Theorem dicvaddcl 36479

Description: Membership in value of the partial isomorphism C is closed under vector sum. (Contributed by NM, 16-Feb-2014.)

Hypotheses

Ref	Expression
dicvaddcl.l	|- .<_ = ( le ` K )
dicvaddcl.a	|- A = ( Atoms ` K )
dicvaddcl.h	|- H = ( LHyp ` K )
dicvaddcl.u	|- U = ( ( DVecH ` K ) ` W )
dicvaddcl.i	|- I = ( ( DIsoC ` K ) ` W )
dicvaddcl.p	|- .+ = ( +g ` U )

Assertion

Ref	Expression
dicvaddcl	|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( X .+ Y ) e. ( I ` Q ) )

Proof of Theorem dicvaddcl

Dummy variables g h s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( K e. HL /\ W e. H ) )
2		dicvaddcl.l	. . . . . . 7 |- .<_ = ( le ` K )
3		dicvaddcl.a	. . . . . . 7 |- A = ( Atoms ` K )
4		dicvaddcl.h	. . . . . . 7 |- H = ( LHyp ` K )
5		dicvaddcl.i	. . . . . . 7 |- I = ( ( DIsoC ` K ) ` W )
6		dicvaddcl.u	. . . . . . 7 |- U = ( ( DVecH ` K ) ` W )
7		eqid 2622	. . . . . . 7 |- ( Base ` U ) = ( Base ` U )
8	2, 3, 4, 5, 6, 7	dicssdvh 36475	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) C_ ( Base ` U ) )
9		eqid 2622	. . . . . . . . 9 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
10		eqid 2622	. . . . . . . . 9 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
11	4, 9, 10, 6, 7	dvhvbase 36376	. . . . . . . 8 |- ( ( K e. HL /\ W e. H ) -> ( Base ` U ) = ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
12	11	eqcomd 2628	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) = ( Base ` U ) )
13	12	adantr 481	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) = ( Base ` U ) )
14	8, 13	sseqtr4d 3642	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) C_ ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
15	14	3adant3 1081	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( I ` Q ) C_ ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
16		simp3l 1089	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> X e. ( I ` Q ) )
17	15, 16	sseldd 3604	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> X e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
18		simp3r 1090	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> Y e. ( I ` Q ) )
19	15, 18	sseldd 3604	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> Y e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
20		eqid 2622	. . . 4 |- (Scalar ` U ) = (Scalar ` U )
21		dicvaddcl.p	. . . 4 |- .+ = ( +g ` U )
22		eqid 2622	. . . 4 |- ( +g ` (Scalar ` U ) ) = ( +g ` (Scalar ` U ) )
23	4, 9, 10, 6, 20, 21, 22	dvhvadd 36381	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) /\ Y e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) ) ) -> ( X .+ Y ) = <. ( ( 1st ` X ) o. ( 1st ` Y ) ) , ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) >. )
24	1, 17, 19, 23	syl12anc 1324	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( X .+ Y ) = <. ( ( 1st ` X ) o. ( 1st ` Y ) ) , ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) >. )
25	2, 3, 4, 10, 5	dicelval2nd 36478	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ X e. ( I ` Q ) ) -> ( 2nd ` X ) e. ( ( TEndo ` K ) ` W ) )
26	25	3adant3r 1323	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( 2nd ` X ) e. ( ( TEndo ` K ) ` W ) )
27	2, 3, 4, 10, 5	dicelval2nd 36478	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Y e. ( I ` Q ) ) -> ( 2nd ` Y ) e. ( ( TEndo ` K ) ` W ) )
28	27	3adant3l 1322	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( 2nd ` Y ) e. ( ( TEndo ` K ) ` W ) )
29		eqid 2622	. . . . . . . 8 |- ( oc ` K ) = ( oc ` K )
30	2, 29, 3, 4	lhpocnel 35304	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) )
31	30	3ad2ant1 1082	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) )
32		simp2 1062	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
33		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 )
34	2, 3, 4, 9, 33	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 ) )
35	1, 31, 32, 34	syl3anc 1326	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) e. ( ( LTrn ` K ) ` W ) )
36		eqid 2622	. . . . . 6 |- ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) = ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) )
37	9, 36	tendospdi2 36311	. . . . 5 |- ( ( ( 2nd ` X ) e. ( ( TEndo ` K ) ` W ) /\ ( 2nd ` Y ) e. ( ( TEndo ` K ) ` W ) /\ ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) e. ( ( LTrn ` K ) ` W ) ) -> ( ( ( 2nd ` X ) ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) = ( ( ( 2nd ` X ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) o. ( ( 2nd ` Y ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) ) )
