Theorem metustexhalf 22361

Description: For any element A of the filter base generated by the metric D, the half element (corresponding to half the distance) is also in this base. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Hypothesis

Ref	Expression
metust.1	|- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )

Assertion

Ref	Expression
metustexhalf	|- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) -> E. v e. F ( v o. v ) C_ A )

Distinct variable groups:   D, a   X, a   A, a   F, a, v   v, A   v, D   v, F   v, X

Proof of Theorem metustexhalf

Dummy variables b p q r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp-4r 807	. . . 4 |- ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> D e. (PsMet ` X ) )
2		simplr 792	. . . . . 6 |- ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> a e. RR+ )
3	2	rphalfcld 11884	. . . . 5 |- ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( a / 2 ) e. RR+ )
4		eqidd 2623	. . . . 5 |- ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( `' D " ( 0 [,) ( a / 2 ) ) ) = ( `' D " ( 0 [,) ( a / 2 ) ) ) )
5		oveq2 6658	. . . . . . . 8 |- ( b = ( a / 2 ) -> ( 0 [,) b ) = ( 0 [,) ( a / 2 ) ) )
6	5	imaeq2d 5466	. . . . . . 7 |- ( b = ( a / 2 ) -> ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) ( a / 2 ) ) ) )
7	6	eqeq2d 2632	. . . . . 6 |- ( b = ( a / 2 ) -> ( ( `' D " ( 0 [,) ( a / 2 ) ) ) = ( `' D " ( 0 [,) b ) ) <-> ( `' D " ( 0 [,) ( a / 2 ) ) ) = ( `' D " ( 0 [,) ( a / 2 ) ) ) ) )
8	7	rspcev 3309	. . . . 5 |- ( ( ( a / 2 ) e. RR+ /\ ( `' D " ( 0 [,) ( a / 2 ) ) ) = ( `' D " ( 0 [,) ( a / 2 ) ) ) ) -> E. b e. RR+ ( `' D " ( 0 [,) ( a / 2 ) ) ) = ( `' D " ( 0 [,) b ) ) )
9	3, 4, 8	syl2anc 693	. . . 4 |- ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> E. b e. RR+ ( `' D " ( 0 [,) ( a / 2 ) ) ) = ( `' D " ( 0 [,) b ) ) )
10		metust.1	. . . . . . 7 |- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
11		oveq2 6658	. . . . . . . . . 10 |- ( a = b -> ( 0 [,) a ) = ( 0 [,) b ) )
12	11	imaeq2d 5466	. . . . . . . . 9 |- ( a = b -> ( `' D " ( 0 [,) a ) ) = ( `' D " ( 0 [,) b ) ) )
13	12	cbvmptv 4750	. . . . . . . 8 |- ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) = ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) )
14	13	rneqi 5352	. . . . . . 7 |- ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) = ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) )
15	10, 14	eqtri 2644	. . . . . 6 |- F = ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) )
16	15	metustel 22355	. . . . 5 |- ( D e. (PsMet ` X ) -> ( ( `' D " ( 0 [,) ( a / 2 ) ) ) e. F <-> E. b e. RR+ ( `' D " ( 0 [,) ( a / 2 ) ) ) = ( `' D " ( 0 [,) b ) ) ) )
17	16	biimpar 502	. . . 4 |- ( ( D e. (PsMet ` X ) /\ E. b e. RR+ ( `' D " ( 0 [,) ( a / 2 ) ) ) = ( `' D " ( 0 [,) b ) ) ) -> ( `' D " ( 0 [,) ( a / 2 ) ) ) e. F )
18	1, 9, 17	syl2anc 693	. . 3 |- ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( `' D " ( 0 [,) ( a / 2 ) ) ) e. F )
19		relco 5633	. . . . 5 |- Rel ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) )
20	19	a1i 11	. . . 4 |- ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> Rel ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) )
21		cossxp 5658	. . . . . . . . . 10 |- ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) C_ ( dom ( `' D " ( 0 [,) ( a / 2 ) ) ) X. ran ( `' D " ( 0 [,) ( a / 2 ) ) ) )
22		cnvimass 5485	. . . . . . . . . . . . . 14 |- ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ dom D
23		psmetf 22111	. . . . . . . . . . . . . . 15 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
24		fdm 6051	. . . . . . . . . . . . . . 15 |- ( D : ( X X. X ) --> RR* -> dom D = ( X X. X ) )
