Theorem tsmspropd 21935

Description: The group sum depends only on the base set, additive operation, and topology components. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 17316 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)

Hypotheses

Ref	Expression
tsmspropd.f	|- ( ph -> F e. V )
tsmspropd.g	|- ( ph -> G e. W )
tsmspropd.h	|- ( ph -> H e. X )
tsmspropd.b	|- ( ph -> ( Base ` G ) = ( Base ` H ) )
tsmspropd.p	|- ( ph -> ( +g ` G ) = ( +g ` H ) )
tsmspropd.j	|- ( ph -> ( TopOpen ` G ) = ( TopOpen ` H ) )

Assertion

Ref	Expression
tsmspropd	|- ( ph -> ( G tsums F ) = ( H tsums F ) )

Proof of Theorem tsmspropd

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tsmspropd.j	. . . 4 |- ( ph -> ( TopOpen ` G ) = ( TopOpen ` H ) )
2	1	oveq1d 6665	. . 3 |- ( ph -> ( ( TopOpen ` G ) fLimf ( ( ~P dom F i^i Fin ) filGen ran ( z e. ( ~P dom F i^i Fin ) |-> { y e. ( ~P dom F i^i Fin ) | z C_ y } ) ) ) = ( ( TopOpen ` H ) fLimf ( ( ~P dom F i^i Fin ) filGen ran ( z e. ( ~P dom F i^i Fin ) |-> { y e. ( ~P dom F i^i Fin ) | z C_ y } ) ) ) )
3		tsmspropd.f	. . . . . 6 |- ( ph -> F e. V )
4		resexg 5442	. . . . . 6 |- ( F e. V -> ( F |` y ) e. _V )
5	3, 4	syl 17	. . . . 5 |- ( ph -> ( F |` y ) e. _V )
6		tsmspropd.g	. . . . 5 |- ( ph -> G e. W )
7		tsmspropd.h	. . . . 5 |- ( ph -> H e. X )
8		tsmspropd.b	. . . . 5 |- ( ph -> ( Base ` G ) = ( Base ` H ) )
9		tsmspropd.p	. . . . 5 |- ( ph -> ( +g ` G ) = ( +g ` H ) )
10	5, 6, 7, 8, 9	gsumpropd 17272	. . . 4 |- ( ph -> ( G gsumg ( F |` y ) ) = ( H gsumg ( F |` y ) ) )
11	10	mpteq2dv 4745	. . 3 |- ( ph -> ( y e. ( ~P dom F i^i Fin ) |-> ( G gsumg ( F |` y ) ) ) = ( y e. ( ~P dom F i^i Fin ) |-> ( H gsumg ( F |` y ) ) ) )
12	2, 11	fveq12d 6197	. 2 |- ( ph -> ( ( ( TopOpen ` G ) fLimf ( ( ~P dom F i^i Fin ) filGen ran ( z e. ( ~P dom F i^i Fin ) |-> { y e. ( ~P dom F i^i Fin ) | z C_ y } ) ) ) ` ( y e. ( ~P dom F i^i Fin ) |-> ( G gsumg ( F |` y ) ) ) ) = ( ( ( TopOpen ` H ) fLimf ( ( ~P dom F i^i Fin ) filGen ran ( z e. ( ~P dom F i^i Fin ) |-> { y e. ( ~P dom F i^i Fin ) | z C_ y } ) ) ) ` ( y e. ( ~P dom F i^i Fin ) |-> ( H gsumg ( F |` y ) ) ) ) )
13		eqid 2622	. . 3 |- ( Base ` G ) = ( Base ` G )
14		eqid 2622	. . 3 |- ( TopOpen ` G ) = ( TopOpen ` G )
15		eqid 2622	. . 3 |- ( ~P dom F i^i Fin ) = ( ~P dom F i^i Fin )
16		eqid 2622	. . 3 |- ran ( z e. ( ~P dom F i^i Fin ) |-> { y e. ( ~P dom F i^i Fin ) | z C_ y } ) = ran ( z e. ( ~P dom F i^i Fin ) |-> { y e. ( ~P dom F i^i Fin ) | z C_ y } )
17		eqidd 2623	. . 3 |- ( ph -> dom F = dom F )
18	13, 14, 15, 16, 6, 3, 17	tsmsval2 21933	. 2 |- ( ph -> ( G tsums F ) = ( ( ( TopOpen ` G ) fLimf ( ( ~P dom F i^i Fin ) filGen ran ( z e. ( ~P dom F i^i Fin ) |-> { y e. ( ~P dom F i^i Fin ) | z C_ y } ) ) ) ` ( y e. ( ~P dom F i^i Fin ) |-> ( G gsumg ( F |` y ) ) ) ) )
19		eqid 2622	. . 3 |- ( Base ` H ) = ( Base ` H )
20		eqid 2622	. . 3 |- ( TopOpen ` H ) = ( TopOpen ` H )
21	19, 20, 15, 16, 7, 3, 17	tsmsval2 21933	. 2 |- ( ph -> ( H tsums F ) = ( ( ( TopOpen ` H ) fLimf ( ( ~P dom F i^i Fin ) filGen ran ( z e. ( ~P dom F i^i Fin ) |-> { y e. ( ~P dom F i^i Fin ) | z C_ y } ) ) ) ` ( y e. ( ~P dom F i^i Fin ) |-> ( H gsumg ( F |` y ) ) ) ) )
22	12, 18, 21	3eqtr4d 2666	1 |- ( ph -> ( G tsums F ) = ( H tsums F ) )

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   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   +g cplusg 15941   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-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802  df-0g 16102  df-gsum 16103  df-tsms 21930

This theorem is referenced by: (None)