Theorem lssex 14519

Description: Existence of a linear subspace. (Contributed by Jim Kingdon, 27-Apr-2025.)

Assertion

Ref	Expression
lssex	|- ( W e. V -> ( LSubSp ` W ) e. _V )

Proof of Theorem lssex

Dummy variables w a b j s x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		basfn 13288	. . . . . 6 |- Base Fn _V
2		vex 2818	. . . . . . 7 |- w e. _V
3	2	a1i 9	. . . . . 6 |- ( W e. V -> w e. _V )
4		funfvex 5689	. . . . . . 7 |- ( ( Fun Base /\ w e. dom Base ) -> ( Base ` w ) e. _V )
5	4	funfni 5460	. . . . . 6 |- ( ( Base Fn _V /\ w e. _V ) -> ( Base ` w ) e. _V )
6	1, 3, 5	sylancr 414	. . . . 5 |- ( W e. V -> ( Base ` w ) e. _V )
7	6	pwexd 4296	. . . 4 |- ( W e. V -> ~P ( Base ` w ) e. _V )
8		rabexg 4257	. . . 4 |- ( ~P ( Base ` w ) e. _V -> { s e. ~P ( Base ` w ) | ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) } e. _V )
9	7, 8	syl 14	. . 3 |- ( W e. V -> { s e. ~P ( Base ` w ) | ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) } e. _V )
10	9	alrimiv 1923	. 2 |- ( W e. V -> A. w { s e. ~P ( Base ` w ) | ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) } e. _V )
11		df-lssm 14518	. . 3 |- LSubSp = ( w e. _V |-> { s e. ~P ( Base ` w ) | ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) } )
12	11	mptfvex 5765	. 2 |- ( ( A. w { s e. ~P ( Base ` w ) | ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) } e. _V /\ W e. V ) -> ( LSubSp ` W ) e. _V )
13	10, 12	mpancom 422	1 |- ( W e. V -> ( LSubSp ` W ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1396   E.wex 1541   e. wcel 2205   A.wral 2522   {crab 2526   _Vcvv 2815   ~Pcpw 3671   Fn wfn 5349   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307  Scalarcsca 13310   .scvsca 13311   LSubSpclss 14517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-inn 9240  df-ndx 13232  df-slot 13233  df-base 13235  df-lssm 14518

This theorem is referenced by:  lidlex  14638