25	23, 24	syl 17	. . . . . . . . . . . . . 14 |- ( D e. (PsMet ` X ) -> dom D = ( X X. X ) )
26	22, 25	syl5sseq 3653	. . . . . . . . . . . . 13 |- ( D e. (PsMet ` X ) -> ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ ( X X. X ) )
27		dmss 5323	. . . . . . . . . . . . . 14 |- ( ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ ( X X. X ) -> dom ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ dom ( X X. X ) )
28		rnss 5354	. . . . . . . . . . . . . 14 |- ( ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ ( X X. X ) -> ran ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ ran ( X X. X ) )
29		xpss12 5225	. . . . . . . . . . . . . 14 |- ( ( dom ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ dom ( X X. X ) /\ ran ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ ran ( X X. X ) ) -> ( dom ( `' D " ( 0 [,) ( a / 2 ) ) ) X. ran ( `' D " ( 0 [,) ( a / 2 ) ) ) ) C_ ( dom ( X X. X ) X. ran ( X X. X ) ) )
30	27, 28, 29	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ ( X X. X ) -> ( dom ( `' D " ( 0 [,) ( a / 2 ) ) ) X. ran ( `' D " ( 0 [,) ( a / 2 ) ) ) ) C_ ( dom ( X X. X ) X. ran ( X X. X ) ) )
31	26, 30	syl 17	. . . . . . . . . . . 12 |- ( D e. (PsMet ` X ) -> ( dom ( `' D " ( 0 [,) ( a / 2 ) ) ) X. ran ( `' D " ( 0 [,) ( a / 2 ) ) ) ) C_ ( dom ( X X. X ) X. ran ( X X. X ) ) )
32	31	adantl 482	. . . . . . . . . . 11 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( dom ( `' D " ( 0 [,) ( a / 2 ) ) ) X. ran ( `' D " ( 0 [,) ( a / 2 ) ) ) ) C_ ( dom ( X X. X ) X. ran ( X X. X ) ) )
33		dmxp 5344	. . . . . . . . . . . . 13 |- ( X =/= (/) -> dom ( X X. X ) = X )
34		rnxp 5564	. . . . . . . . . . . . 13 |- ( X =/= (/) -> ran ( X X. X ) = X )
35	33, 34	xpeq12d 5140	. . . . . . . . . . . 12 |- ( X =/= (/) -> ( dom ( X X. X ) X. ran ( X X. X ) ) = ( X X. X ) )
36	35	adantr 481	. . . . . . . . . . 11 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( dom ( X X. X ) X. ran ( X X. X ) ) = ( X X. X ) )
37	32, 36	sseqtrd 3641	. . . . . . . . . 10 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( dom ( `' D " ( 0 [,) ( a / 2 ) ) ) X. ran ( `' D " ( 0 [,) ( a / 2 ) ) ) ) C_ ( X X. X ) )
38	21, 37	syl5ss 3614	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) C_ ( X X. X ) )
39	38	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) C_ ( X X. X ) )
40	39	sselda 3603	. . . . . . 7 |- ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) -> <. p , q >. e. ( X X. X ) )
41		opelxp 5146	. . . . . . 7 |- ( <. p , q >. e. ( X X. X ) <-> ( p e. X /\ q e. X ) )
42	40, 41	sylib 208	. . . . . 6 |- ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) -> ( p e. X /\ q e. X ) )
43		simpll 790	. . . . . . 7 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ ( p e. X /\ q e. X ) ) -> ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) )
44		simprl 794	. . . . . . 7 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ ( p e. X /\ q e. X ) ) -> p e. X )
45		simprr 796	. . . . . . 7 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ ( p e. X /\ q e. X ) ) -> q e. X )
46		simplr 792	. . . . . . 7 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ ( p e. X /\ q e. X ) ) -> <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) )
47		simplll 798	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) )
48	47	simp1d 1073	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) )
49	48, 1	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> D e. (PsMet ` X ) )
50	48, 2	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> a e. RR+ )
51	49, 50	jca 554	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( D e. (PsMet ` X ) /\ a e. RR+ ) )
