( H |` z ) ) )
80	79	adantr 465	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( ( F oF .+ H ) |` z ) = ( ( F |` z ) oF .+ ( H |` z ) ) )
81	80	oveq2d 6291	. . . . . 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 15734	. . . . . . 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 661	. . . . . 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 2511	. . . . 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 4526	. . . 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 5861	. . 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 2501	. 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 2551	1 |- ( ph -> ( X .+ Y ) e. ( G tsums ( F oF .+ H ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1374   e. wcel 1762   {crab 2811   _Vcvv 3106   i^i cin 3468   C_ wss 3469   ~Pcpw 4003   <.cop 4026   U.cuni 4238   |-> cmpt 4498   X. cxp 4990   ran crn 4993   |` cres 4994   -->wf 5575   ` cfv 5579  (class class class)co 6275   oFcof 6513   Fincfn 7506   Basecbs 14479   +g cplusg 14544   TopOpenctopn 14666   0gc0g 14684   gsum_g cgsu 14685   Mndcmnd 15715   +fcplusf 15718  CMndccmn 16587   fBascfbas 18170   filGencfg 18171  TopOnctopon 19155   TopSpctps 19157   Cn ccn 19484   CnP ccnp 19485   tX ctx 19789   Filcfil 20074   fLimf cflf 20164  TopMndctmd 20297   tsums ctsu 20352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-gsum 14687  df-topgen 14688  df-mnd 15721  df-plusf 15722  df-submnd 15771  df-cntz 16143  df-cmn 16589  df-fbas 18180  df-fg 18181  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-ntr 19280  df-nei 19358  df-cn 19487  df-cnp 19488  df-tx 19791  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-tmd 20299  df-tsms 20353

This theorem is referenced by:  tsmssub  20379  tsmssplit  20382  esumadd  27554  esumaddf  27559