Theorem cdlemg28b 36845

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 36846.) (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 1024	. 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 1026	. 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 1028	. 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 1115	. 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 1116	. 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 994	. . 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 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 ) ) ) -> z e. A )
8		simp21 1027	. . 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 1141	. . . 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 530	. . 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 1119	. . 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 1143	. . 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 1121	. . 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 36838	. . 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 1249	. 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 1142	. . . 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 530	. . 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 1120	. . 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 1122	. . 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 36838	. . 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 1249	. 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 36833	. 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 1249	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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14794   joincjn 15775   meetcmee 15776   Atomscatm 35404   HLchlt 35491   LHypclh 36124   LTrncltrn 36241   trLctrl 36299

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  ax-riotaBAD 35100

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  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-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  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-iin 4318  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-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  df-map 7414  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-p1 15872  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-llines 35638  df-lplanes 35639  df-lvols 35640  df-lines 35641  df-psubsp 35643  df-pmap 35644  df-padd 35936  df-lhyp 36128  df-laut 36129  df-ldil 36244  df-ltrn 36245  df-trl 36300

This theorem is referenced by:  cdlemg28  36846