Theorem lfl1dim 32687

Description: Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)

Hypotheses

Ref	Expression
lfl1dim.v	|- V = ( Base ` W )
lfl1dim.d	|- D = (Scalar ` W )
lfl1dim.f	|- F = (LFnl ` W )
lfl1dim.l	|- L = (LKer ` W )
lfl1dim.k	|- K = ( Base ` D )
lfl1dim.t	|- .x. = ( .r ` D )
lfl1dim.w	|- ( ph -> W e. LVec )
lfl1dim.g	|- ( ph -> G e. F )

Assertion

Ref	Expression
lfl1dim	|- ( ph -> { g e. F | ( L ` G ) C_ ( L ` g ) } = { g | E. k e. K g = ( G oF .x. ( V X. { k } ) ) } )

Distinct variable groups:   D, k   k, F   k, G   k, K   k, L   k, V   k, W   g, k, ph   .x. , k

Allowed substitution hints:   D( g)   .x. ( g)   F( g)   G( g)   K( g)   L( g)   V( g)   W( g)

Proof of Theorem lfl1dim

Step	Hyp	Ref	Expression
1		df-rab 2746	. 2 |- { g e. F | ( L ` G ) C_ ( L ` g ) } = { g | ( g e. F /\ ( L ` G ) C_ ( L ` g ) ) }
2		lfl1dim.w	. . . . . . . . . . . 12 |- ( ph -> W e. LVec )
3		lveclmod 18329	. . . . . . . . . . . 12 |- ( W e. LVec -> W e. LMod )
4	2, 3	syl 17	. . . . . . . . . . 11 |- ( ph -> W e. LMod )
5		lfl1dim.d	. . . . . . . . . . . 12 |- D = (Scalar ` W )
6		lfl1dim.k	. . . . . . . . . . . 12 |- K = ( Base ` D )
7		eqid 2451	. . . . . . . . . . . 12 |- ( 0g ` D ) = ( 0g ` D )
8	5, 6, 7	lmod0cl 18117	. . . . . . . . . . 11 |- ( W e. LMod -> ( 0g ` D ) e. K )
9	4, 8	syl 17	. . . . . . . . . 10 |- ( ph -> ( 0g ` D ) e. K )
10	9	ad2antrr 732	. . . . . . . . 9 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> ( 0g ` D ) e. K )
11		simpr 463	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> g = ( V X. { ( 0g ` D ) } ) )
12		lfl1dim.v	. . . . . . . . . . 11 |- V = ( Base ` W )
13		lfl1dim.f	. . . . . . . . . . 11 |- F = (LFnl ` W )
14		lfl1dim.t	. . . . . . . . . . 11 |- .x. = ( .r ` D )
15	4	ad2antrr 732	. . . . . . . . . . 11 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> W e. LMod )
16		lfl1dim.g	. . . . . . . . . . . 12 |- ( ph -> G e. F )
17	16	ad2antrr 732	. . . . . . . . . . 11 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> G e. F )
18	12, 5, 13, 6, 14, 7, 15, 17	lfl0sc 32648	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> ( G oF .x. ( V X. { ( 0g ` D ) } ) ) = ( V X. { ( 0g ` D ) } ) )
19	11, 18	eqtr4d 2488	. . . . . . . . 9 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> g = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) )
20		sneq 3978	. . . . . . . . . . . . 13 |- ( k = ( 0g ` D ) -> { k } = { ( 0g ` D ) } )
21	20	xpeq2d 4858	. . . . . . . . . . . 12 |- ( k = ( 0g ` D ) -> ( V X. { k } ) = ( V X. { ( 0g ` D ) } ) )
22	21	oveq2d 6306	. . . . . . . . . . 11 |- ( k = ( 0g ` D ) -> ( G oF .x. ( V X. { k } ) ) = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) )
23	22	eqeq2d 2461	. . . . . . . . . 10 |- ( k = ( 0g ` D ) -> ( g = ( G oF .x. ( V X. { k } ) ) <-> g = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) )
24	23	rspcev 3150	. . . . . . . . 9 |- ( ( ( 0g ` D ) e. K /\ g = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) )
25	10, 19, 24	syl2anc 667	. . . . . . . 8 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) )
26	25	a1d 26	. . . . . . 7 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> ( ( L ` G ) C_ ( L ` g ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) )
27	9	ad3antrrr 736	. . . . . . . . 9 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> ( 0g ` D ) e. K )
28		lfl1dim.l	. . . . . . . . . . . . 13 |- L = (LKer ` W )
29	4	ad3antrrr 736	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> W e. LMod )
30		simpllr 769	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> g e. F )
31	12, 13, 28, 29, 30	lkrssv 32662	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> ( L ` g ) C_ V )
32	4	adantr 467	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ g e. F ) -> W e. LMod )
33	16	adantr 467	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ g e. F ) -> G e. F )
34	5, 7, 12, 13, 28	lkr0f 32660	. . . . . . . . . . . . . . . 16 |- ( ( W e. LMod /\ G e. F ) -> ( ( L ` G ) = V <-> G = ( V X. { ( 0g ` D ) } ) ) )
35	32, 33, 34	syl2anc 667	. . . . . . . . . . . . . . 15 |- ( ( ph /\ g e. F ) -> ( ( L ` G ) = V <-> G = ( V X. { ( 0g ` D ) } ) ) )
