Theorem lcvbr 34308

Description: The covers relation for a left vector space (or a left module). (cvbr 29141 analog.) (Contributed by NM, 9-Jan-2015.)

Hypotheses

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

Assertion

Ref	Expression
lcvbr	|- ( ph -> ( T C U <-> ( T C. U /\ -. E. s e. S ( T C. s /\ s C. U ) ) ) )

Distinct variable groups:   S, s   W, s   T, s   U, s

Allowed substitution hints:   ph( s)   C( s)   X( s)

Proof of Theorem lcvbr

Dummy variables t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcvfbr.t	. . 3 |- ( ph -> T e. S )
2		lcvfbr.u	. . 3 |- ( ph -> U e. S )
3		eleq1 2689	. . . . . 6 |- ( t = T -> ( t e. S <-> T e. S ) )
4	3	anbi1d 741	. . . . 5 |- ( t = T -> ( ( t e. S /\ u e. S ) <-> ( T e. S /\ u e. S ) ) )
5		psseq1 3694	. . . . . 6 |- ( t = T -> ( t C. u <-> T C. u ) )
6		psseq1 3694	. . . . . . . . 9 |- ( t = T -> ( t C. s <-> T C. s ) )
7	6	anbi1d 741	. . . . . . . 8 |- ( t = T -> ( ( t C. s /\ s C. u ) <-> ( T C. s /\ s C. u ) ) )
8	7	rexbidv 3052	. . . . . . 7 |- ( t = T -> ( E. s e. S ( t C. s /\ s C. u ) <-> E. s e. S ( T C. s /\ s C. u ) ) )
9	8	notbid 308	. . . . . 6 |- ( t = T -> ( -. E. s e. S ( t C. s /\ s C. u ) <-> -. E. s e. S ( T C. s /\ s C. u ) ) )
10	5, 9	anbi12d 747	. . . . 5 |- ( t = T -> ( ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) <-> ( T C. u /\ -. E. s e. S ( T C. s /\ s C. u ) ) ) )
11	4, 10	anbi12d 747	. . . 4 |- ( t = T -> ( ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( 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 ) ) ) ) )
12		eleq1 2689	. . . . . 6 |- ( u = U -> ( u e. S <-> U e. S ) )
13	12	anbi2d 740	. . . . 5 |- ( u = U -> ( ( T e. S /\ u e. S ) <-> ( T e. S /\ U e. S ) ) )
14		psseq2 3695	. . . . . 6 |- ( u = U -> ( T C. u <-> T C. U ) )
15		psseq2 3695	. . . . . . . . 9 |- ( u = U -> ( s C. u <-> s C. U ) )
16	15	anbi2d 740	. . . . . . . 8 |- ( u = U -> ( ( T C. s /\ s C. u ) <-> ( T C. s /\ s C. U ) ) )
17	16	rexbidv 3052	. . . . . . 7 |- ( u = U -> ( E. s e. S ( T C. s /\ s C. u ) <-> E. s e. S ( T C. s /\ s C. U ) ) )
18	17	notbid 308	. . . . . 6 |- ( u = U -> ( -. E. s e. S ( T C. s /\ s C. u ) <-> -. E. s e. S ( T C. s /\ s C. U ) ) )
19	14, 18	anbi12d 747	. . . . 5 |- ( u = U -> ( ( T C. u /\ -. E. s e. S ( T C. s /\ s C. u ) ) <-> ( T C. U /\ -. E. s e. S ( T C. s /\ s C. U ) ) ) )
20	13, 19	anbi12d 747	. . . 4 |- ( u = U -> ( ( ( T e. S /\ u e. S ) /\ ( T C. u /\ -. E. s e. S ( 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 ) ) ) ) )
21		eqid 2622	. . . 4 |- { <. t , u >. | ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( 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 ) ) ) }
22	11, 20, 21	brabg 4994	. . 3 |- ( ( T e. S /\ U e. S ) -> ( T { <. t , u >. | ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) } U <-> ( ( T e. S /\ U e. S ) /\ ( T C. U /\ -. E. s e. S ( T C. s /\ s C. U ) ) ) ) )
23	1, 2, 22	syl2anc 693	. 2 |- ( ph -> ( T { <. t , u >. | ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) } U <-> ( ( T e. S /\ U e. S ) /\ ( T C. U /\ -. E. s e. S ( T C. s /\ s C. U ) ) ) ) )
24		lcvfbr.s	. . . 4 |- S = ( LSubSp ` W )
25		lcvfbr.c	. . . 4 |- C = ( <oLL ` W )
26		lcvfbr.w	. . . 4 |- ( ph -> W e. X )
27	24, 25, 26	lcvfbr 34307	. . 3 |- ( ph -> C = { <. t , u >. | ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) } )
28	27	breqd 4664	. 2 |- ( ph -> ( T C U <-> T { <. t , u >. | ( ( t e. S /\ u e. S ) /\ ( t C. u /\ -. E. s e. S ( t C. s /\ s C. u ) ) ) } U ) )
29	1, 2	jca 554	. . 3 |- ( ph -> ( T e. S /\ U e. S ) )
30	29	biantrurd 529	. 2 |- ( ph -> ( ( T C. U /\ -. E. s e. S ( 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 ) ) ) ) )
31	23, 28, 30	3bitr4d 300	1 |- ( ph -> ( T C U <-> ( T C. U /\ -. E. s e. S ( T C. s /\ s C. U ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C. wpss 3575   class class class wbr 4653   {copab 4712   ` 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-ne 2795  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-pss 3590  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:  lcvbr2  34309  lcvbr3  34310  lcvpss  34311  lcvnbtwn  34312  lsatcv0  34318  mapdcv  36949