Theorem cdlemg28b 33978

Description: Part of proof of Lemma G of [Crawley] p. 116. Second equality of the equation of line 14 on p. 117. Note that -. z .<_ W is redundant here (but simplifies cdlemg28 33979.) (Contributed by NM, 29-May-2013.)

Hypotheses

Ref	Expression
cdlemg12.l	|- .<_ = ( le ` K )
cdlemg12.j	|- .\/ = ( join ` K )
cdlemg12.m	|- ./\ = ( meet ` K )
cdlemg12.a	|- A = ( Atoms ` K )
cdlemg12.h	|- H = ( LHyp ` K )
cdlemg12.t	|- T = ( ( LTrn ` K ) ` W )
cdlemg12b.r	|- R = ( ( trL ` K ) ` W )
cdlemg31.n	|- N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
cdlemg33.o	|- O = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` G ) ) )

Assertion

Ref	Expression
cdlemg28b	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) = ( ( z .\/ ( F ` ( G ` z ) ) ) ./\ W ) )

Distinct variable groups:   z, A   z, F   z, H   z, .\/   z, K   z, .<_   z, N   z, P   z, Q   z, R   z, T   z, W   z, v   z, G   z, O

Allowed substitution hints:   A( v)   P( v)   Q( v)   R( v)   T( v)   F( v)   G( v)   H( v)   .\/ ( v)   K( v)   .<_ ( v)   ./\ ( z, v)   N( v)   O( v)   W( v)

Proof of Theorem cdlemg28b

Step	Hyp	Ref	Expression
1		simp11 1035	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( K e. HL /\ W e. H ) )
2		simp13 1037	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
3		simp22 1039	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( z e. A /\ -. z .<_ W ) )
4		simp23l 1126	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> F e. T )
5		simp23r 1127	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> G e. T )
6		simp1 1005	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
7		simp22l 1124	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> z e. A )
8		simp21 1038	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( v e. A /\ v .<_ W ) )
9		simp311 1152	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> z =/= N )
10	4, 9	jca 534	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( F e. T /\ z =/= N ) )
11		simp32l 1130	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> v =/= ( R ` F ) )
12		simp313 1154	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> z .<_ ( P .\/ v ) )
13		simp33l 1132	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( F ` P ) =/= P )
14		cdlemg12.l	. . . 4 |- .<_ = ( le ` K )
15		cdlemg12.j	. . . 4 |- .\/ = ( join ` K )
16		cdlemg12.m	. . . 4 |- ./\ = ( meet ` K )
17		cdlemg12.a	. . . 4 |- A = ( Atoms ` K )
18		cdlemg12.h	. . . 4 |- H = ( LHyp ` K )
19		cdlemg12.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
20		cdlemg12b.r	. . . 4 |- R = ( ( trL ` K ) ` W )
21		cdlemg31.n	. . . 4 |- N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
22	14, 15, 16, 17, 18, 19, 20, 21	cdlemg27b 33971	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. ( R ` F ) .<_ ( Q .\/ z ) )
23	6, 7, 8, 10, 11, 12, 13, 22	syl133anc 1287	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> -. ( R ` F ) .<_ ( Q .\/ z ) )
24		simp312 1153	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> z =/= O )
25	5, 24	jca 534	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( G e. T /\ z =/= O ) )
26		simp32r 1131	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> v =/= ( R ` G ) )
27		simp33r 1133	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( G ` P ) =/= P )
28		cdlemg33.o	. . . 4 |- O = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` G ) ) )
29	14, 15, 16, 17, 18, 19, 20, 28	cdlemg27b 33971	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( G e. T /\ z =/= O ) ) /\ ( v =/= ( R ` G ) /\ z .<_ ( P .\/ v ) /\ ( G ` P ) =/= P ) ) -> -. ( R ` G ) .<_ ( Q .\/ z ) )
30	6, 7, 8, 25, 26, 12, 27, 29	syl133anc 1287	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> -. ( R ` G ) .<_ ( Q .\/ z ) )
31	14, 15, 16, 17, 18, 19, 20	cdlemg26zz 33966	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( Q .\/ z ) /\ -. ( R ` G ) .<_ ( Q .\/ z ) ) ) -> ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) = ( ( z .\/ ( F ` ( G ` z ) ) ) ./\ W ) )
32	1, 2, 3, 4, 5, 23, 30, 31	syl133anc 1287	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) = ( ( z .\/ ( F ` ( G ` z ) ) ) ./\ W ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   lecple 15159   joincjn 16140   meetcmee 16141   Atomscatm 32537   HLchlt 32624   LHypclh 33257   LTrncltrn 33374   trLctrl 33432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-riotaBAD 32233

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-undef 7028  df-map 7482  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32450  df-ol 32452  df-oml 32453  df-covers 32540  df-ats 32541  df-atl 32572  df-cvlat 32596  df-hlat 32625  df-llines 32771  df-lplanes 32772  df-lvols 32773  df-lines 32774  df-psubsp 32776  df-pmap 32777  df-padd 33069  df-lhyp 33261  df-laut 33262  df-ldil 33377  df-ltrn 33378  df-trl 33433

This theorem is referenced by:  cdlemg28  33979