Theorem cdleml3N 36266

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 )
cdleml3.o	|- .0. = ( g e. T |-> ( _I |` B ) )

Assertion

Ref	Expression
cdleml3N	|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> E. s e. E ( s o. U ) = V )

Distinct variable groups:   E, s   K, s   R, s   T, s   U, s   V, s   W, s, f, g   B, g, s   f, E   f, g, H, s   f, K, g   .0. , f, s   T, f, g   U, f   f, V   f, W, g

Allowed substitution hints:   B( f)   R( f, g)   U( g)   E( g)   V( g)   .0. ( g)

Proof of Theorem cdleml3N

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> ( K e. HL /\ W e. H ) )
2		simp2 1062	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> ( U e. E /\ V e. E /\ f e. T ) )
3		simp31 1097	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> f =/= ( _I |` B ) )
4		simp32 1098	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> U =/= .0. )
5		simp21 1094	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> U e. E )
6		simp23 1096	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> f e. T )
7		cdleml1.b	. . . . . . 7 |- B = ( Base ` K )
8		cdleml1.h	. . . . . . 7 |- H = ( LHyp ` K )
9		cdleml1.t	. . . . . . 7 |- T = ( ( LTrn ` K ) ` W )
10		cdleml1.e	. . . . . . 7 |- E = ( ( TEndo ` K ) ` W )
11		cdleml3.o	. . . . . . 7 |- .0. = ( g e. T |-> ( _I |` B ) )
12	7, 8, 9, 10, 11	tendoid0 36113	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ ( f e. T /\ f =/= ( _I |` B ) ) ) -> ( ( U ` f ) = ( _I |` B ) <-> U = .0. ) )
13	1, 5, 6, 3, 12	syl112anc 1330	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> ( ( U ` f ) = ( _I |` B ) <-> U = .0. ) )
14	13	necon3bid 2838	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> ( ( U ` f ) =/= ( _I |` B ) <-> U =/= .0. ) )
15	4, 14	mpbird 247	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> ( U ` f ) =/= ( _I |` B ) )
16		simp33 1099	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> V =/= .0. )
17		simp22 1095	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> V e. E )
18	7, 8, 9, 10, 11	tendoid0 36113	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ V e. E /\ ( f e. T /\ f =/= ( _I |` B ) ) ) -> ( ( V ` f ) = ( _I |` B ) <-> V = .0. ) )
19	1, 17, 6, 3, 18	syl112anc 1330	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> ( ( V ` f ) = ( _I |` B ) <-> V = .0. ) )
20	19	necon3bid 2838	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> ( ( V ` f ) =/= ( _I |` B ) <-> V =/= .0. ) )
21	16, 20	mpbird 247	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> ( V ` f ) =/= ( _I |` B ) )
22		cdleml1.r	. . . 4 |- R = ( ( trL ` K ) ` W )
23	7, 8, 9, 22, 10	cdleml2N 36265	. . 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 ) ) ) -> E. s e. E ( s ` ( U ` f ) ) = ( V ` f ) )
24	1, 2, 3, 15, 21, 23	syl113anc 1338	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> E. s e. E ( s ` ( U ` f ) ) = ( V ` f ) )
25		simpl1 1064	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E ) -> ( K e. HL /\ W e. H ) )
26		simpr 477	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E ) -> s e. E )
27		simpl21 1139	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E ) -> U e. E )
28		simpl23 1141	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E ) -> f e. T )
29	8, 9, 10	tendocoval 36054	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( s e. E /\ U e. E ) /\ f e. T ) -> ( ( s o. U ) ` f ) = ( s ` ( U ` f ) ) )
30	25, 26, 27, 28, 29	syl121anc 1331	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E ) -> ( ( s o. U ) ` f ) = ( s ` ( U ` f ) ) )
31	30	eqeq1d 2624	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E ) -> ( ( ( s o. U ) ` f ) = ( V ` f ) <-> ( s ` ( U ` f ) ) = ( V ` f ) ) )
32		simp11 1091	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E /\ ( ( s o. U ) ` f ) = ( V ` f ) ) -> ( K e. HL /\ W e. H ) )
33		simp2 1062	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E /\ ( ( s o. U ) ` f ) = ( V ` f ) ) -> s e. E )
34		simp121 1193	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E /\ ( ( s o. U ) ` f ) = ( V ` f ) ) -> U e. E )
35	8, 10	tendococl 36060	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ s e. E /\ U e. E ) -> ( s o. U ) e. E )
36	32, 33, 34, 35	syl3anc 1326	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E /\ ( ( s o. U ) ` f ) = ( V ` f ) ) -> ( s o. U ) e. E )
37		simp122 1194	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E /\ ( ( s o. U ) ` f ) = ( V ` f ) ) -> V e. E )
38		simp3 1063	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E /\ ( ( s o. U ) ` f ) = ( V ` f ) ) -> ( ( s o. U ) ` f ) = ( V ` f ) )
39		simp123 1195	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E /\ ( ( s o. U ) ` f ) = ( V ` f ) ) -> f e. T )
40		simp131 1196	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E /\ ( ( s o. U ) ` f ) = ( V ` f ) ) -> f =/= ( _I |` B ) )
41	7, 8, 9, 10	tendocan 36112	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( s o. U ) e. E /\ V e. E /\ ( ( s o. U ) ` f ) = ( V ` f ) ) /\ ( f e. T /\ f =/= ( _I |` B ) ) ) -> ( s o. U ) = V )
42	32, 36, 37, 38, 39, 40, 41	syl132anc 1344	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E /\ ( ( s o. U ) ` f ) = ( V ` f ) ) -> ( s o. U ) = V )
43	42	3expia 1267	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E ) -> ( ( ( s o. U ) ` f ) = ( V ` f ) -> ( s o. U ) = V ) )
44	31, 43	sylbird 250	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) /\ s e. E ) -> ( ( s ` ( U ` f ) ) = ( V ` f ) -> ( s o. U ) = V ) )
45	44	reximdva 3017	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> ( E. s e. E ( s ` ( U ` f ) ) = ( V ` f ) -> E. s e. E ( s o. U ) = V ) )
46	24, 45	mpd 15	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> E. s e. E ( s o. U ) = V )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   |-> cmpt 4729   _I cid 5023   |` cres 5116   o. ccom 5118   ` cfv 5888   Basecbs 15857   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  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-fal 1489  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  df-tendo 36043

This theorem is referenced by:  cdleml4N  36267