Theorem txmetcnp 22352

Description: Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
metcn.2	|- J = ( MetOpen ` C )
metcn.4	|- K = ( MetOpen ` D )
txmetcnp.4	|- L = ( MetOpen ` E )

Assertion

Ref	Expression
txmetcnp	|- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( F e. ( ( ( J tX K ) CnP L ) ` <. A , B >. ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) ) )

Distinct variable groups:   v, u, w, z, F   u, J, v, w, z   u, K, v, w, z   u, X, v, w, z   u, Y, v, w, z   u, Z, v, w, z   u, A, v, w, z   u, C, v, w, z   u, D, v, w, z   u, B, v, w, z   u, E, v, w, z   w, L, z

Allowed substitution hints:   L( v, u)

Proof of Theorem txmetcnp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) = ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) )
2		simpl1 1064	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> C e. ( *Met ` X ) )
3		simpl2 1065	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> D e. ( *Met ` Y ) )
4	1, 2, 3	tmsxps 22341	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) e. ( *Met ` ( X X. Y ) ) )
5		simpl3 1066	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> E e. ( *Met ` Z ) )
6		opelxpi 5148	. . . 4 |- ( ( A e. X /\ B e. Y ) -> <. A , B >. e. ( X X. Y ) )
7	6	adantl 482	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> <. A , B >. e. ( X X. Y ) )
8		eqid 2622	. . . 4 |- ( MetOpen ` ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) ) = ( MetOpen ` ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) )
9		txmetcnp.4	. . . 4 |- L = ( MetOpen ` E )
10	8, 9	metcnp 22346	. . 3 |- ( ( ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) e. ( *Met ` ( X X. Y ) ) /\ E e. ( *Met ` Z ) /\ <. A , B >. e. ( X X. Y ) ) -> ( F e. ( ( ( MetOpen ` ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) ) CnP L ) ` <. A , B >. ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. x e. ( X X. Y ) ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) x ) < w -> ( ( F ` <. A , B >. ) E ( F ` x ) ) < z ) ) ) )
11	4, 5, 7, 10	syl3anc 1326	. 2 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( F e. ( ( ( MetOpen ` ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) ) CnP L ) ` <. A , B >. ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. x e. ( X X. Y ) ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) x ) < w -> ( ( F ` <. A , B >. ) E ( F ` x ) ) < z ) ) ) )
12		metcn.2	. . . . . 6 |- J = ( MetOpen ` C )
13		metcn.4	. . . . . 6 |- K = ( MetOpen ` D )
14	1, 2, 3, 12, 13, 8	tmsxpsmopn 22342	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( MetOpen ` ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) ) = ( J tX K ) )
15	14	oveq1d 6665	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( ( MetOpen ` ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) ) CnP L ) = ( ( J tX K ) CnP L ) )
16	15	fveq1d 6193	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( ( ( MetOpen ` ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) ) CnP L ) ` <. A , B >. ) = ( ( ( J tX K ) CnP L ) ` <. A , B >. ) )
17	16	eleq2d 2687	. 2 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( F e. ( ( ( MetOpen ` ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) ) CnP L ) ` <. A , B >. ) <-> F e. ( ( ( J tX K ) CnP L ) ` <. A , B >. ) ) )
18		oveq2 6658	. . . . . . . . 9 |- ( x = <. u , v >. -> ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) x ) = ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) <. u , v >. ) )
19	18	breq1d 4663	. . . . . . . 8 |- ( x = <. u , v >. -> ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) x ) < w <-> ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) <. u , v >. ) < w ) )
20		df-ov 6653	. . . . . . . . . . 11 |- ( A F B ) = ( F ` <. A , B >. )
21	20	oveq1i 6660	. . . . . . . . . 10 |- ( ( A F B ) E ( F ` x ) ) = ( ( F ` <. A , B >. ) E ( F ` x ) )
22		fveq2 6191	. . . . . . . . . . . 12 |- ( x = <. u , v >. -> ( F ` x ) = ( F ` <. u , v >. ) )
23		df-ov 6653	. . . . . . . . . . . 12 |- ( u F v ) = ( F ` <. u , v >. )
24	22, 23	syl6eqr 2674	. . . . . . . . . . 11 |- ( x = <. u , v >. -> ( F ` x ) = ( u F v ) )
25	24	oveq2d 6666	. . . . . . . . . 10 |- ( x = <. u , v >. -> ( ( A F B ) E ( F ` x ) ) = ( ( A F B ) E ( u F v ) ) )
26	21, 25	syl5eqr 2670	. . . . . . . . 9 |- ( x = <. u , v >. -> ( ( F ` <. A , B >. ) E ( F ` x ) ) = ( ( A F B ) E ( u F v ) ) )
27	26	breq1d 4663	. . . . . . . 8 |- ( x = <. u , v >. -> ( ( ( F ` <. A , B >. ) E ( F ` x ) ) < z <-> ( ( A F B ) E ( u F v ) ) < z ) )
28	19, 27	imbi12d 334	. . . . . . 7 |- ( x = <. u , v >. -> ( ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) x ) < w -> ( ( F ` <. A , B >. ) E ( F ` x ) ) < z ) <-> ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) <. u , v >. ) < w -> ( ( A F B ) E ( u F v ) ) < z ) ) )
29	28	ralxp 5263	. . . . . 6 |- ( A. x e. ( X X. Y ) ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) x ) < w -> ( ( F ` <. A , B >. ) E ( F ` x ) ) < z ) <-> A. u e. X A. v e. Y ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) <. u , v >. ) < w -> ( ( A F B ) E ( u F v ) ) < z ) )
