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Theorem cdlemg28a 35845
Description: Part of proof of Lemma G of [Crawley] p. 116. First equality of the equation of line 14 on p. 117. (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 )
Assertion
Ref Expression
cdlemg28a |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( z .\/ ( F ` ( G ` z ) ) ) ./\ W ) )
Proof of Theorem cdlemg28a
StepHypRef Expression
1 simp11 1026 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( K e. HL /\ W e. H ) )
2 simp12 1027 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( P e. A /\ -. P .<_ W ) )
3 simp21 1029 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( z e. A /\ -. z .<_ W ) )
4 simp22 1030 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> F e. T )
5 simp23 1031 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> G e. T )
6 simp1 996 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) )
7 simp21l 1113 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> z e. A )
8 simp31l 1119 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> v =/= ( R ` F ) )
9 simp32 1033 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> z .<_ ( P .\/ v ) )
10 simp33l 1123 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( F ` P ) =/= P )
11 cdlemg12.l . . . 4 |- .<_ = ( le ` K )
12 cdlemg12.j . . . 4 |- .\/ = ( join ` K )
13 cdlemg12.m . . . 4 |- ./\ = ( meet ` K )
14 cdlemg12.a . . . 4 |- A = ( Atoms ` K )
15 cdlemg12.h . . . 4 |- H = ( LHyp ` K )
16 cdlemg12.t . . . 4 |- T = ( ( LTrn ` K ) ` W )
17 cdlemg12b.r . . . 4 |- R = ( ( trL ` K ) ` W )
1811, 12, 13, 14, 15, 16, 17cdlemg27a 35844 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. ( R ` F ) .<_ ( P .\/ z ) )
196, 7, 4, 8, 9, 10, 18syl123anc 1245 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> -. ( R ` F ) .<_ ( P .\/ z ) )
20 simp31r 1120 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> v =/= ( R ` G ) )
21 simp33r 1124 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( G ` P ) =/= P )
2211, 12, 13, 14, 15, 16, 17cdlemg27a 35844 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ G e. T ) /\ ( v =/= ( R ` G ) /\ z .<_ ( P .\/ v ) /\ ( G ` P ) =/= P ) ) -> -. ( R ` G ) .<_ ( P .\/ z ) )
236, 7, 5, 20, 9, 21, 22syl123anc 1245 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> -. ( R ` G ) .<_ ( P .\/ z ) )
2411, 12, 13, 14, 15, 16, 17cdlemg25zz 35842 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ z ) /\ -. ( R ` G ) .<_ ( P .\/ z ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( z .\/ ( F ` ( G ` z ) ) ) ./\ W ) )
251, 2, 3, 4, 5, 19, 23, 24syl133anc 1251 1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( z .\/ ( F ` ( G ` z ) ) ) ./\ W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   lecple 14579   joincjn 15448   meetcmee 15449   Atomscatm 34416   HLchlt 34503   LHypclh 35136   LTrncltrn 35253   trLctrl 35310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-riotaBAD 34112
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-undef 7014  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34329  df-ol 34331  df-oml 34332  df-covers 34419  df-ats 34420  df-atl 34451  df-cvlat 34475  df-hlat 34504  df-llines 34650  df-lplanes 34651  df-lvols 34652  df-lines 34653  df-psubsp 34655  df-pmap 34656  df-padd 34948  df-lhyp 35140  df-laut 35141  df-ldil 35256  df-ltrn 35257  df-trl 35311
This theorem is referenced by:  cdlemg28  35856
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