36	35	biimpar 488	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) -> ( L ` G ) = V )
37	36	sseq1d 3459	. . . . . . . . . . . . 13 |- ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) -> ( ( L ` G ) C_ ( L ` g ) <-> V C_ ( L ` g ) ) )
38	37	biimpa 487	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> V C_ ( L ` g ) )
39	31, 38	eqssd 3449	. . . . . . . . . . 11 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> ( L ` g ) = V )
40	5, 7, 12, 13, 28	lkr0f 32660	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ g e. F ) -> ( ( L ` g ) = V <-> g = ( V X. { ( 0g ` D ) } ) ) )
41	29, 30, 40	syl2anc 667	. . . . . . . . . . 11 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> ( ( L ` g ) = V <-> g = ( V X. { ( 0g ` D ) } ) ) )
42	39, 41	mpbid 214	. . . . . . . . . 10 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> g = ( V X. { ( 0g ` D ) } ) )
43	16	ad3antrrr 736	. . . . . . . . . . 11 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> G e. F )
44	12, 5, 13, 6, 14, 7, 29, 43	lfl0sc 32648	. . . . . . . . . 10 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> ( G oF .x. ( V X. { ( 0g ` D ) } ) ) = ( V X. { ( 0g ` D ) } ) )
45	42, 44	eqtr4d 2488	. . . . . . . . 9 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> g = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) )
46	27, 45, 24	syl2anc 667	. . . . . . . 8 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) )
47	46	ex 436	. . . . . . 7 |- ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) -> ( ( L ` G ) C_ ( L ` g ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) )
48		eqid 2451	. . . . . . . . 9 |- (LSHyp ` W ) = (LSHyp ` W )
49	2	ad2antrr 732	. . . . . . . . 9 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> W e. LVec )
50	16	ad2antrr 732	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> G e. F )
51		simprr 766	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> G =/= ( V X. { ( 0g ` D ) } ) )
52	12, 5, 7, 48, 13, 28	lkrshp 32671	. . . . . . . . . 10 |- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { ( 0g ` D ) } ) ) -> ( L ` G ) e. (LSHyp ` W ) )
53	49, 50, 51, 52	syl3anc 1268	. . . . . . . . 9 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> ( L ` G ) e. (LSHyp ` W ) )
54		simplr 762	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> g e. F )
55		simprl 764	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> g =/= ( V X. { ( 0g ` D ) } ) )
56	12, 5, 7, 48, 13, 28	lkrshp 32671	. . . . . . . . . 10 |- ( ( W e. LVec /\ g e. F /\ g =/= ( V X. { ( 0g ` D ) } ) ) -> ( L ` g ) e. (LSHyp ` W ) )
57	49, 54, 55, 56	syl3anc 1268	. . . . . . . . 9 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> ( L ` g ) e. (LSHyp ` W ) )
58	48, 49, 53, 57	lshpcmp 32554	. . . . . . . 8 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> ( ( L ` G ) C_ ( L ` g ) <-> ( L ` G ) = ( L ` g ) ) )
59	2	ad3antrrr 736	. . . . . . . . . 10 |- ( ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) /\ ( L ` G ) = ( L ` g ) ) -> W e. LVec )
60	16	ad3antrrr 736	. . . . . . . . . 10 |- ( ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) /\ ( L ` G ) = ( L ` g ) ) -> G e. F )
61		simpllr 769	. . . . . . . . . 10 |- ( ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) /\ ( L ` G ) = ( L ` g ) ) -> g e. F )
62		simpr 463	. . . . . . . . . 10 |- ( ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) /\ ( L ` G ) = ( L ` g ) ) -> ( L ` G ) = ( L ` g ) )
63	5, 6, 14, 12, 13, 28	eqlkr2 32666	. . . . . . . . . 10 |- ( ( W e. LVec /\ ( G e. F /\ g e. F ) /\ ( L ` G ) = ( L ` g ) ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) )
64	59, 60, 61, 62, 63	syl121anc 1273	. . . . . . . . 9 |- ( ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) /\ ( L ` G ) = ( L ` g ) ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) )
65	64	ex 436	. . . . . . . 8 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> ( ( L ` G ) = ( L ` g ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) )
