Theorem lcvfbr 34307

Description: The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)

Hypotheses

Ref	Expression
lcvfbr.s	|- S = ( LSubSp ` W )
lcvfbr.c	|- C = ( <oLL ` W )
lcvfbr.w	|- ( ph -> W e. X )

Assertion

Ref	Expression
lcvfbr	|- ( ph -> C = { <. t , u >. | ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) } )

Distinct variable groups:   t, s, u, S   W, s, t, u

Allowed substitution hints:   ph( u, t, s)   C( u, t, s)   X( u, t, s)

Proof of Theorem lcvfbr

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcvfbr.c	. 2 |- C = ( <oLL ` W )
2		lcvfbr.w	. . 3 |- ( ph -> W e. X )
3		elex 3212	. . 3 |- ( W e. X -> W e. _V )
4		fveq2 6191	. . . . . . . . 9 |- ( w = W -> ( LSubSp ` w ) = ( LSubSp ` W ) )
5		lcvfbr.s	. . . . . . . . 9 |- S = ( LSubSp ` W )
6	4, 5	syl6eqr 2674	. . . . . . . 8 |- ( w = W -> ( LSubSp ` w ) = S )
7	6	eleq2d 2687	. . . . . . 7 |- ( w = W -> ( t e. ( LSubSp ` w ) <-> t e. S ) )
8	6	eleq2d 2687	. . . . . . 7 |- ( w = W -> ( u e. ( LSubSp ` w ) <-> u e. S ) )
9	7, 8	anbi12d 747	. . . . . 6 |- ( w = W -> ( ( t e. ( LSubSp ` w ) /\ u e. ( LSubSp ` w ) ) <-> ( t e. S /\ u e. S ) ) )
10	6	rexeqdv 3145	. . . . . . . 8 |- ( w = W -> ( E. s e. ( LSubSp ` w ) ( t C. s /\ s C. u ) <-> E. s e. S ( t C. s /\ s C. u ) ) )
11	10	notbid 308	. . . . . . 7 |- ( w = W -> ( -. E. s e. ( LSubSp ` w ) ( t C. s /\ s C. u ) <-> -. E. s e. S ( t C. s /\ s C. u ) ) )
12	11	anbi2d 740	. . . . . 6 |- ( w = W -> ( ( t C. u /\ -. E. s e. ( LSubSp ` w ) ( t C. s /\ s C. u ) ) <-> ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) )
13	9, 12	anbi12d 747	. . . . 5 |- ( w = W -> ( ( ( t e. ( LSubSp ` w ) /\ u e. ( LSubSp ` w ) ) /\ ( t C. u /\ -. E. s e. ( LSubSp ` w ) ( t C. s /\ s C. u ) ) ) <-> ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) ) )
14	13	opabbidv 4716	. . . 4 |- ( w = W -> { <. t , u >. | ( ( t e. ( LSubSp ` w ) /\ u e. ( LSubSp ` w ) ) /\ ( t C. u /\ -. E. s e. ( LSubSp ` w ) ( t C. s /\ s C. u ) ) ) } = { <. t , u >. | ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) } )
15		df-lcv 34306	. . . 4 |- <oLL = ( w e. _V |-> { <. t , u >. | ( ( t e. ( LSubSp ` w ) /\ u e. ( LSubSp ` w ) ) /\ ( t C. u /\ -. E. s e. ( LSubSp ` w ) ( t C. s /\ s C. u ) ) ) } )
16		fvex 6201	. . . . . . 7 |- ( LSubSp ` W ) e. _V
17	5, 16	eqeltri 2697	. . . . . 6 |- S e. _V
18	17, 17	xpex 6962	. . . . 5 |- ( S X. S ) e. _V
19		opabssxp 5193	. . . . 5 |- { <. t , u >. | ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) } C_ ( S X. S )
20	18, 19	ssexi 4803	. . . 4 |- { <. t , u >. | ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) } e. _V
21	14, 15, 20	fvmpt 6282	. . 3 |- ( W e. _V -> ( <oLL ` W ) = { <. t , u >. | ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) } )
22	2, 3, 21	3syl 18	. 2 |- ( ph -> ( <oLL ` W ) = { <. t , u >. | ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) } )
23	1, 22	syl5eq 2668	1 |- ( ph -> C = { <. t , u >. | ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C. wpss 3575   {copab 4712   X. cxp 5112   ` cfv 5888   LSubSpclss 18932   <oL_L clcv 34305

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-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-iota 5851  df-fun 5890  df-fv 5896  df-lcv 34306

This theorem is referenced by:  lcvbr  34308