Theorem cdlemg28b 31185

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 31186.) (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 987	. 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 989	. 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 991	. 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 1078	. 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 1079	. 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 957	. . 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 1076	. . 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 990	. . 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 1104	. . . 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 519	. . 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 1082	. . 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 1106	. . 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 1084	. . 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 31178	. . 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 1207	. 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 1105	. . . 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 519	. . 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 1083	. . 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 1085	. . 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 31178	. . 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 1207	. 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 31173	. 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 1207	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 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640

This theorem is referenced by:  cdlemg28  31186

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641