Theorem lcvbr3 34310

Description: The covers relation for a left vector space (or a left module). (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
lcvbr3	|- ( ph -> ( T C U <-> ( T C. U /\ A. s e. S ( ( T C_ s /\ s C_ U ) -> ( s = T \/ s = U ) ) ) ) )

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

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

Proof of Theorem lcvbr3

Step	Hyp	Ref	Expression
1		lcvfbr.s	. . 3 |- S = ( LSubSp ` W )
2		lcvfbr.c	. . 3 |- C = ( <oLL ` W )
3		lcvfbr.w	. . 3 |- ( ph -> W e. X )
4		lcvfbr.t	. . 3 |- ( ph -> T e. S )
5		lcvfbr.u	. . 3 |- ( ph -> U e. S )
6	1, 2, 3, 4, 5	lcvbr 34308	. 2 |- ( ph -> ( T C U <-> ( T C. U /\ -. E. s e. S ( T C. s /\ s C. U ) ) ) )
7		iman 440	. . . . . 6 |- ( ( ( T C_ s /\ s C_ U ) -> ( s = T \/ s = U ) ) <-> -. ( ( T C_ s /\ s C_ U ) /\ -. ( s = T \/ s = U ) ) )
8		df-pss 3590	. . . . . . . . 9 |- ( T C. s <-> ( T C_ s /\ T =/= s ) )
9		necom 2847	. . . . . . . . . 10 |- ( T =/= s <-> s =/= T )
10	9	anbi2i 730	. . . . . . . . 9 |- ( ( T C_ s /\ T =/= s ) <-> ( T C_ s /\ s =/= T ) )
11	8, 10	bitri 264	. . . . . . . 8 |- ( T C. s <-> ( T C_ s /\ s =/= T ) )
12		df-pss 3590	. . . . . . . 8 |- ( s C. U <-> ( s C_ U /\ s =/= U ) )
13	11, 12	anbi12i 733	. . . . . . 7 |- ( ( T C. s /\ s C. U ) <-> ( ( T C_ s /\ s =/= T ) /\ ( s C_ U /\ s =/= U ) ) )
14		an4 865	. . . . . . . 8 |- ( ( ( T C_ s /\ s =/= T ) /\ ( s C_ U /\ s =/= U ) ) <-> ( ( T C_ s /\ s C_ U ) /\ ( s =/= T /\ s =/= U ) ) )
15		neanior 2886	. . . . . . . . 9 |- ( ( s =/= T /\ s =/= U ) <-> -. ( s = T \/ s = U ) )
16	15	anbi2i 730	. . . . . . . 8 |- ( ( ( T C_ s /\ s C_ U ) /\ ( s =/= T /\ s =/= U ) ) <-> ( ( T C_ s /\ s C_ U ) /\ -. ( s = T \/ s = U ) ) )
17	14, 16	bitri 264	. . . . . . 7 |- ( ( ( T C_ s /\ s =/= T ) /\ ( s C_ U /\ s =/= U ) ) <-> ( ( T C_ s /\ s C_ U ) /\ -. ( s = T \/ s = U ) ) )
18	13, 17	bitri 264	. . . . . 6 |- ( ( T C. s /\ s C. U ) <-> ( ( T C_ s /\ s C_ U ) /\ -. ( s = T \/ s = U ) ) )
19	7, 18	xchbinxr 325	. . . . 5 |- ( ( ( T C_ s /\ s C_ U ) -> ( s = T \/ s = U ) ) <-> -. ( T C. s /\ s C. U ) )
20	19	ralbii 2980	. . . 4 |- ( A. s e. S ( ( T C_ s /\ s C_ U ) -> ( s = T \/ s = U ) ) <-> A. s e. S -. ( T C. s /\ s C. U ) )
21		ralnex 2992	. . . 4 |- ( A. s e. S -. ( T C. s /\ s C. U ) <-> -. E. s e. S ( T C. s /\ s C. U ) )
22	20, 21	bitri 264	. . 3 |- ( A. s e. S ( ( T C_ s /\ s C_ U ) -> ( s = T \/ s = U ) ) <-> -. E. s e. S ( T C. s /\ s C. U ) )
23	22	anbi2i 730	. 2 |- ( ( T C. U /\ A. s e. S ( ( T C_ s /\ s C_ U ) -> ( s = T \/ s = U ) ) ) <-> ( T C. U /\ -. E. s e. S ( T C. s /\ s C. U ) ) )
24	6, 23	syl6bbr 278	1 |- ( ph -> ( T C U <-> ( T C. U /\ A. s e. S ( ( T C_ s /\ s C_ U ) -> ( s = T \/ s = U ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   C. wpss 3575   class class class wbr 4653   ` 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:  lcvexchlem4  34324  lcvexchlem5  34325