Theorem tsmsadd 19837

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 19763	. . . . . . 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 19821	. . . . 5 |- ( ph -> ( G tsums F ) C_ B )
9		tsmsadd.x	. . . . 5 |- ( ph -> X e. ( G tsums F ) )
10	8, 9	sseldd 3455	. . . 4 |- ( ph -> X e. B )
11		tsmsadd.h	. . . . . 6 |- ( ph -> H : A --> B )
12	1, 2, 5, 6, 11	tsmscl 19821	. . . . 5 |- ( ph -> ( G tsums H ) C_ B )
13		tsmsadd.y	. . . . 5 |- ( ph -> Y e. ( G tsums H ) )
14	12, 13	sseldd 3455	. . . 4 |- ( ph -> Y e. B )
15		tsmsadd.p	. . . . 5 |- .+ = ( +g ` G )
16		eqid 2451	. . . . 5 |- ( +f ` G ) = ( +f ` G )
17	1, 15, 16	plusfval 15530	. . . 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 2451	. . . . . 6 |- ( TopOpen ` G ) = ( TopOpen ` G )
20	1, 19	istps 18657	. . . . 5 |- ( G e. TopSp <-> ( TopOpen ` G ) e. (TopOn ` B ) )
21	5, 20	sylib 196	. . . 4 |- ( ph -> ( TopOpen ` G ) e. (TopOn ` B ) )
22		eqid 2451	. . . . . 6 |- ( ~P A i^i Fin ) = ( ~P A i^i Fin )
23		eqid 2451	. . . . . 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 2451	. . . . . 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 19814	. . . . 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 19567	. . . . 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 19815	. . . 4 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` z ) ) e. B )
29	1, 22, 2, 6, 11	tsmslem1 19815	. . . 4 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsumg ( H |` z ) ) e. B )
30	1, 19, 22, 24, 2, 6, 7	tsmsval 19817	. . . . 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 2541	. . . 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 19817	. . . . 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 2541	. . . 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 19770	. . . . . 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 4969	. . . . . . 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 19280	. . . . . . . 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 18648	. . . . . . 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 2541	. . . . 5 |- ( ph -> <. X , Y >. e. U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) )
43		eqid 2451	. . . . . 6 |- U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) = U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) )
44	43	cncnpi 18998	. . . . 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 19696	. . 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 2540	. 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 16396	. . . . . . 7 |- ( G e. CMnd -> G e. Mnd )
49	2, 48	syl 16	. . . . . 6 |- ( ph -> G e. Mnd )
50	1, 15	mndcl 15522	. . . . . . 7 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
51	50	3expb 1189	. . . . . 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 3657	. . . . 5 |- ( A i^i A ) = A
54	52, 7, 11, 6, 6, 53	off 6434	. . . 4 |- ( ph -> ( F oF .+ H ) : A --> B )
55	1, 19, 22, 24, 2, 6, 54	tsmsval 19817	. . 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 2451	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
57	2	adantr 465	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> G e. CMnd )
58		elfpw 7714	. . . . . . . . 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 5676	. . . . . . . 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 5676	. . . . . . . 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 5799	. . . . . . . . 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 7731	. . . . . . 7 |- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( F |` z ) finSupp ( 0g ` G ) )
69	65, 60, 67	fdmfifsupp 7731	. . . . . . 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 16516	. . . . . 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 5799	. . . . . . . . . . . 12 |- ( Base ` G ) e. _V
72	1, 71	eqeltri 2535	. . . . . . . . . . 11 |- B e. _V
73	72	a1i 11	. . . . . . . . . 10 |- ( ph -> B e. _V )
74		fex2 6632	. . . . . . . . . 10 |- ( ( F : A --> B /\ A e. V /\ B e. _V ) -> F e. _V )
75	7, 6, 73, 74	syl3anc 1219	. . . . . . . . 9 |- ( ph -> F e. _V )
76		fex2 6632	. . . . . . . . . 10 |- ( ( H : A --> B /\ A e. V /\ B e. _V ) -> H e. _V )
77	11, 6, 73, 76	syl3anc 1219	. . . . . . . . 9 |- ( ph -> H e. _V )
78		offres 6672	. . . . . . . . 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 6206	. . . . . 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 15530	. . . . . . 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 2502	. . . . 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 4476	. . . 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 5793	. . 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 2492	. 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 2542	1 |- ( ph -> ( X .+ Y ) e. ( G tsums ( F oF .+ H ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   {crab 2799   _Vcvv 3068   i^i cin 3425   C_ wss 3426   ~Pcpw 3958   <.cop 3981   U.cuni 4189   |-> cmpt 4448   X. cxp 4936   ran crn 4939   |` cres 4940   -->wf 5512   ` cfv 5516  (class class class)co 6190   oFcof 6418   Fincfn 7410   Basecbs 14276   +g cplusg 14340   TopOpenctopn 14462   0gc0g 14480   gsum_g cgsu 14481   Mndcmnd 15511   +fcplusf 15514  CMndccmn 16381   fBascfbas 17913   filGencfg 17914  TopOnctopon 18615   TopSpctps 18617   Cn ccn 18944   CnP ccnp 18945   tX ctx 19249   Filcfil 19534   fLimf cflf 19624  TopMndctmd 19757   tsums ctsu 19812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-fzo 11650  df-seq 11908  df-hash 12205  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-0g 14482  df-gsum 14483  df-topgen 14484  df-mnd 15517  df-plusf 15518  df-submnd 15567  df-cntz 15937  df-cmn 16383  df-fbas 17923  df-fg 17924  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-ntr 18740  df-nei 18818  df-cn 18947  df-cnp 18948  df-tx 19251  df-fil 19535  df-fm 19627  df-flim 19628  df-flf 19629  df-tmd 19759  df-tsms 19813

This theorem is referenced by:  tsmssub  19839  tsmssplit  19842  esumadd  26641  esumaddf  26646