Theorem cdlemg41 34670

Description: Convert cdlemg40 34669 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 34669	. 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 988	. . . . 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 990	. . . . 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 1055	. . . . 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 34097	. . . . 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 1219	. . . 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 6208	. . 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 6207	. 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 1057	. . . . 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 34097	. . . . 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 1219	. . . 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 6208	. . 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 6207	. 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 2502	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 965   = wceq 1370   e. wcel 1758   class class class wbr 4392   o. ccom 4944   ` cfv 5518  (class class class)co 6192   lecple 14349   joincjn 15218   meetcmee 15219   Atomscatm 33216   HLchlt 33303   LHypclh 33936   LTrncltrn 34053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-riotaBAD 32912

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-undef 6894  df-map 7318  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-p1 15314  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452  df-lines 33453  df-psubsp 33455  df-pmap 33456  df-padd 33748  df-lhyp 33940  df-laut 33941  df-ldil 34056  df-ltrn 34057  df-trl 34111

This theorem is referenced by:  ltrnco  34671