30	2	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> C e. ( *Met ` X ) )
31	3	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> D e. ( *Met ` Y ) )
32		simpllr 799	. . . . . . . . . . . . 13 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> ( A e. X /\ B e. Y ) )
33	32	simpld 475	. . . . . . . . . . . 12 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> A e. X )
34	32	simprd 479	. . . . . . . . . . . 12 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> B e. Y )
35		simprrl 804	. . . . . . . . . . . 12 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> u e. X )
36		simprrr 805	. . . . . . . . . . . 12 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> v e. Y )
37	1, 30, 31, 33, 34, 35, 36	tmsxpsval2 22344	. . . . . . . . . . 11 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) <. u , v >. ) = if ( ( A C u ) <_ ( B D v ) , ( B D v ) , ( A C u ) ) )
38	37	breq1d 4663	. . . . . . . . . 10 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) <. u , v >. ) < w <-> if ( ( A C u ) <_ ( B D v ) , ( B D v ) , ( A C u ) ) < w ) )
39		xmetcl 22136	. . . . . . . . . . . 12 |- ( ( C e. ( *Met ` X ) /\ A e. X /\ u e. X ) -> ( A C u ) e. RR* )
40	30, 33, 35, 39	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> ( A C u ) e. RR* )
41		xmetcl 22136	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` Y ) /\ B e. Y /\ v e. Y ) -> ( B D v ) e. RR* )
42	31, 34, 36, 41	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> ( B D v ) e. RR* )
43		rpxr 11840	. . . . . . . . . . . 12 |- ( w e. RR+ -> w e. RR* )
44	43	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> w e. RR* )
45		xrmaxlt 12012	. . . . . . . . . . 11 |- ( ( ( A C u ) e. RR* /\ ( B D v ) e. RR* /\ w e. RR* ) -> ( if ( ( A C u ) <_ ( B D v ) , ( B D v ) , ( A C u ) ) < w <-> ( ( A C u ) < w /\ ( B D v ) < w ) ) )
46	40, 42, 44, 45	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> ( if ( ( A C u ) <_ ( B D v ) , ( B D v ) , ( A C u ) ) < w <-> ( ( A C u ) < w /\ ( B D v ) < w ) ) )
47	38, 46	bitrd 268	. . . . . . . . 9 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) <. u , v >. ) < w <-> ( ( A C u ) < w /\ ( B D v ) < w ) ) )
48	47	imbi1d 331	. . . . . . . 8 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ ( w e. RR+ /\ ( u e. X /\ v e. Y ) ) ) -> ( ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) <. u , v >. ) < w -> ( ( A F B ) E ( u F v ) ) < z ) <-> ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) )
49	48	anassrs 680	. . . . . . 7 |- ( ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ w e. RR+ ) /\ ( u e. X /\ v e. Y ) ) -> ( ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) <. u , v >. ) < w -> ( ( A F B ) E ( u F v ) ) < z ) <-> ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) )
50	49	2ralbidva 2988	. . . . . 6 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ w e. RR+ ) -> ( A. u e. X A. v e. Y ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) <. u , v >. ) < w -> ( ( A F B ) E ( u F v ) ) < z ) <-> A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) )
51	29, 50	syl5bb 272	. . . . 5 |- ( ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) /\ w e. RR+ ) -> ( A. x e. ( X X. Y ) ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) x ) < w -> ( ( F ` <. A , B >. ) E ( F ` x ) ) < z ) <-> A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) )
52	51	rexbidva 3049	. . . 4 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) -> ( E. w e. RR+ A. x e. ( X X. Y ) ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) x ) < w -> ( ( F ` <. A , B >. ) E ( F ` x ) ) < z ) <-> E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) )
53	52	ralbidv 2986	. . 3 |- ( ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) /\ F : ( X X. Y ) --> Z ) -> ( A. z e. RR+ E. w e. RR+ A. x e. ( X X. Y ) ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) x ) < w -> ( ( F ` <. A , B >. ) E ( F ` x ) ) < z ) <-> A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) )
54	53	pm5.32da 673	. 2 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. x e. ( X X. Y ) ( ( <. A , B >. ( dist ` ( (toMetSp ` C ) X.s (toMetSp ` D ) ) ) x ) < w -> ( ( F ` <. A , B >. ) E ( F ` x ) ) < z ) ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) ) )
55	11, 17, 54	3bitr3d 298	1 |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( F e. ( ( ( J tX K ) CnP L ) ` <. A , B >. ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   ifcif 4086   <.cop 4183   class class class wbr 4653   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   RR*cxr 10073   < clt 10074   <_ cle 10075   RR+crp 11832   distcds 15950   X._s cxps 16166   *Metcxmt 19731   MetOpencmopn 19736   CnP ccnp 21029   tX ctx 21363  toMetSpctmt 22124

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-inf2 8538  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-pre-sup 10014

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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-tms 22127

This theorem is referenced by:  txmetcn  22353  cxpcn3  24489