Theorem cdlemg17g 36845

Description: TODO: fix comment. (Contributed by NM, 9-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
cdlemg17g	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` ( F ` P ) ) .<_ ( ( F ` P ) .\/ ( F ` Q ) ) )

Distinct variable groups:   A, r   G, r   .\/ , r   .<_ , r   P, r   Q, r   W, r

Allowed substitution hints:   R( r)   T( r)   F( r)   H( r)   K( r)   ./\ ( r)

Proof of Theorem cdlemg17g

Step	Hyp	Ref	Expression
1		simp11l 1105	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> K e. HL )
2		simp11 1024	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( K e. HL /\ W e. H ) )
3		simp21 1027	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> F e. T )
4		simp12l 1107	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P e. A )
5		cdlemg12.l	. . . . 5 |- .<_ = ( le ` K )
6		cdlemg12.a	. . . . 5 |- A = ( Atoms ` K )
7		cdlemg12.h	. . . . 5 |- H = ( LHyp ` K )
8		cdlemg12.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
9	5, 6, 7, 8	ltrnat 36316	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A )
10	2, 3, 4, 9	syl3anc 1226	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F ` P ) e. A )
11		simp22 1028	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> G e. T )
12	5, 6, 7, 8	ltrncoat 36320	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ F e. T ) /\ P e. A ) -> ( G ` ( F ` P ) ) e. A )
13	2, 11, 3, 4, 12	syl121anc 1231	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` ( F ` P ) ) e. A )
14		cdlemg12.j	. . . 4 |- .\/ = ( join ` K )
15	5, 14, 6	hlatlej2 35552	. . 3 |- ( ( K e. HL /\ ( F ` P ) e. A /\ ( G ` ( F ` P ) ) e. A ) -> ( G ` ( F ` P ) ) .<_ ( ( F ` P ) .\/ ( G ` ( F ` P ) ) ) )
16	1, 10, 13, 15	syl3anc 1226	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` ( F ` P ) ) .<_ ( ( F ` P ) .\/ ( G ` ( F ` P ) ) ) )
17		cdlemg12.m	. . 3 |- ./\ = ( meet ` K )
18		cdlemg12b.r	. . 3 |- R = ( ( trL ` K ) ` W )
19	5, 14, 17, 6, 7, 8, 18	cdlemg17f 36844	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` P ) .\/ ( G ` ( F ` P ) ) ) )
20	16, 19	breqtrrd 4410	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` ( F ` P ) ) .<_ ( ( F ` P ) .\/ ( F ` Q ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   /\ w3a 971   = wceq 1399   e. wcel 1836   =/= wne 2591   E.wrex 2747   class class class wbr 4384   ` cfv 5513  (class class class)co 6218   lecple 14732   joincjn 15713   meetcmee 15714   Atomscatm 35440   HLchlt 35527   LHypclh 36160   LTrncltrn 36277   trLctrl 36335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-riotaBAD 35136

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-iin 4263  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-1st 6721  df-2nd 6722  df-undef 6942  df-map 7362  df-preset 15697  df-poset 15715  df-plt 15728  df-lub 15744  df-glb 15745  df-join 15746  df-meet 15747  df-p0 15809  df-p1 15810  df-lat 15816  df-clat 15878  df-oposet 35353  df-ol 35355  df-oml 35356  df-covers 35443  df-ats 35444  df-atl 35475  df-cvlat 35499  df-hlat 35528  df-llines 35674  df-lplanes 35675  df-lvols 35676  df-lines 35677  df-psubsp 35679  df-pmap 35680  df-padd 35972  df-lhyp 36164  df-laut 36165  df-ldil 36280  df-ltrn 36281  df-trl 36336

This theorem is referenced by:  cdlemg17i  36847