Theorem rrxfsupp 23185

Description: Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.)

Hypotheses

Ref	Expression
rrxmval.1	|- X = { h e. ( RR ^m I ) | h finSupp 0 }
rrxf.1	|- ( ph -> F e. X )

Assertion

Ref	Expression
rrxfsupp	|- ( ph -> ( F supp 0 ) e. Fin )

Distinct variable groups:   h, F   h, I

Allowed substitution hints:   ph( h)   X( h)

Proof of Theorem rrxfsupp

Step	Hyp	Ref	Expression
1		rrxf.1	. . . . 5 |- ( ph -> F e. X )
2		rrxmval.1	. . . . 5 |- X = { h e. ( RR ^m I ) | h finSupp 0 }
3	1, 2	syl6eleq 2711	. . . 4 |- ( ph -> F e. { h e. ( RR ^m I ) | h finSupp 0 } )
4		breq1 4656	. . . . 5 |- ( h = F -> ( h finSupp 0 <-> F finSupp 0 ) )
5	4	elrab 3363	. . . 4 |- ( F e. { h e. ( RR ^m I ) | h finSupp 0 } <-> ( F e. ( RR ^m I ) /\ F finSupp 0 ) )
6	3, 5	sylib 208	. . 3 |- ( ph -> ( F e. ( RR ^m I ) /\ F finSupp 0 ) )
7	6	simprd 479	. 2 |- ( ph -> F finSupp 0 )
8	7	fsuppimpd 8282	1 |- ( ph -> ( F supp 0 ) e. Fin )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653  (class class class)co 6650   supp csupp 7295   ^m cmap 7857   Fincfn 7955   finSupp cfsupp 8275   RRcr 9935   0cc0 9936

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-fsupp 8276

This theorem is referenced by:  rrxmval  23188  rrxmet  23191  rrxdstprj1  23192