Theorem cdlemg8a 35915

Description: TODO: FIX COMMENT. (Contributed by NM, 29-Apr-2013.)

Hypotheses

Ref	Expression
cdlemg8.l	|- .<_ = ( le ` K )
cdlemg8.j	|- .\/ = ( join ` K )
cdlemg8.m	|- ./\ = ( meet ` K )
cdlemg8.a	|- A = ( Atoms ` K )
cdlemg8.h	|- H = ( LHyp ` K )
cdlemg8.t	|- T = ( ( LTrn ` K ) ` W )

Assertion

Ref	Expression
cdlemg8a	|- ( ( ( 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 ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )

Proof of Theorem cdlemg8a

Step	Hyp	Ref	Expression
1		simp1 1061	. . 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 ) ) -> ( K e. HL /\ W e. H ) )
2		simp2r 1088	. . 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 e. A /\ -. Q .<_ W ) )
3		cdlemg8.l	. . . 4 |- .<_ = ( le ` K )
4		cdlemg8.m	. . . 4 |- ./\ = ( meet ` K )
5		eqid 2622	. . . 4 |- ( 0. ` K ) = ( 0. ` K )
6		cdlemg8.a	. . . 4 |- A = ( Atoms ` K )
7		cdlemg8.h	. . . 4 |- H = ( LHyp ` K )
8	3, 4, 5, 6, 7	lhpmat 35316	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Q ./\ W ) = ( 0. ` K ) )
9	1, 2, 8	syl2anc 693	. 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 ./\ W ) = ( 0. ` K ) )
10		cdlemg8.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
11	3, 6, 7, 10	cdlemg6 35911	. . . . 5 |- ( ( ( 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 )
12	11	oveq2d 6666	. . . 4 |- ( ( ( 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 .\/ ( F ` ( G ` Q ) ) ) = ( Q .\/ Q ) )
13		simp1l 1085	. . . . 5 |- ( ( ( 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 ) ) -> K e. HL )
14		simp2rl 1130	. . . . 5 |- ( ( ( 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 e. A )
15		cdlemg8.j	. . . . . 6 |- .\/ = ( join ` K )
16	15, 6	hlatjidm 34655	. . . . 5 |- ( ( K e. HL /\ Q e. A ) -> ( Q .\/ Q ) = Q )
17	13, 14, 16	syl2anc 693	. . . 4 |- ( ( ( 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 .\/ Q ) = Q )
18	12, 17	eqtrd 2656	. . 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 .\/ ( F ` ( G ` Q ) ) ) = Q )
19	18	oveq1d 6665	. 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 .\/ ( F ` ( G ` Q ) ) ) ./\ W ) = ( Q ./\ W ) )
20		simp33 1099	. . . . . 6 |- ( ( ( 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 ` P ) ) = P )
21	20	oveq2d 6666	. . . . 5 |- ( ( ( 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 ) ) -> ( P .\/ ( F ` ( G ` P ) ) ) = ( P .\/ P ) )
22		simp2ll 1128	. . . . . 6 |- ( ( ( 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 ) ) -> P e. A )
23	15, 6	hlatjidm 34655	. . . . . 6 |- ( ( K e. HL /\ P e. A ) -> ( P .\/ P ) = P )
24	13, 22, 23	syl2anc 693	. . . . 5 |- ( ( ( 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 ) ) -> ( P .\/ P ) = P )
25	21, 24	eqtrd 2656	. . . 4 |- ( ( ( 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 ) ) -> ( P .\/ ( F ` ( G ` P ) ) ) = P )
26	25	oveq1d 6665	. . 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 ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( P ./\ W ) )
27		simp2l 1087	. . . 4 |- ( ( ( 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 ) ) -> ( P e. A /\ -. P .<_ W ) )
28	3, 4, 5, 6, 7	lhpmat 35316	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = ( 0. ` K ) )
29	1, 27, 28	syl2anc 693	. . 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 ) ) -> ( P ./\ W ) = ( 0. ` K ) )
30	26, 29	eqtrd 2656	. 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 ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( 0. ` K ) )
31	9, 19, 30	3eqtr4rd 2667	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 ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   meetcmee 16945   0.cp0 17037   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by:  cdlemg8  35919