Theorem lfl1dim 35243

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 2813	. 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 17947	. . . . . . . . . . . 12 |- ( W e. LVec -> W e. LMod )
4	2, 3	syl 16	. . . . . . . . . . 11 |- ( ph -> W e. LMod )
5		lfl1dim.d	. . . . . . . . . . . 12 |- D = (Scalar ` W )
6		lfl1dim.k	. . . . . . . . . . . 12 |- K = ( Base ` D )
7		eqid 2454	. . . . . . . . . . . 12 |- ( 0g ` D ) = ( 0g ` D )
8	5, 6, 7	lmod0cl 17733	. . . . . . . . . . 11 |- ( W e. LMod -> ( 0g ` D ) e. K )
9	4, 8	syl 16	. . . . . . . . . 10 |- ( ph -> ( 0g ` D ) e. K )
10	9	ad2antrr 723	. . . . . . . . 9 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> ( 0g ` D ) e. K )
11		simpr 459	. . . . . . . . . 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 723	. . . . . . . . . . 11 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> W e. LMod )
16		lfl1dim.g	. . . . . . . . . . . 12 |- ( ph -> G e. F )
17	16	ad2antrr 723	. . . . . . . . . . 11 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> G e. F )
18	12, 5, 13, 6, 14, 7, 15, 17	lfl0sc 35204	. . . . . . . . . 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 2498	. . . . . . . . 9 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> g = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) )
20		sneq 4026	. . . . . . . . . . . . 13 |- ( k = ( 0g ` D ) -> { k } = { ( 0g ` D ) } )
21	20	xpeq2d 5012	. . . . . . . . . . . 12 |- ( k = ( 0g ` D ) -> ( V X. { k } ) = ( V X. { ( 0g ` D ) } ) )
22	21	oveq2d 6286	. . . . . . . . . . 11 |- ( k = ( 0g ` D ) -> ( G oF .x. ( V X. { k } ) ) = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) )
23	22	eqeq2d 2468	. . . . . . . . . 10 |- ( k = ( 0g ` D ) -> ( g = ( G oF .x. ( V X. { k } ) ) <-> g = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) )
24	23	rspcev 3207	. . . . . . . . 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 659	. . . . . . . 8 |- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) )
26	25	a1d 25	. . . . . . 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 727	. . . . . . . . 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 727	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> W e. LMod )
30		simpllr 758	. . . . . . . . . . . . 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 35218	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> ( L ` g ) C_ V )
32	4	adantr 463	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ g e. F ) -> W e. LMod )
33	16	adantr 463	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ g e. F ) -> G e. F )
34	5, 7, 12, 13, 28	lkr0f 35216	. . . . . . . . . . . . . . . 16 |- ( ( W e. LMod /\ G e. F ) -> ( ( L ` G ) = V <-> G = ( V X. { ( 0g ` D ) } ) ) )
35	32, 33, 34	syl2anc 659	. . . . . . . . . . . . . . 15 |- ( ( ph /\ g e. F ) -> ( ( L ` G ) = V <-> G = ( V X. { ( 0g ` D ) } ) ) )
36	35	biimpar 483	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) -> ( L ` G ) = V )
37	36	sseq1d 3516	. . . . . . . . . . . . 13 |- ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) -> ( ( L ` G ) C_ ( L ` g ) <-> V C_ ( L ` g ) ) )
38	37	biimpa 482	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> V C_ ( L ` g ) )
39	31, 38	eqssd 3506	. . . . . . . . . . 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 35216	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ g e. F ) -> ( ( L ` g ) = V <-> g = ( V X. { ( 0g ` D ) } ) ) )
41	29, 30, 40	syl2anc 659	. . . . . . . . . . 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 210	. . . . . . . . . 10 |- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> g = ( V X. { ( 0g ` D ) } ) )
43	16	ad3antrrr 727	. . . . . . . . . . 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 35204	. . . . . . . . . 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 2498	. . . . . . . . 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 659	. . . . . . . 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 432	. . . . . . 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 2454	. . . . . . . . 9 |- (LSHyp ` W ) = (LSHyp ` W )
49	2	ad2antrr 723	. . . . . . . . 9 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> W e. LVec )
50	16	ad2antrr 723	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> G e. F )
51		simprr 755	. . . . . . . . . 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 35227	. . . . . . . . . 10 |- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { ( 0g ` D ) } ) ) -> ( L ` G ) e. (LSHyp ` W ) )
