Theorem dicn0 36481

Description: The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.)

Hypotheses

Ref	Expression
dicn0.l	|- .<_ = ( le ` K )
dicn0.a	|- A = ( Atoms ` K )
dicn0.h	|- H = ( LHyp ` K )
dicn0.i	|- I = ( ( DIsoC ` K ) ` W )

Assertion

Ref	Expression
dicn0	|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) =/= (/) )

Proof of Theorem dicn0

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

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( K e. HL /\ W e. H ) )
2		dicn0.l	. . . . . . . 8 |- .<_ = ( le ` K )
3		eqid 2622	. . . . . . . 8 |- ( oc ` K ) = ( oc ` K )
4		dicn0.a	. . . . . . . 8 |- A = ( Atoms ` K )
5		dicn0.h	. . . . . . . 8 |- H = ( LHyp ` K )
6	2, 3, 4, 5	lhpocnel 35304	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) )
7	6	adantr 481	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) )
8		simpr 477	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Q e. A /\ -. Q .<_ W ) )
9		eqid 2622	. . . . . . 7 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
10		eqid 2622	. . . . . . 7 |- ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) = ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q )
11	2, 4, 5, 9, 10	ltrniotacl 35867	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) e. ( ( LTrn ` K ) ` W ) )
12	1, 7, 8, 11	syl3anc 1326	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) e. ( ( LTrn ` K ) ` W ) )
13		eqid 2622	. . . . . 6 |- ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) )
14		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
15	13, 14	tendo02 36075	. . . . 5 |- ( ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) e. ( ( LTrn ` K ) ` W ) -> ( ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) = ( _I |` ( Base ` K ) ) )
16	12, 15	syl 17	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) = ( _I |` ( Base ` K ) ) )
17	16	eqcomd 2628	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( _I |` ( Base ` K ) ) = ( ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) )
18		eqid 2622	. . . . 5 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
19	14, 5, 9, 18, 13	tendo0cl 36078	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) e. ( ( TEndo ` K ) ` W ) )
20	19	adantr 481	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) e. ( ( TEndo ` K ) ` W ) )
21		eqid 2622	. . . 4 |- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
22		dicn0.i	. . . 4 |- I = ( ( DIsoC ` K ) ` W )
23		fvex 6201	. . . . 5 |- ( Base ` K ) e. _V
24		resiexg 7102	. . . . 5 |- ( ( Base ` K ) e. _V -> ( _I |` ( Base ` K ) ) e. _V )
25	23, 24	ax-mp 5	. . . 4 |- ( _I |` ( Base ` K ) ) e. _V
26		fvex 6201	. . . . 5 |- ( ( LTrn ` K ) ` W ) e. _V
27	26	mptex 6486	. . . 4 |- ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) e. _V
28	2, 4, 5, 21, 9, 18, 22, 25, 27	dicopelval 36466	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. ( _I |` ( Base ` K ) ) , ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) >. e. ( I ` Q ) <-> ( ( _I |` ( Base ` K ) ) = ( ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) /\ ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) e. ( ( TEndo ` K ) ` W ) ) ) )
29	17, 20, 28	mpbir2and 957	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> <. ( _I |` ( Base ` K ) ) , ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) >. e. ( I ` Q ) )
30		ne0i 3921	. 2 |- ( <. ( _I |` ( Base ` K ) ) , ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) >. e. ( I ` Q ) -> ( I ` Q ) =/= (/) )
31	29, 30	syl 17	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   |` cres 5116   ` cfv 5888   iota_crio 6610   Basecbs 15857   lecple 15948   occoc 15949   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040   DIsoCcdic 36461

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-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  df-dic 36462

This theorem is referenced by:  diclss  36482