Theorem cdlemn9 36494

Description: Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)

Hypotheses

Ref	Expression
cdlemn8.b	|- B = ( Base ` K )
cdlemn8.l	|- .<_ = ( le ` K )
cdlemn8.a	|- A = ( Atoms ` K )
cdlemn8.h	|- H = ( LHyp ` K )
cdlemn8.p	|- P = ( ( oc ` K ) ` W )
cdlemn8.o	|- O = ( h e. T |-> ( _I |` B ) )
cdlemn8.t	|- T = ( ( LTrn ` K ) ` W )
cdlemn8.e	|- E = ( ( TEndo ` K ) ` W )
cdlemn8.u	|- U = ( ( DVecH ` K ) ` W )
cdlemn8.s	|- .+ = ( +g ` U )
cdlemn8.f	|- F = ( iota_ h e. T ( h ` P ) = Q )
cdlemn8.g	|- G = ( iota_ h e. T ( h ` P ) = R )

Assertion

Ref	Expression
cdlemn9	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( g ` Q ) = R )

Distinct variable groups:   .<_ , h   A, h   B, h   h, H   h, K   T, h   P, h   Q, h   h, W   R, h

Allowed substitution hints:   A( g, s)   B( g, s)   P( g, s)   .+ ( g, h, s)   Q( g, s)   R( g, s)   T( g, s)   U( g, h, s)   E( g, h, s)   F( g, h, s)   G( g, h, s)   H( g, s)   K( g, s)   .<_ ( g, s)   O( g, h, s)   W( g, s)

Proof of Theorem cdlemn9

Step	Hyp	Ref	Expression
1		cdlemn8.b	. . . 4 |- B = ( Base ` K )
2		cdlemn8.l	. . . 4 |- .<_ = ( le ` K )
3		cdlemn8.a	. . . 4 |- A = ( Atoms ` K )
4		cdlemn8.h	. . . 4 |- H = ( LHyp ` K )
5		cdlemn8.p	. . . 4 |- P = ( ( oc ` K ) ` W )
6		cdlemn8.o	. . . 4 |- O = ( h e. T |-> ( _I |` B ) )
7		cdlemn8.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
8		cdlemn8.e	. . . 4 |- E = ( ( TEndo ` K ) ` W )
9		cdlemn8.u	. . . 4 |- U = ( ( DVecH ` K ) ` W )
10		cdlemn8.s	. . . 4 |- .+ = ( +g ` U )
11		cdlemn8.f	. . . 4 |- F = ( iota_ h e. T ( h ` P ) = Q )
12		cdlemn8.g	. . . 4 |- G = ( iota_ h e. T ( h ` P ) = R )
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdlemn8 36493	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> g = ( G o. `' F ) )
14	13	fveq1d 6193	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( g ` Q ) = ( ( G o. `' F ) ` Q ) )
15		simp1 1061	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( K e. HL /\ W e. H ) )
16	2, 3, 4, 5	lhpocnel2 35305	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> ( P e. A /\ -. P .<_ W ) )
17	16	3ad2ant1 1082	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( P e. A /\ -. P .<_ W ) )
18		simp2l 1087	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
19	2, 3, 4, 7, 11	ltrniotacl 35867	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F e. T )
20	15, 17, 18, 19	syl3anc 1326	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> F e. T )
21	1, 4, 7	ltrn1o 35410	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F : B -1-1-onto-> B )
22	15, 20, 21	syl2anc 693	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> F : B -1-1-onto-> B )
23		f1ocnv 6149	. . . 4 |- ( F : B -1-1-onto-> B -> `' F : B -1-1-onto-> B )
24		f1of 6137	. . . 4 |- ( `' F : B -1-1-onto-> B -> `' F : B --> B )
25	22, 23, 24	3syl 18	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> `' F : B --> B )
26		simp2ll 1128	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> Q e. A )
27	1, 3	atbase 34576	. . . 4 |- ( Q e. A -> Q e. B )
28	26, 27	syl 17	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> Q e. B )
29		fvco3 6275	. . 3 |- ( ( `' F : B --> B /\ Q e. B ) -> ( ( G o. `' F ) ` Q ) = ( G ` ( `' F ` Q ) ) )
30	25, 28, 29	syl2anc 693	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( ( G o. `' F ) ` Q ) = ( G ` ( `' F ` Q ) ) )
31	2, 3, 4, 7, 11	ltrniotacnvval 35870	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( `' F ` Q ) = P )
32	15, 17, 18, 31	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( `' F ` Q ) = P )
33	32	fveq2d 6195	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( G ` ( `' F ` Q ) ) = ( G ` P ) )
34		simp2r 1088	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( R e. A /\ -. R .<_ W ) )
35	2, 3, 4, 7, 12	ltrniotaval 35869	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( G ` P ) = R )
36	15, 17, 34, 35	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( G ` P ) = R )
37	33, 36	eqtrd 2656	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( G ` ( `' F ` Q ) ) = R )
38	14, 30, 37	3eqtrd 2660	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( g ` Q ) = R )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   lecple 15948   occoc 15949   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040   DVecHcdvh 36367

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-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  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-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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-undef 7399  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  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  df-edring 36045  df-dvech 36368

This theorem is referenced by:  cdlemn11pre  36499