Theorem cdleme02N 36344

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 36343	. 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 1018	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> K e. HL )
9		hlcvl 35481	. . . . 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 1061	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P e. A )
12		simp2rl 1063	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
13		simp1 994	. . . . 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 1020	. . . . 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 996	. . . . 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 36166	. . . . 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 1230	. . . 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 35465	. . . 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 1239	. . 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 702	. 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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14791   joincjn 15772   meetcmee 15773   Atomscatm 35385   CvLatclc 35387   HLchlt 35472   LHypclh 36105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-lhyp 36109

This theorem is referenced by:  cdleme0moN  36347