52	47	simp2d 1074	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> p e. X )
53	47	simp3d 1075	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> q e. X )
54	51, 52, 53	3jca 1242	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) )
55		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> r e. X )
56		simprl 794	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> p ( `' D " ( 0 [,) ( a / 2 ) ) ) r )
57		simprr 796	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> r ( `' D " ( 0 [,) ( a / 2 ) ) ) q )
58		simpll 790	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) )
59	58	simp1d 1073	. . . . . . . . . . . . . 14 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( D e. (PsMet ` X ) /\ a e. RR+ ) )
60	59	simpld 475	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> D e. (PsMet ` X ) )
61		ffun 6048	. . . . . . . . . . . . . 14 |- ( D : ( X X. X ) --> RR* -> Fun D )
62	23, 61	syl 17	. . . . . . . . . . . . 13 |- ( D e. (PsMet ` X ) -> Fun D )
63	60, 62	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> Fun D )
64	58	simp2d 1074	. . . . . . . . . . . . . 14 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> p e. X )
65	58	simp3d 1075	. . . . . . . . . . . . . 14 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> q e. X )
66	64, 65, 41	sylanbrc 698	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> <. p , q >. e. ( X X. X ) )
67	60, 25	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> dom D = ( X X. X ) )
68	66, 67	eleqtrrd 2704	. . . . . . . . . . . 12 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> <. p , q >. e. dom D )
69		0xr 10086	. . . . . . . . . . . . . 14 |- 0 e. RR*
70	69	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> 0 e. RR* )
71	59	simprd 479	. . . . . . . . . . . . . 14 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> a e. RR+ )
72	71	rpxrd 11873	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> a e. RR* )
73	60, 23	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> D : ( X X. X ) --> RR* )
74	73, 66	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( D ` <. p , q >. ) e. RR* )
75		psmetge0 22117	. . . . . . . . . . . . . . 15 |- ( ( D e. (PsMet ` X ) /\ p e. X /\ q e. X ) -> 0 <_ ( p D q ) )
76	60, 64, 65, 75	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> 0 <_ ( p D q ) )
77		df-ov 6653	. . . . . . . . . . . . . 14 |- ( p D q ) = ( D ` <. p , q >. )
78	76, 77	syl6breq 4694	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> 0 <_ ( D ` <. p , q >. ) )
79	77, 74	syl5eqel 2705	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( p D q ) e. RR* )
80		0red 10041	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> 0 e. RR )
81	71	rpred 11872	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> a e. RR )
82	81	rehalfcld 11279	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( a / 2 ) e. RR )
83	82	rexrd 10089	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( a / 2 ) e. RR* )
84		df-ov 6653	. . . . . . . . . . . . . . . . . . . 20 |- ( p D r ) = ( D ` <. p , r >. )
85		simplr 792	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> r e. X )
86		opelxp 5146	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( <. p , r >. e. ( X X. X ) <-> ( p e. X /\ r e. X ) )
87	64, 85, 86	sylanbrc 698	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> <. p , r >. e. ( X X. X ) )
88	87, 67	eleqtrrd 2704	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> <. p , r >. e. dom D )
89		simprl 794	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> p ( `' D " ( 0 [,) ( a / 2 ) ) ) r )
