Theorem tsmsval 21934

Description: Definition of the topological group sum(s) of a collection F ( x ) of values in the group with index set A. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
tsmsval.b	|- B = ( Base ` G )
tsmsval.j	|- J = ( TopOpen ` G )
tsmsval.s	|- S = ( ~P A i^i Fin )
tsmsval.l	|- L = ran ( z e. S |-> { y e. S | z C_ y } )
tsmsval.g	|- ( ph -> G e. V )
tsmsval.a	|- ( ph -> A e. W )
tsmsval.f	|- ( ph -> F : A --> B )

Assertion

Ref	Expression
tsmsval	|- ( ph -> ( G tsums F ) = ( ( J fLimf ( S filGen L ) ) ` ( y e. S |-> ( G gsumg ( F |` y ) ) ) ) )

Distinct variable groups:   y, z, F   y, G, z   ph, y, z   y, S

Allowed substitution hints:   A( y, z)   B( y, z)   S( z)   J( y, z)   L( y, z)   V( y, z)   W( y, z)

Proof of Theorem tsmsval

Step	Hyp	Ref	Expression
1		tsmsval.b	. 2 |- B = ( Base ` G )
2		tsmsval.j	. 2 |- J = ( TopOpen ` G )
3		tsmsval.s	. 2 |- S = ( ~P A i^i Fin )
4		tsmsval.l	. 2 |- L = ran ( z e. S |-> { y e. S | z C_ y } )
5		tsmsval.g	. 2 |- ( ph -> G e. V )
6		tsmsval.f	. . 3 |- ( ph -> F : A --> B )
7		tsmsval.a	. . 3 |- ( ph -> A e. W )
8		fvex 6201	. . . . 5 |- ( Base ` G ) e. _V
9	1, 8	eqeltri 2697	. . . 4 |- B e. _V
10	9	a1i 11	. . 3 |- ( ph -> B e. _V )
11		fex2 7121	. . 3 |- ( ( F : A --> B /\ A e. W /\ B e. _V ) -> F e. _V )
12	6, 7, 10, 11	syl3anc 1326	. 2 |- ( ph -> F e. _V )
13		fdm 6051	. . 3 |- ( F : A --> B -> dom F = A )
14	6, 13	syl 17	. 2 |- ( ph -> dom F = A )
15	1, 2, 3, 4, 5, 12, 14	tsmsval2 21933	1 |- ( ph -> ( G tsums F ) = ( ( J fLimf ( S filGen L ) ) ` ( y e. S |-> ( G gsumg ( F |` y ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   TopOpenctopn 16082   gsum_g cgsu 16101   filGencfg 19735   fLimf cflf 21739   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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-tsms 21930

This theorem is referenced by:  eltsms  21936  haustsms  21939  tsmscls  21941  tsmsmhm  21949  tsmsadd  21950