Theorem cdlemg28b 34670

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 34671.) (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 1018	. 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 1020	. 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 1022	. 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 1109	. 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 1110	. 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 988	. . 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 1107	. . 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 1021	. . 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 1135	. . . 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 532	. . 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 1113	. . 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 1137	. . 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 1115	. . 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 34663	. . 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 1242	. 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 1136	. . . 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 532	. . 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 1114	. . 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 1116	. . 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 34663	. . 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 1242	. 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 34658	. 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 1242	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 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2647   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   lecple 14363   joincjn 15232   meetcmee 15233   Atomscatm 33231   HLchlt 33318   LHypclh 33951   LTrncltrn 34068   trLctrl 34125

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 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-riotaBAD 32927

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-undef 6901  df-map 7325  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-p1 15328  df-lat 15334  df-clat 15396  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319  df-llines 33465  df-lplanes 33466  df-lvols 33467  df-lines 33468  df-psubsp 33470  df-pmap 33471  df-padd 33763  df-lhyp 33955  df-laut 33956  df-ldil 34071  df-ltrn 34072  df-trl 34126

This theorem is referenced by:  cdlemg28  34671