90		df-br 4654	. . . . . . . . . . . . . . . . . . . . . 22 |- ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r <-> <. p , r >. e. ( `' D " ( 0 [,) ( a / 2 ) ) ) )
91	89, 90	sylib 208	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> <. p , r >. e. ( `' D " ( 0 [,) ( a / 2 ) ) ) )
92		fvimacnv 6332	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( Fun D /\ <. p , r >. e. dom D ) -> ( ( D ` <. p , r >. ) e. ( 0 [,) ( a / 2 ) ) <-> <. p , r >. e. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) )
93	92	biimpar 502	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( Fun D /\ <. p , r >. e. dom D ) /\ <. p , r >. e. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) -> ( D ` <. p , r >. ) e. ( 0 [,) ( a / 2 ) ) )
94	63, 88, 91, 93	syl21anc 1325	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( D ` <. p , r >. ) e. ( 0 [,) ( a / 2 ) ) )
95	84, 94	syl5eqel 2705	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( p D r ) e. ( 0 [,) ( a / 2 ) ) )
96		elico2 12237	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( 0 e. RR /\ ( a / 2 ) e. RR* ) -> ( ( p D r ) e. ( 0 [,) ( a / 2 ) ) <-> ( ( p D r ) e. RR /\ 0 <_ ( p D r ) /\ ( p D r ) < ( a / 2 ) ) ) )
97	96	biimpa 501	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( 0 e. RR /\ ( a / 2 ) e. RR* ) /\ ( p D r ) e. ( 0 [,) ( a / 2 ) ) ) -> ( ( p D r ) e. RR /\ 0 <_ ( p D r ) /\ ( p D r ) < ( a / 2 ) ) )
98	97	simp1d 1073	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( 0 e. RR /\ ( a / 2 ) e. RR* ) /\ ( p D r ) e. ( 0 [,) ( a / 2 ) ) ) -> ( p D r ) e. RR )
99	80, 83, 95, 98	syl21anc 1325	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( p D r ) e. RR )
100		df-ov 6653	. . . . . . . . . . . . . . . . . . . 20 |- ( r D q ) = ( D ` <. r , q >. )
101		opelxp 5146	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( <. r , q >. e. ( X X. X ) <-> ( r e. X /\ q e. X ) )
102	85, 65, 101	sylanbrc 698	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> <. r , q >. e. ( X X. X ) )
103	102, 67	eleqtrrd 2704	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> <. r , q >. e. dom D )
104		simprr 796	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> r ( `' D " ( 0 [,) ( a / 2 ) ) ) q )
105		df-br 4654	. . . . . . . . . . . . . . . . . . . . . 22 |- ( r ( `' D " ( 0 [,) ( a / 2 ) ) ) q <-> <. r , q >. e. ( `' D " ( 0 [,) ( a / 2 ) ) ) )
106	104, 105	sylib 208	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> <. r , q >. e. ( `' D " ( 0 [,) ( a / 2 ) ) ) )
107		fvimacnv 6332	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( Fun D /\ <. r , q >. e. dom D ) -> ( ( D ` <. r , q >. ) e. ( 0 [,) ( a / 2 ) ) <-> <. r , q >. e. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) )
108	107	biimpar 502	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( Fun D /\ <. r , q >. e. dom D ) /\ <. r , q >. e. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) -> ( D ` <. r , q >. ) e. ( 0 [,) ( a / 2 ) ) )
109	63, 103, 106, 108	syl21anc 1325	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( D ` <. r , q >. ) e. ( 0 [,) ( a / 2 ) ) )
110	100, 109	syl5eqel 2705	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( r D q ) e. ( 0 [,) ( a / 2 ) ) )
111		elico2 12237	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( 0 e. RR /\ ( a / 2 ) e. RR* ) -> ( ( r D q ) e. ( 0 [,) ( a / 2 ) ) <-> ( ( r D q ) e. RR /\ 0 <_ ( r D q ) /\ ( r D q ) < ( a / 2 ) ) ) )
112	111	biimpa 501	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( 0 e. RR /\ ( a / 2 ) e. RR* ) /\ ( r D q ) e. ( 0 [,) ( a / 2 ) ) ) -> ( ( r D q ) e. RR /\ 0 <_ ( r D q ) /\ ( r D q ) < ( a / 2 ) ) )
