Theorem dihordlem7b 36504

Description: Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Hypotheses

Ref	Expression
dihordlem8.b	|- B = ( Base ` K )
dihordlem8.l	|- .<_ = ( le ` K )
dihordlem8.a	|- A = ( Atoms ` K )
dihordlem8.h	|- H = ( LHyp ` K )
dihordlem8.p	|- P = ( ( oc ` K ) ` W )
dihordlem8.o	|- O = ( h e. T |-> ( _I |` B ) )
dihordlem8.t	|- T = ( ( LTrn ` K ) ` W )
dihordlem8.e	|- E = ( ( TEndo ` K ) ` W )
dihordlem8.u	|- U = ( ( DVecH ` K ) ` W )
dihordlem8.s	|- .+ = ( +g ` U )
dihordlem8.g	|- G = ( iota_ h e. T ( h ` P ) = R )

Assertion

Ref	Expression
dihordlem7b	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( f = g /\ O = s ) )

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

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

Proof of Theorem dihordlem7b

Step	Hyp	Ref	Expression
1		dihordlem8.b	. . . . 5 |- B = ( Base ` K )
2		dihordlem8.l	. . . . 5 |- .<_ = ( le ` K )
3		dihordlem8.a	. . . . 5 |- A = ( Atoms ` K )
4		dihordlem8.h	. . . . 5 |- H = ( LHyp ` K )
5		dihordlem8.p	. . . . 5 |- P = ( ( oc ` K ) ` W )
6		dihordlem8.o	. . . . 5 |- O = ( h e. T |-> ( _I |` B ) )
7		dihordlem8.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
8		dihordlem8.e	. . . . 5 |- E = ( ( TEndo ` K ) ` W )
9		dihordlem8.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
10		dihordlem8.s	. . . . 5 |- .+ = ( +g ` U )
11		dihordlem8.g	. . . . 5 |- G = ( iota_ h e. T ( h ` P ) = R )
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	dihordlem7 36503	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( f = ( ( s ` G ) o. g ) /\ O = s ) )
13	12	simpld 475	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> f = ( ( s ` G ) o. g ) )
14	12	simprd 479	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> O = s )
15	14	fveq1d 6193	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( O ` G ) = ( s ` G ) )
16		simp1 1061	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( K e. HL /\ W e. H ) )
17	2, 3, 4, 5	lhpocnel2 35305	. . . . . . . 8 |- ( ( K e. HL /\ W e. H ) -> ( P e. A /\ -. P .<_ W ) )
18	17	3ad2ant1 1082	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( P e. A /\ -. P .<_ W ) )
19		simp2r 1088	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( R e. A /\ -. R .<_ W ) )
20	2, 3, 4, 7, 11	ltrniotacl 35867	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) -> G e. T )
21	16, 18, 19, 20	syl3anc 1326	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> G e. T )
22	6, 1	tendo02 36075	. . . . . 6 |- ( G e. T -> ( O ` G ) = ( _I |` B ) )
23	21, 22	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( O ` G ) = ( _I |` B ) )
24	15, 23	eqtr3d 2658	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( s ` G ) = ( _I |` B ) )
25	24	coeq1d 5283	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( ( s ` G ) o. g ) = ( ( _I |` B ) o. g ) )
26		simp32 1098	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> g e. T )
27	1, 4, 7	ltrn1o 35410	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ g e. T ) -> g : B -1-1-onto-> B )
28	16, 26, 27	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 /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> g : B -1-1-onto-> B )
29		f1of 6137	. . . 4 |- ( g : B -1-1-onto-> B -> g : B --> B )
30		fcoi2 6079	. . . 4 |- ( g : B --> B -> ( ( _I |` B ) o. g ) = g )
31	28, 29, 30	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 /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( ( _I |` B ) o. g ) = g )
32	13, 25, 31	3eqtrd 2660	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> f = g )
33	32, 14	jca 554	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( f = g /\ O = s ) )

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   |` 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:  dihord10  36512