Theorem tsmsval2 21933

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 )
tsmsval2.f	|- ( ph -> F e. W )
tsmsval2.a	|- ( ph -> dom F = A )

Assertion

Ref	Expression
tsmsval2	|- ( 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 tsmsval2

Dummy variables f s w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tsms 21930	. . 3 |- tsums = ( w e. _V , f e. _V |-> [_ ( ~P dom f i^i Fin ) / s ]_ ( ( ( TopOpen ` w ) fLimf ( s filGen ran ( z e. s |-> { y e. s | z C_ y } ) ) ) ` ( y e. s |-> ( w gsumg ( f |` y ) ) ) ) )
2	1	a1i 11	. 2 |- ( ph -> tsums = ( w e. _V , f e. _V |-> [_ ( ~P dom f i^i Fin ) / s ]_ ( ( ( TopOpen ` w ) fLimf ( s filGen ran ( z e. s |-> { y e. s | z C_ y } ) ) ) ` ( y e. s |-> ( w gsumg ( f |` y ) ) ) ) ) )
3		vex 3203	. . . . . . 7 |- f e. _V
4	3	dmex 7099	. . . . . 6 |- dom f e. _V
5	4	pwex 4848	. . . . 5 |- ~P dom f e. _V
6	5	inex1 4799	. . . 4 |- ( ~P dom f i^i Fin ) e. _V
7	6	a1i 11	. . 3 |- ( ( ph /\ ( w = G /\ f = F ) ) -> ( ~P dom f i^i Fin ) e. _V )
8		simplrl 800	. . . . . . 7 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> w = G )
9	8	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> ( TopOpen ` w ) = ( TopOpen ` G ) )
10		tsmsval.j	. . . . . 6 |- J = ( TopOpen ` G )
11	9, 10	syl6eqr 2674	. . . . 5 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> ( TopOpen ` w ) = J )
12		id 22	. . . . . . 7 |- ( s = ( ~P dom f i^i Fin ) -> s = ( ~P dom f i^i Fin ) )
13		simprr 796	. . . . . . . . . . . 12 |- ( ( ph /\ ( w = G /\ f = F ) ) -> f = F )
14	13	dmeqd 5326	. . . . . . . . . . 11 |- ( ( ph /\ ( w = G /\ f = F ) ) -> dom f = dom F )
15		tsmsval2.a	. . . . . . . . . . . 12 |- ( ph -> dom F = A )
16	15	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( w = G /\ f = F ) ) -> dom F = A )
17	14, 16	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ ( w = G /\ f = F ) ) -> dom f = A )
18	17	pweqd 4163	. . . . . . . . 9 |- ( ( ph /\ ( w = G /\ f = F ) ) -> ~P dom f = ~P A )
19	18	ineq1d 3813	. . . . . . . 8 |- ( ( ph /\ ( w = G /\ f = F ) ) -> ( ~P dom f i^i Fin ) = ( ~P A i^i Fin ) )
20		tsmsval.s	. . . . . . . 8 |- S = ( ~P A i^i Fin )
21	19, 20	syl6eqr 2674	. . . . . . 7 |- ( ( ph /\ ( w = G /\ f = F ) ) -> ( ~P dom f i^i Fin ) = S )
22	12, 21	sylan9eqr 2678	. . . . . 6 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> s = S )
23		rabeq 3192	. . . . . . . . . 10 |- ( s = S -> { y e. s | z C_ y } = { y e. S | z C_ y } )
24	22, 23	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> { y e. s | z C_ y } = { y e. S | z C_ y } )
25	22, 24	mpteq12dv 4733	. . . . . . . 8 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> ( z e. s |-> { y e. s | z C_ y } ) = ( z e. S |-> { y e. S | z C_ y } ) )
26	25	rneqd 5353	. . . . . . 7 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> ran ( z e. s |-> { y e. s | z C_ y } ) = ran ( z e. S |-> { y e. S | z C_ y } ) )
27		tsmsval.l	. . . . . . 7 |- L = ran ( z e. S |-> { y e. S | z C_ y } )
28	26, 27	syl6eqr 2674	. . . . . 6 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> ran ( z e. s |-> { y e. s | z C_ y } ) = L )
29	22, 28	oveq12d 6668	. . . . 5 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> ( s filGen ran ( z e. s |-> { y e. s | z C_ y } ) ) = ( S filGen L ) )
30	11, 29	oveq12d 6668	. . . 4 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> ( ( TopOpen ` w ) fLimf ( s filGen ran ( z e. s |-> { y e. s | z C_ y } ) ) ) = ( J fLimf ( S filGen L ) ) )
31		simplrr 801	. . . . . . 7 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> f = F )
32	31	reseq1d 5395	. . . . . 6 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> ( f |` y ) = ( F |` y ) )
33	8, 32	oveq12d 6668	. . . . 5 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> ( w gsumg ( f |` y ) ) = ( G gsumg ( F |` y ) ) )
34	22, 33	mpteq12dv 4733	. . . 4 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> ( y e. s |-> ( w gsumg ( f |` y ) ) ) = ( y e. S |-> ( G gsumg ( F |` y ) ) ) )
35	30, 34	fveq12d 6197	. . 3 |- ( ( ( ph /\ ( w = G /\ f = F ) ) /\ s = ( ~P dom f i^i Fin ) ) -> ( ( ( TopOpen ` w ) fLimf ( s filGen ran ( z e. s |-> { y e. s | z C_ y } ) ) ) ` ( y e. s |-> ( w gsumg ( f |` y ) ) ) ) = ( ( J fLimf ( S filGen L ) ) ` ( y e. S |-> ( G gsumg ( F |` y ) ) ) ) )
36	7, 35	csbied 3560	. 2 |- ( ( ph /\ ( w = G /\ f = F ) ) -> [_ ( ~P dom f i^i Fin ) / s ]_ ( ( ( TopOpen ` w ) fLimf ( s filGen ran ( z e. s |-> { y e. s | z C_ y } ) ) ) ` ( y e. s |-> ( w gsumg ( f |` y ) ) ) ) = ( ( J fLimf ( S filGen L ) ) ` ( y e. S |-> ( G gsumg ( F |` y ) ) ) ) )
37		tsmsval.g	. . 3 |- ( ph -> G e. V )
38		elex 3212	. . 3 |- ( G e. V -> G e. _V )
39	37, 38	syl 17	. 2 |- ( ph -> G e. _V )
40		tsmsval2.f	. . 3 |- ( ph -> F e. W )
41		elex 3212	. . 3 |- ( F e. W -> F e. _V )
42	40, 41	syl 17	. 2 |- ( ph -> F e. _V )
43		fvexd 6203	. 2 |- ( ph -> ( ( J fLimf ( S filGen L ) ) ` ( y e. S |-> ( G gsumg ( F |` y ) ) ) ) e. _V )
44	2, 36, 39, 42, 43	ovmpt2d 6788	1 |- ( ph -> ( G tsums F ) = ( ( J fLimf ( S filGen L ) ) ` ( y e. S |-> ( G gsumg ( F |` y ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   [_csb 3533   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-tsms 21930

This theorem is referenced by:  tsmsval  21934  tsmspropd  21935