Theorem cdlemj3 36111

Description: Part of proof of Lemma J of [Crawley] p. 118. Eliminate g. (Contributed by NM, 20-Jun-2013.)

Hypotheses

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

Assertion

Ref	Expression
cdlemj3	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) -> ( U ` h ) = ( V ` h ) )

Proof of Theorem cdlemj3

Dummy variables g u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1064	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) -> ( K e. HL /\ W e. H ) )
2		eqid 2622	. . . 4 |- ( le ` K ) = ( le ` K )
3		eqid 2622	. . . 4 |- ( Atoms ` K ) = ( Atoms ` K )
4		cdlemj.h	. . . 4 |- H = ( LHyp ` K )
5	2, 3, 4	lhpexle2 35296	. . 3 |- ( ( K e. HL /\ W e. H ) -> E. u e. ( Atoms ` K ) ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) )
6	1, 5	syl 17	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) -> E. u e. ( Atoms ` K ) ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) )
7		simpl1l 1112	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) -> K e. HL )
8	7	adantr 481	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) ) -> K e. HL )
9		simpl1r 1113	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) -> W e. H )
10	9	adantr 481	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) ) -> W e. H )
11		simprl 794	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) ) -> u e. ( Atoms ` K ) )
12		simprr1 1109	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) ) -> u ( le ` K ) W )
13		cdlemj.b	. . . . 5 |- B = ( Base ` K )
14		cdlemj.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
15		cdlemj.r	. . . . 5 |- R = ( ( trL ` K ) ` W )
16	13, 2, 3, 4, 14, 15	cdlemfnid 35852	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( u e. ( Atoms ` K ) /\ u ( le ` K ) W ) ) -> E. g e. T ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) )
17	8, 10, 11, 12, 16	syl22anc 1327	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) ) -> E. g e. T ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) )
18		simp1l 1085	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) /\ ( g e. T /\ ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) )
19		simp1r 1086	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) /\ ( g e. T /\ ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) ) ) -> h =/= ( _I |` B ) )
20		simp3l 1089	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) /\ ( g e. T /\ ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) ) ) -> g e. T )
21		simp3rr 1135	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) /\ ( g e. T /\ ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) ) ) -> g =/= ( _I |` B ) )
22		simp2r2 1164	. . . . . . . . 9 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) /\ ( g e. T /\ ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) ) ) -> u =/= ( R ` F ) )
23	22	necomd 2849	. . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) /\ ( g e. T /\ ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) ) ) -> ( R ` F ) =/= u )
24		simp3rl 1134	. . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) /\ ( g e. T /\ ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) ) ) -> ( R ` g ) = u )
25	23, 24	neeqtrrd 2868	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) /\ ( g e. T /\ ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) ) ) -> ( R ` F ) =/= ( R ` g ) )
26		simp2r3 1165	. . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) /\ ( g e. T /\ ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) ) ) -> u =/= ( R ` h ) )
27	24, 26	eqnetrd 2861	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) /\ ( g e. T /\ ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) ) ) -> ( R ` g ) =/= ( R ` h ) )
28		cdlemj.e	. . . . . . . 8 |- E = ( ( TEndo ` K ) ` W )
29	13, 4, 14, 15, 28	cdlemj2 36110	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ ( h =/= ( _I |` B ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( ( R ` F ) =/= ( R ` g ) /\ ( R ` g ) =/= ( R ` h ) ) ) -> ( U ` h ) = ( V ` h ) )
30	18, 19, 20, 21, 25, 27, 29	syl132anc 1344	. . . . . 6 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) /\ ( g e. T /\ ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) ) ) -> ( U ` h ) = ( V ` h ) )
31	30	3expia 1267	. . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) ) -> ( ( g e. T /\ ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) ) -> ( U ` h ) = ( V ` h ) ) )
32	31	expd 452	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) ) -> ( g e. T -> ( ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) -> ( U ` h ) = ( V ` h ) ) ) )
33	32	rexlimdv 3030	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) ) -> ( E. g e. T ( ( R ` g ) = u /\ g =/= ( _I |` B ) ) -> ( U ` h ) = ( V ` h ) ) )
34	17, 33	mpd 15	. 2 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ u =/= ( R ` F ) /\ u =/= ( R ` h ) ) ) ) -> ( U ` h ) = ( V ` h ) )
35	6, 34	rexlimddv 3035	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ ( U ` F ) = ( V ` F ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ h e. T ) ) /\ h =/= ( _I |` B ) ) -> ( U ` h ) = ( V ` h ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   _I cid 5023   |` cres 5116   ` cfv 5888   Basecbs 15857   lecple 15948   Atomscatm 34550   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:  tendocan  36112