53	49, 50, 51, 52	syl3anc 1226	. . . . . . . . 9 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> ( L ` G ) e. (LSHyp ` W ) )
54		simplr 753	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> g e. F )
55		simprl 754	. . . . . . . . . 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 35227	. . . . . . . . . 10 |- ( ( W e. LVec /\ g e. F /\ g =/= ( V X. { ( 0g ` D ) } ) ) -> ( L ` g ) e. (LSHyp ` W ) )
57	49, 54, 55, 56	syl3anc 1226	. . . . . . . . 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 35110	. . . . . . . 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 727	. . . . . . . . . 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 727	. . . . . . . . . 10 |- ( ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) /\ ( L ` G ) = ( L ` g ) ) -> G e. F )
61		simpllr 758	. . . . . . . . . 10 |- ( ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) /\ ( L ` G ) = ( L ` g ) ) -> g e. F )
62		simpr 459	. . . . . . . . . 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 35222	. . . . . . . . . 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 1231	. . . . . . . . 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 432	. . . . . . . 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 215	. . . . . . 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 2773	. . . . . 6 |- ( ( ph /\ g e. F ) -> ( ( L ` G ) C_ ( L ` g ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) )
68	2	ad2antrr 723	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ k e. K ) -> W e. LVec )
69	16	ad2antrr 723	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ k e. K ) -> G e. F )
70		simpr 459	. . . . . . . . . 10 |- ( ( ( ph /\ g e. F ) /\ k e. K ) -> k e. K )
71	12, 5, 6, 14, 13, 28, 68, 69, 70	lkrscss 35220	. . . . . . . . 9 |- ( ( ( ph /\ g e. F ) /\ k e. K ) -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { k } ) ) ) )
72	71	ex 432	. . . . . . . 8 |- ( ( ph /\ g e. F ) -> ( k e. K -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { k } ) ) ) ) )
73		fveq2 5848	. . . . . . . . . 10 |- ( g = ( G oF .x. ( V X. { k } ) ) -> ( L ` g ) = ( L ` ( G oF .x. ( V X. { k } ) ) ) )
74	73	sseq2d 3517	. . . . . . . . 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 225	. . . . . . . 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 33	. . . . . . 7 |- ( ( ph /\ g e. F ) -> ( k e. K -> ( g = ( G oF .x. ( V X. { k } ) ) -> ( L ` G ) C_ ( L ` g ) ) ) )
77	76	rexlimdv 2944	. . . . . 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 191	. . . . 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 639	. . . 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 463	. . . . . . . . 9 |- ( ( ph /\ k e. K ) -> W e. LMod )
81	16	adantr 463	. . . . . . . . 9 |- ( ( ph /\ k e. K ) -> G e. F )
82		simpr 459	. . . . . . . . 9 |- ( ( ph /\ k e. K ) -> k e. K )
83	12, 5, 6, 14, 13, 80, 81, 82	lflvscl 35199	. . . . . . . 8 |- ( ( ph /\ k e. K ) -> ( G oF .x. ( V X. { k } ) ) e. F )
84		eleq1a 2537	. . . . . . . 8 |- ( ( G oF .x. ( V X. { k } ) ) e. F -> ( g = ( G oF .x. ( V X. { k } ) ) -> g e. F ) )
85	83, 84	syl 16	. . . . . . 7 |- ( ( ph /\ k e. K ) -> ( g = ( G oF .x. ( V X. { k } ) ) -> g e. F ) )
86	85	pm4.71rd 633	. . . . . 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 2962	. . . . 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 3009	. . . . 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 262	. . . 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 253	. . 3 |- ( ph -> ( ( g e. F /\ ( L ` G ) C_ ( L ` g ) ) <-> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) )
91	90	abbidv 2590	. 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 2507	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 184   /\ wa 367   = wceq 1398   e. wcel 1823   {cab 2439   =/= wne 2649   E.wrex 2805   {crab 2808   C_ wss 3461   {csn 4016   X. cxp 4986   ` cfv 5570  (class class class)co 6270   oFcof 6511   Basecbs 14716   .rcmulr 14785  Scalarcsca 14787   0gc0g 14929   LModclmod 17707   LVecclvec 17943  LSHypclsh 35097  LFnlclfn 35179  LKerclk 35207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-cntz 16554  df-lsm 16855  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-drng 17593  df-lmod 17709  df-lss 17774  df-lsp 17813  df-lvec 17944  df-lshyp 35099  df-lfl 35180  df-lkr 35208

This theorem is referenced by:  ldual1dim  35288