38	26, 28, 35, 37	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( ( 2nd ` X ) ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) = ( ( ( 2nd ` X ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) o. ( ( 2nd ` Y ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) ) )
39	4, 9, 10, 6, 20, 36, 22	dvhfplusr 36373	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> ( +g ` (Scalar ` U ) ) = ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) )
40	39	3ad2ant1 1082	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( +g ` (Scalar ` U ) ) = ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) )
41	40	oveqd 6667	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) = ( ( 2nd ` X ) ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ( 2nd ` Y ) ) )
42	41	fveq1d 6193	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) = ( ( ( 2nd ` X ) ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) )
43		eqid 2622	. . . . . . 7 |- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
44	2, 3, 4, 43, 9, 5	dicelval1sta 36476	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ X e. ( I ` Q ) ) -> ( 1st ` X ) = ( ( 2nd ` X ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) )
45	44	3adant3r 1323	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( 1st ` X ) = ( ( 2nd ` X ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) )
46	2, 3, 4, 43, 9, 5	dicelval1sta 36476	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Y e. ( I ` Q ) ) -> ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) )
47	46	3adant3l 1322	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) )
48	45, 47	coeq12d 5286	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( 1st ` X ) o. ( 1st ` Y ) ) = ( ( ( 2nd ` X ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) o. ( ( 2nd ` Y ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) ) )
49	38, 42, 48	3eqtr4rd 2667	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( 1st ` X ) o. ( 1st ` Y ) ) = ( ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) )
50	4, 9, 10, 36	tendoplcl 36069	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( 2nd ` X ) e. ( ( TEndo ` K ) ` W ) /\ ( 2nd ` Y ) e. ( ( TEndo ` K ) ` W ) ) -> ( ( 2nd ` X ) ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ( 2nd ` Y ) ) e. ( ( TEndo ` K ) ` W ) )
51	1, 26, 28, 50	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( 2nd ` X ) ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ( 2nd ` Y ) ) e. ( ( TEndo ` K ) ` W ) )
52	41, 51	eqeltrd 2701	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) e. ( ( TEndo ` K ) ` W ) )
53		fvex 6201	. . . . . 6 |- ( 1st ` X ) e. _V
54		fvex 6201	. . . . . 6 |- ( 1st ` Y ) e. _V
55	53, 54	coex 7118	. . . . 5 |- ( ( 1st ` X ) o. ( 1st ` Y ) ) e. _V
56		ovex 6678	. . . . 5 |- ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) e. _V
57	2, 3, 4, 43, 9, 10, 5, 55, 56	dicopelval 36466	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. ( ( 1st ` X ) o. ( 1st ` Y ) ) , ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) >. e. ( I ` Q ) <-> ( ( ( 1st ` X ) o. ( 1st ` Y ) ) = ( ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) /\ ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) e. ( ( TEndo ` K ) ` W ) ) ) )
58	57	3adant3 1081	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( <. ( ( 1st ` X ) o. ( 1st ` Y ) ) , ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) >. e. ( I ` Q ) <-> ( ( ( 1st ` X ) o. ( 1st ` Y ) ) = ( ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) /\ ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) e. ( ( TEndo ` K ) ` W ) ) ) )
59	49, 52, 58	mpbir2and 957	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> <. ( ( 1st ` X ) o. ( 1st ` Y ) ) , ( ( 2nd ` X ) ( +g ` (Scalar ` U ) ) ( 2nd ` Y ) ) >. e. ( I ` Q ) )
60	24, 59	eqeltrd 2701	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( X .+ Y ) e. ( I ` Q ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   lecple 15948   occoc 15949   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040   DVecHcdvh 36367   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-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  df-dic 36462

This theorem is referenced by:  diclss  36482