Theorem atcvr0eq 34712

Description: The covers relation is not transitive. (atcv0eq 29238 analog.) (Contributed by NM, 29-Nov-2011.)

Hypotheses

Ref	Expression
atcvr0eq.j	|- .\/ = ( join ` K )
atcvr0eq.z	|- .0. = ( 0. ` K )
atcvr0eq.c	|- C = ( <o ` K )
atcvr0eq.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
atcvr0eq	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( .0. C ( P .\/ Q ) <-> P = Q ) )

Proof of Theorem atcvr0eq

Step	Hyp	Ref	Expression
1		atcvr0eq.j	. . . . . 6 |- .\/ = ( join ` K )
2		atcvr0eq.c	. . . . . 6 |- C = ( <o ` K )
3		atcvr0eq.a	. . . . . 6 |- A = ( Atoms ` K )
4	1, 2, 3	atcvr1 34703	. . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )
5		atcvr0eq.z	. . . . . . . 8 |- .0. = ( 0. ` K )
6	5, 2, 3	atcvr0 34575	. . . . . . 7 |- ( ( K e. HL /\ P e. A ) -> .0. C P )
7	6	3adant3 1081	. . . . . 6 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> .0. C P )
8	7	biantrurd 529	. . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P C ( P .\/ Q ) <-> ( .0. C P /\ P C ( P .\/ Q ) ) ) )
9	4, 8	bitrd 268	. . . 4 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( .0. C P /\ P C ( P .\/ Q ) ) ) )
10		simp1 1061	. . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> K e. HL )
11		hlop 34649	. . . . . . 7 |- ( K e. HL -> K e. OP )
12	11	3ad2ant1 1082	. . . . . 6 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> K e. OP )
13		eqid 2622	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
14	13, 5	op0cl 34471	. . . . . 6 |- ( K e. OP -> .0. e. ( Base ` K ) )
15	12, 14	syl 17	. . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> .0. e. ( Base ` K ) )
16	13, 3	atbase 34576	. . . . . 6 |- ( P e. A -> P e. ( Base ` K ) )
17	16	3ad2ant2 1083	. . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> P e. ( Base ` K ) )
18	13, 1, 3	hlatjcl 34653	. . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
19	13, 2	cvrntr 34711	. . . . 5 |- ( ( K e. HL /\ ( .0. e. ( Base ` K ) /\ P e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( .0. C P /\ P C ( P .\/ Q ) ) -> -. .0. C ( P .\/ Q ) ) )
20	10, 15, 17, 18, 19	syl13anc 1328	. . . 4 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( ( .0. C P /\ P C ( P .\/ Q ) ) -> -. .0. C ( P .\/ Q ) ) )
21	9, 20	sylbid 230	. . 3 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q -> -. .0. C ( P .\/ Q ) ) )
22	21	necon4ad 2813	. 2 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( .0. C ( P .\/ Q ) -> P = Q ) )
23	1, 3	hlatjidm 34655	. . . . 5 |- ( ( K e. HL /\ P e. A ) -> ( P .\/ P ) = P )
24	23	3adant3 1081	. . . 4 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ P ) = P )
25	7, 24	breqtrrd 4681	. . 3 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> .0. C ( P .\/ P ) )
26		oveq2 6658	. . . 4 |- ( P = Q -> ( P .\/ P ) = ( P .\/ Q ) )
27	26	breq2d 4665	. . 3 |- ( P = Q -> ( .0. C ( P .\/ P ) <-> .0. C ( P .\/ Q ) ) )
28	25, 27	syl5ibcom 235	. 2 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P = Q -> .0. C ( P .\/ Q ) ) )
29	22, 28	impbid 202	1 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( .0. C ( P .\/ Q ) <-> P = Q ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   joincjn 16944   0.cp0 17037   OPcops 34459   <o ccvr 34549   Atomscatm 34550   HLchlt 34637

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638

This theorem is referenced by:  atcvrj0  34714