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Theorem cdlemg28a 34676
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 1018 . 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 1019 . 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 1021 . 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 1022 . 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 1023 . 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 988 . . 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 1105 . . 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 1111 . . 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 1025 . . 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 1115 . . 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 34675 . . 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 1236 . 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 1112 . . 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 1116 . . 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 34675 . . 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 1236 . 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 34673 . 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 1242 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 965   = wceq 1370   e. wcel 1758   =/= wne 2648   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   lecple 14365   joincjn 15234   meetcmee 15235   Atomscatm 33247   HLchlt 33334   LHypclh 33967   LTrncltrn 34084   trLctrl 34141
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 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-riotaBAD 32943
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 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-undef 6903  df-map 7327  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481  df-lplanes 33482  df-lvols 33483  df-lines 33484  df-psubsp 33486  df-pmap 33487  df-padd 33779  df-lhyp 33971  df-laut 33972  df-ldil 34087  df-ltrn 34088  df-trl 34142
This theorem is referenced by:  cdlemg28  34687
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