Theorem cdleml1N 36264

Description: Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleml1.b	|- B = ( Base ` K )
cdleml1.h	|- H = ( LHyp ` K )
cdleml1.t	|- T = ( ( LTrn ` K ) ` W )
cdleml1.r	|- R = ( ( trL ` K ) ` W )
cdleml1.e	|- E = ( ( TEndo ` K ) ` W )

Assertion

Ref	Expression
cdleml1N	|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( R ` ( U ` f ) ) = ( R ` ( V ` f ) ) )

Proof of Theorem cdleml1N

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( K e. HL /\ W e. H ) )
2		simp21 1094	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> U e. E )
3		simp23 1096	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> f e. T )
4		eqid 2622	. . . . 5 |- ( le ` K ) = ( le ` K )
5		cdleml1.h	. . . . 5 |- H = ( LHyp ` K )
6		cdleml1.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
7		cdleml1.r	. . . . 5 |- R = ( ( trL ` K ) ` W )
8		cdleml1.e	. . . . 5 |- E = ( ( TEndo ` K ) ` W )
9	4, 5, 6, 7, 8	tendotp 36049	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ f e. T ) -> ( R ` ( U ` f ) ) ( le ` K ) ( R ` f ) )
10	1, 2, 3, 9	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( R ` ( U ` f ) ) ( le ` K ) ( R ` f ) )
11		simp1l 1085	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> K e. HL )
12		hlatl 34647	. . . . 5 |- ( K e. HL -> K e. AtLat )
13	11, 12	syl 17	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> K e. AtLat )
14	5, 6, 8	tendocl 36055	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ f e. T ) -> ( U ` f ) e. T )
15	1, 2, 3, 14	syl3anc 1326	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( U ` f ) e. T )
16		simp32 1098	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( U ` f ) =/= ( _I |` B ) )
17		cdleml1.b	. . . . . 6 |- B = ( Base ` K )
18		eqid 2622	. . . . . 6 |- ( Atoms ` K ) = ( Atoms ` K )
19	17, 18, 5, 6, 7	trlnidat 35460	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U ` f ) e. T /\ ( U ` f ) =/= ( _I |` B ) ) -> ( R ` ( U ` f ) ) e. ( Atoms ` K ) )
20	1, 15, 16, 19	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( R ` ( U ` f ) ) e. ( Atoms ` K ) )
21		simp31 1097	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> f =/= ( _I |` B ) )
22	17, 18, 5, 6, 7	trlnidat 35460	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. T /\ f =/= ( _I |` B ) ) -> ( R ` f ) e. ( Atoms ` K ) )
23	1, 3, 21, 22	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( R ` f ) e. ( Atoms ` K ) )
24	4, 18	atcmp 34598	. . . 4 |- ( ( K e. AtLat /\ ( R ` ( U ` f ) ) e. ( Atoms ` K ) /\ ( R ` f ) e. ( Atoms ` K ) ) -> ( ( R ` ( U ` f ) ) ( le ` K ) ( R ` f ) <-> ( R ` ( U ` f ) ) = ( R ` f ) ) )
25	13, 20, 23, 24	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( ( R ` ( U ` f ) ) ( le ` K ) ( R ` f ) <-> ( R ` ( U ` f ) ) = ( R ` f ) ) )
26	10, 25	mpbid 222	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( R ` ( U ` f ) ) = ( R ` f ) )
27		simp22 1095	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> V e. E )
28	4, 5, 6, 7, 8	tendotp 36049	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ V e. E /\ f e. T ) -> ( R ` ( V ` f ) ) ( le ` K ) ( R ` f ) )
29	1, 27, 3, 28	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( R ` ( V ` f ) ) ( le ` K ) ( R ` f ) )
30	5, 6, 8	tendocl 36055	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ V e. E /\ f e. T ) -> ( V ` f ) e. T )
31	1, 27, 3, 30	syl3anc 1326	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( V ` f ) e. T )
32		simp33 1099	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( V ` f ) =/= ( _I |` B ) )
33	17, 18, 5, 6, 7	trlnidat 35460	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( V ` f ) e. T /\ ( V ` f ) =/= ( _I |` B ) ) -> ( R ` ( V ` f ) ) e. ( Atoms ` K ) )
34	1, 31, 32, 33	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( R ` ( V ` f ) ) e. ( Atoms ` K ) )
35	4, 18	atcmp 34598	. . . 4 |- ( ( K e. AtLat /\ ( R ` ( V ` f ) ) e. ( Atoms ` K ) /\ ( R ` f ) e. ( Atoms ` K ) ) -> ( ( R ` ( V ` f ) ) ( le ` K ) ( R ` f ) <-> ( R ` ( V ` f ) ) = ( R ` f ) ) )
36	13, 34, 23, 35	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( ( R ` ( V ` f ) ) ( le ` K ) ( R ` f ) <-> ( R ` ( V ` f ) ) = ( R ` f ) ) )
37	29, 36	mpbid 222	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( R ` ( V ` f ) ) = ( R ` f ) )
38	26, 37	eqtr4d 2659	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( R ` ( U ` f ) ) = ( R ` ( V ` f ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   _I cid 5023   |` cres 5116   ` cfv 5888   Basecbs 15857   lecple 15948   Atomscatm 34550   AtLatcal 34551   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   TEndoctendo 36040

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-reu 2919  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-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-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-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043

This theorem is referenced by:  cdleml2N  36265