Theorem cdlemg41 36146

Description: Convert cdlemg40 36145 to function composition. TODO: Fix comment. (Contributed by NM, 31-May-2013.)

Hypotheses

Ref	Expression
cdlemg35.l	|- .<_ = ( le ` K )
cdlemg35.j	|- .\/ = ( join ` K )
cdlemg35.m	|- ./\ = ( meet ` K )
cdlemg35.a	|- A = ( Atoms ` K )
cdlemg35.h	|- H = ( LHyp ` K )
cdlemg35.t	|- T = ( ( LTrn ` K ) ` W )

Assertion

Ref	Expression
cdlemg41	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( ( P .\/ ( ( F o. G ) ` P ) ) ./\ W ) = ( ( Q .\/ ( ( F o. G ) ` Q ) ) ./\ W ) )

Proof of Theorem cdlemg41

Step	Hyp	Ref	Expression
1		cdlemg35.l	. . 3 |- .<_ = ( le ` K )
2		cdlemg35.j	. . 3 |- .\/ = ( join ` K )
3		cdlemg35.m	. . 3 |- ./\ = ( meet ` K )
4		cdlemg35.a	. . 3 |- A = ( Atoms ` K )
5		cdlemg35.h	. . 3 |- H = ( LHyp ` K )
6		cdlemg35.t	. . 3 |- T = ( ( LTrn ` K ) ` W )
7	1, 2, 3, 4, 5, 6	cdlemg40 36145	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
8		simp1 995	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( K e. HL /\ W e. H ) )
9		simp3 997	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( F e. T /\ G e. T ) )
10		simp2ll 1062	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> P e. A )
11	1, 4, 5, 6	ltrncoval 35571	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> ( ( F o. G ) ` P ) = ( F ` ( G ` P ) ) )
12	8, 9, 10, 11	syl3anc 1227	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( ( F o. G ) ` P ) = ( F ` ( G ` P ) ) )
13	12	oveq2d 6293	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( P .\/ ( ( F o. G ) ` P ) ) = ( P .\/ ( F ` ( G ` P ) ) ) )
14	13	oveq1d 6292	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( ( P .\/ ( ( F o. G ) ` P ) ) ./\ W ) = ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) )
15		simp2rl 1064	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> Q e. A )
16	1, 4, 5, 6	ltrncoval 35571	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ Q e. A ) -> ( ( F o. G ) ` Q ) = ( F ` ( G ` Q ) ) )
17	8, 9, 15, 16	syl3anc 1227	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( ( F o. G ) ` Q ) = ( F ` ( G ` Q ) ) )
18	17	oveq2d 6293	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( Q .\/ ( ( F o. G ) ` Q ) ) = ( Q .\/ ( F ` ( G ` Q ) ) ) )
19	18	oveq1d 6292	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( ( Q .\/ ( ( F o. G ) ` Q ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
20	7, 14, 19	3eqtr4d 2492	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( ( P .\/ ( ( F o. G ) ` P ) ) ./\ W ) = ( ( Q .\/ ( ( F o. G ) ` Q ) ) ./\ W ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 972   = wceq 1381   e. wcel 1802   class class class wbr 4433   o. ccom 4989   ` cfv 5574  (class class class)co 6277   lecple 14576   joincjn 15442   meetcmee 15443   Atomscatm 34690   HLchlt 34777   LHypclh 35410   LTrncltrn 35527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-riotaBAD 34386

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-undef 7000  df-map 7420  df-preset 15426  df-poset 15444  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-p1 15539  df-lat 15545  df-clat 15607  df-oposet 34603  df-ol 34605  df-oml 34606  df-covers 34693  df-ats 34694  df-atl 34725  df-cvlat 34749  df-hlat 34778  df-llines 34924  df-lplanes 34925  df-lvols 34926  df-lines 34927  df-psubsp 34929  df-pmap 34930  df-padd 35222  df-lhyp 35414  df-laut 35415  df-ldil 35530  df-ltrn 35531  df-trl 35586

This theorem is referenced by:  ltrnco  36147