66	58, 65	sylbid 219	. . . . . . 7 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> ( ( L ` G ) C_ ( L ` g ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) )
67	26, 47, 66	pm2.61da2ne 2712	. . . . . 6 |- ( ( ph /\ g e. F ) -> ( ( L ` G ) C_ ( L ` g ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) )
68	2	ad2antrr 732	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ k e. K ) -> W e. LVec )
69	16	ad2antrr 732	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ k e. K ) -> G e. F )
70		simpr 463	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ k e. K ) -> k e. K )
71	12, 5, 6, 14, 13, 28, 68, 69, 70	lkrscss 32664	. . . . . . . . 9 |- ( ( ( ph /\ g e. F ) /\ k e. K ) -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { k } ) ) ) )
72	71	ex 436	. . . . . . . 8 |- ( ( ph /\ g e. F ) -> ( k e. K -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { k } ) ) ) ) )
73		fveq2 5865	. . . . . . . . . 10 |- ( g = ( G oF .x. ( V X. { k } ) ) -> ( L ` g ) = ( L ` ( G oF .x. ( V X. { k } ) ) ) )
74	73	sseq2d 3460	. . . . . . . . 9 |- ( g = ( G oF .x. ( V X. { k } ) ) -> ( ( L ` G ) C_ ( L ` g ) <-> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { k } ) ) ) ) )
75	74	biimprcd 229	. . . . . . . 8 |- ( ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { k } ) ) ) -> ( g = ( G oF .x. ( V X. { k } ) ) -> ( L ` G ) C_ ( L ` g ) ) )
76	72, 75	syl6 34	. . . . . . 7 |- ( ( ph /\ g e. F ) -> ( k e. K -> ( g = ( G oF .x. ( V X. { k } ) ) -> ( L ` G ) C_ ( L ` g ) ) ) )
77	76	rexlimdv 2877	. . . . . 6 |- ( ( ph /\ g e. F ) -> ( E. k e. K g = ( G oF .x. ( V X. { k } ) ) -> ( L ` G ) C_ ( L ` g ) ) )
78	67, 77	impbid 194	. . . . 5 |- ( ( ph /\ g e. F ) -> ( ( L ` G ) C_ ( L ` g ) <-> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) )
79	78	pm5.32da 647	. . . 4 |- ( ph -> ( ( g e. F /\ ( L ` G ) C_ ( L ` g ) ) <-> ( g e. F /\ E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) ) )
80	4	adantr 467	. . . . . . . . 9 |- ( ( ph /\ k e. K ) -> W e. LMod )
81	16	adantr 467	. . . . . . . . 9 |- ( ( ph /\ k e. K ) -> G e. F )
82		simpr 463	. . . . . . . . 9 |- ( ( ph /\ k e. K ) -> k e. K )
83	12, 5, 6, 14, 13, 80, 81, 82	lflvscl 32643	. . . . . . . 8 |- ( ( ph /\ k e. K ) -> ( G oF .x. ( V X. { k } ) ) e. F )
84		eleq1a 2524	. . . . . . . 8 |- ( ( G oF .x. ( V X. { k } ) ) e. F -> ( g = ( G oF .x. ( V X. { k } ) ) -> g e. F ) )
85	83, 84	syl 17	. . . . . . 7 |- ( ( ph /\ k e. K ) -> ( g = ( G oF .x. ( V X. { k } ) ) -> g e. F ) )
86	85	pm4.71rd 641	. . . . . 6 |- ( ( ph /\ k e. K ) -> ( g = ( G oF .x. ( V X. { k } ) ) <-> ( g e. F /\ g = ( G oF .x. ( V X. { k } ) ) ) ) )
87	86	rexbidva 2898	. . . . 5 |- ( ph -> ( E. k e. K g = ( G oF .x. ( V X. { k } ) ) <-> E. k e. K ( g e. F /\ g = ( G oF .x. ( V X. { k } ) ) ) ) )
88		r19.42v 2945	. . . . 5 |- ( E. k e. K ( g e. F /\ g = ( G oF .x. ( V X. { k } ) ) ) <-> ( g e. F /\ E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) )
89	87, 88	syl6rbb 266	. . . 4 |- ( ph -> ( ( g e. F /\ E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) <-> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) )
90	79, 89	bitrd 257	. . 3 |- ( ph -> ( ( g e. F /\ ( L ` G ) C_ ( L ` g ) ) <-> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) )
91	90	abbidv 2569	. 2 |- ( ph -> { g | ( g e. F /\ ( L ` G ) C_ ( L ` g ) ) } = { g | E. k e. K g = ( G oF .x. ( V X. { k } ) ) } )
92	1, 91	syl5eq 2497	1 |- ( ph -> { g e. F | ( L ` G ) C_ ( L ` g ) } = { g | E. k e. K g = ( G oF .x. ( V X. { k } ) ) } )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   {cab 2437   =/= wne 2622   E.wrex 2738   {crab 2741   C_ wss 3404   {csn 3968   X. cxp 4832   ` cfv 5582  (class class class)co 6290   oFcof 6529   Basecbs 15121   .rcmulr 15191  Scalarcsca 15193   0gc0g 15338   LModclmod 18091   LVecclvec 18325  LSHypclsh 32541  LFnlclfn 32623  LKerclk 32651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-subg 16814  df-cntz 16971  df-lsm 17288  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-drng 17977  df-lmod 18093  df-lss 18156  df-lsp 18195  df-lvec 18326  df-lshyp 32543  df-lfl 32624  df-lkr 32652

This theorem is referenced by:  ldual1dim  32732