Theorem cdlemg7fvN 35912

Description: Value of a translation composition in terms of an associated atom. (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemg7fv.b	|- B = ( Base ` K )
cdlemg7fv.l	|- .<_ = ( le ` K )
cdlemg7fv.j	|- .\/ = ( join ` K )
cdlemg7fv.m	|- ./\ = ( meet ` K )
cdlemg7fv.a	|- A = ( Atoms ` K )
cdlemg7fv.h	|- H = ( LHyp ` K )
cdlemg7fv.t	|- T = ( ( LTrn ` K ) ` W )

Assertion

Ref	Expression
cdlemg7fvN	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( F ` ( G ` X ) ) = ( ( F ` ( G ` P ) ) .\/ ( X ./\ W ) ) )

Proof of Theorem cdlemg7fvN

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( K e. HL /\ W e. H ) )
2		simp32 1098	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> G e. T )
3		simp2l 1087	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( P e. A /\ -. P .<_ W ) )
4		cdlemg7fv.l	. . . . 5 |- .<_ = ( le ` K )
5		cdlemg7fv.a	. . . . 5 |- A = ( Atoms ` K )
6		cdlemg7fv.h	. . . . 5 |- H = ( LHyp ` K )
7		cdlemg7fv.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
8	4, 5, 6, 7	ltrnel 35425	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
9	1, 2, 3, 8	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
10		simp2r 1088	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( X e. B /\ -. X .<_ W ) )
11		cdlemg7fv.b	. . . . 5 |- B = ( Base ` K )
12	4, 5, 6, 7, 11	cdlemg7fvbwN 35895	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ G e. T ) -> ( ( G ` X ) e. B /\ -. ( G ` X ) .<_ W ) )
13	1, 10, 2, 12	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( G ` X ) e. B /\ -. ( G ` X ) .<_ W ) )
14		simp31 1097	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> F e. T )
15		simp33 1099	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( P .\/ ( X ./\ W ) ) = X )
16		cdlemg7fv.j	. . . . . . . . 9 |- .\/ = ( join ` K )
17		cdlemg7fv.m	. . . . . . . . 9 |- ./\ = ( meet ` K )
18	6, 7, 4, 16, 5, 17, 11	cdlemg2fv 35887	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( G ` X ) = ( ( G ` P ) .\/ ( X ./\ W ) ) )
19	1, 3, 10, 2, 15, 18	syl122anc 1335	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( G ` X ) = ( ( G ` P ) .\/ ( X ./\ W ) ) )
20	19	oveq1d 6665	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( G ` X ) ./\ W ) = ( ( ( G ` P ) .\/ ( X ./\ W ) ) ./\ W ) )
21		simp2rl 1130	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> X e. B )
22	11, 4, 16, 17, 5, 6	lhpelim 35323	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) /\ X e. B ) -> ( ( ( G ` P ) .\/ ( X ./\ W ) ) ./\ W ) = ( X ./\ W ) )
23	1, 9, 21, 22	syl3anc 1326	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( ( G ` P ) .\/ ( X ./\ W ) ) ./\ W ) = ( X ./\ W ) )
24	20, 23	eqtrd 2656	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( G ` X ) ./\ W ) = ( X ./\ W ) )
25	24	oveq2d 6666	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( G ` P ) .\/ ( ( G ` X ) ./\ W ) ) = ( ( G ` P ) .\/ ( X ./\ W ) ) )
26	25, 19	eqtr4d 2659	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( G ` P ) .\/ ( ( G ` X ) ./\ W ) ) = ( G ` X ) )
27	6, 7, 4, 16, 5, 17, 11	cdlemg2fv 35887	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) /\ ( ( G ` X ) e. B /\ -. ( G ` X ) .<_ W ) ) /\ ( F e. T /\ ( ( G ` P ) .\/ ( ( G ` X ) ./\ W ) ) = ( G ` X ) ) ) -> ( F ` ( G ` X ) ) = ( ( F ` ( G ` P ) ) .\/ ( ( G ` X ) ./\ W ) ) )
28	1, 9, 13, 14, 26, 27	syl122anc 1335	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( F ` ( G ` X ) ) = ( ( F ` ( G ` P ) ) .\/ ( ( G ` X ) ./\ W ) ) )
29	24	oveq2d 6666	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( F ` ( G ` P ) ) .\/ ( ( G ` X ) ./\ W ) ) = ( ( F ` ( G ` P ) ) .\/ ( X ./\ W ) ) )
30	28, 29	eqtrd 2656	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( F ` ( G ` X ) ) = ( ( F ` ( G ` P ) ) .\/ ( X ./\ 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   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   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:  cdlemg7aN  35913