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Theorem cdlemg6 34236
Description: TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013.)
Hypotheses
Ref Expression
cdlemg6.l |- .<_ = ( le ` K )
cdlemg6.a |- A = ( Atoms ` K )
cdlemg6.h |- H = ( LHyp ` K )
cdlemg6.t |- T = ( ( LTrn ` K ) ` W )
Assertion
Ref Expression
cdlemg6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` Q ) ) = Q )
Proof of Theorem cdlemg6
StepHypRef Expression
1 simpl1 1017 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> ( K e. HL /\ W e. H ) )
2 simpl2l 1067 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> ( P e. A /\ -. P .<_ W ) )
3 simpl2r 1068 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
4 simpl31 1095 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> F e. T )
5 simpl32 1096 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> G e. T )
6 simpr 467 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) )
7 simpl33 1097 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> ( F ` ( G ` P ) ) = P )
8 cdlemg6.l . . . 4 |- .<_ = ( le ` K )
9 cdlemg6.a . . . 4 |- A = ( Atoms ` K )
10 cdlemg6.h . . . 4 |- H = ( LHyp ` K )
11 cdlemg6.t . . . 4 |- T = ( ( LTrn ` K ) ` W )
12 eqid 2462 . . . 4 |- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
13 eqid 2462 . . . 4 |- ( join ` K ) = ( join ` K )
14 eqid 2462 . . . 4 |- ( ( ( trL ` K ) ` W ) ` G ) = ( ( ( trL ` K ) ` W ) ` G )
158, 9, 10, 11, 12, 13, 14cdlemg6e 34235 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` Q ) ) = Q )
161, 2, 3, 4, 5, 6, 7, 15syl133anc 1299 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> ( F ` ( G ` Q ) ) = Q )
17 simpl1 1017 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ -. Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> ( K e. HL /\ W e. H ) )
18 simpl2l 1067 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ -. Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> ( P e. A /\ -. P .<_ W ) )
19 simpl2r 1068 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ -. Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
20 simpl31 1095 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ -. Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> F e. T )
21 simpl32 1096 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ -. Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> G e. T )
22 simpr 467 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ -. Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> -. Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) )
23 simpl33 1097 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ -. Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> ( F ` ( G ` P ) ) = P )
248, 9, 10, 11, 12, 13, 14cdlemg4 34230 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` Q ) ) = Q )
2517, 18, 19, 20, 21, 22, 23, 24syl133anc 1299 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ -. Q .<_ ( P ( join ` K ) ( ( ( trL ` K ) ` W ) ` G ) ) ) -> ( F ` ( G ` Q ) ) = Q )
2616, 25pm2.61dan 805 1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` Q ) ) = Q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   class class class wbr 4418   ` cfv 5605  (class class class)co 6320   lecple 15252   joincjn 16244   Atomscatm 32875   HLchlt 32962   LHypclh 33595   LTrncltrn 33712   trLctrl 33770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-riotaBAD 32571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-iin 4295  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-1st 6825  df-2nd 6826  df-undef 7051  df-map 7505  df-preset 16228  df-poset 16246  df-plt 16259  df-lub 16275  df-glb 16276  df-join 16277  df-meet 16278  df-p0 16340  df-p1 16341  df-lat 16347  df-clat 16409  df-oposet 32788  df-ol 32790  df-oml 32791  df-covers 32878  df-ats 32879  df-atl 32910  df-cvlat 32934  df-hlat 32963  df-llines 33109  df-lplanes 33110  df-lvols 33111  df-lines 33112  df-psubsp 33114  df-pmap 33115  df-padd 33407  df-lhyp 33599  df-laut 33600  df-ldil 33715  df-ltrn 33716  df-trl 33771
This theorem is referenced by:  cdlemg7aN  34238  cdlemg8a  34240  cdlemg8c  34242  cdlemg11a  34250
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