Theorem cdleme02N 34185

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme0.l	|- .<_ = ( le ` K )
cdleme0.j	|- .\/ = ( join ` K )
cdleme0.m	|- ./\ = ( meet ` K )
cdleme0.a	|- A = ( Atoms ` K )
cdleme0.h	|- H = ( LHyp ` K )
cdleme0.u	|- U = ( ( P .\/ Q ) ./\ W )

Assertion

Ref	Expression
cdleme02N	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( ( P .\/ U ) = ( Q .\/ U ) /\ U .<_ W ) )

Proof of Theorem cdleme02N

Step	Hyp	Ref	Expression
1		cdleme0.l	. . 3 |- .<_ = ( le ` K )
2		cdleme0.j	. . 3 |- .\/ = ( join ` K )
3		cdleme0.m	. . 3 |- ./\ = ( meet ` K )
4		cdleme0.a	. . 3 |- A = ( Atoms ` K )
5		cdleme0.h	. . 3 |- H = ( LHyp ` K )
6		cdleme0.u	. . 3 |- U = ( ( P .\/ Q ) ./\ W )
7	1, 2, 3, 4, 5, 6	cdleme01N 34184	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( ( U =/= P /\ U =/= Q /\ U .<_ ( P .\/ Q ) ) /\ U .<_ W ) )
8		simp1l 1012	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> K e. HL )
9		hlcvl 33323	. . . . 5 |- ( K e. HL -> K e. CvLat )
10	8, 9	syl 16	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> K e. CvLat )
11		simp2ll 1055	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P e. A )
12		simp2rl 1057	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
13		simp1 988	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) )
14		simp2l 1014	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( P e. A /\ -. P .<_ W ) )
15		simp3 990	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P =/= Q )
16	1, 2, 3, 4, 5, 6	lhpat2 34008	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
17	13, 14, 12, 15, 16	syl112anc 1223	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> U e. A )
18	4, 1, 2	cvlsupr2 33307	. . . 4 |- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ U e. A ) /\ P =/= Q ) -> ( ( P .\/ U ) = ( Q .\/ U ) <-> ( U =/= P /\ U =/= Q /\ U .<_ ( P .\/ Q ) ) ) )
19	10, 11, 12, 17, 15, 18	syl131anc 1232	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( ( P .\/ U ) = ( Q .\/ U ) <-> ( U =/= P /\ U =/= Q /\ U .<_ ( P .\/ Q ) ) ) )
20	19	anbi1d 704	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( ( ( P .\/ U ) = ( Q .\/ U ) /\ U .<_ W ) <-> ( ( U =/= P /\ U =/= Q /\ U .<_ ( P .\/ Q ) ) /\ U .<_ W ) ) )
21	7, 20	mpbird 232	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( ( P .\/ U ) = ( Q .\/ U ) /\ U .<_ W ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   lecple 14359   joincjn 15228   meetcmee 15229   Atomscatm 33227   CvLatclc 33229   HLchlt 33314   LHypclh 33947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-lhyp 33951

This theorem is referenced by:  cdleme0moN  34188