113	112	simp1d 1073	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( 0 e. RR /\ ( a / 2 ) e. RR* ) /\ ( r D q ) e. ( 0 [,) ( a / 2 ) ) ) -> ( r D q ) e. RR )
114	80, 83, 110, 113	syl21anc 1325	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( r D q ) e. RR )
115		rexadd 12063	. . . . . . . . . . . . . . . . . 18 |- ( ( ( p D r ) e. RR /\ ( r D q ) e. RR ) -> ( ( p D r ) +e ( r D q ) ) = ( ( p D r ) + ( r D q ) ) )
116	99, 114, 115	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( ( p D r ) +e ( r D q ) ) = ( ( p D r ) + ( r D q ) ) )
117	99, 114	readdcld 10069	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( ( p D r ) + ( r D q ) ) e. RR )
118	116, 117	eqeltrd 2701	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( ( p D r ) +e ( r D q ) ) e. RR )
119	118	rexrd 10089	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( ( p D r ) +e ( r D q ) ) e. RR* )
120		psmettri 22116	. . . . . . . . . . . . . . . 16 |- ( ( D e. (PsMet ` X ) /\ ( p e. X /\ q e. X /\ r e. X ) ) -> ( p D q ) <_ ( ( p D r ) +e ( r D q ) ) )
121	60, 64, 65, 85, 120	syl13anc 1328	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( p D q ) <_ ( ( p D r ) +e ( r D q ) ) )
122	97	simp3d 1075	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 0 e. RR /\ ( a / 2 ) e. RR* ) /\ ( p D r ) e. ( 0 [,) ( a / 2 ) ) ) -> ( p D r ) < ( a / 2 ) )
123	80, 83, 95, 122	syl21anc 1325	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( p D r ) < ( a / 2 ) )
124	112	simp3d 1075	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 0 e. RR /\ ( a / 2 ) e. RR* ) /\ ( r D q ) e. ( 0 [,) ( a / 2 ) ) ) -> ( r D q ) < ( a / 2 ) )
125	80, 83, 110, 124	syl21anc 1325	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( r D q ) < ( a / 2 ) )
126	99, 114, 81, 123, 125	lt2halvesd 11280	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( ( p D r ) + ( r D q ) ) < a )
127	116, 126	eqbrtrd 4675	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( ( p D r ) +e ( r D q ) ) < a )
128	79, 119, 72, 121, 127	xrlelttrd 11991	. . . . . . . . . . . . . 14 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( p D q ) < a )
129	77, 128	syl5eqbrr 4689	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( D ` <. p , q >. ) < a )
130		elico1 12218	. . . . . . . . . . . . . 14 |- ( ( 0 e. RR* /\ a e. RR* ) -> ( ( D ` <. p , q >. ) e. ( 0 [,) a ) <-> ( ( D ` <. p , q >. ) e. RR* /\ 0 <_ ( D ` <. p , q >. ) /\ ( D ` <. p , q >. ) < a ) ) )
131	130	biimpar 502	. . . . . . . . . . . . 13 |- ( ( ( 0 e. RR* /\ a e. RR* ) /\ ( ( D ` <. p , q >. ) e. RR* /\ 0 <_ ( D ` <. p , q >. ) /\ ( D ` <. p , q >. ) < a ) ) -> ( D ` <. p , q >. ) e. ( 0 [,) a ) )
132	70, 72, 74, 78, 129, 131	syl23anc 1333	. . . . . . . . . . . 12 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( D ` <. p , q >. ) e. ( 0 [,) a ) )
133		fvimacnv 6332	. . . . . . . . . . . . . 14 |- ( ( Fun D /\ <. p , q >. e. dom D ) -> ( ( D ` <. p , q >. ) e. ( 0 [,) a ) <-> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) )
134	133	biimpa 501	. . . . . . . . . . . . 13 |- ( ( ( Fun D /\ <. p , q >. e. dom D ) /\ ( D ` <. p , q >. ) e. ( 0 [,) a ) ) -> <. p , q >. e. ( `' D " ( 0 [,) a ) ) )
135		df-br 4654	. . . . . . . . . . . . 13 |- ( p ( `' D " ( 0 [,) a ) ) q <-> <. p , q >. e. ( `' D " ( 0 [,) a ) ) )
136	134, 135	sylibr 224	. . . . . . . . . . . 12 |- ( ( ( Fun D /\ <. p , q >. e. dom D ) /\ ( D ` <. p , q >. ) e. ( 0 [,) a ) ) -> p ( `' D " ( 0 [,) a ) ) q )
137	63, 68, 132, 136	syl21anc 1325	. . . . . . . . . . 11 |- ( ( ( ( ( D e. (PsMet ` X ) /\ a e. RR+ ) /\ p e. X /\ q e. X ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> p ( `' D " ( 0 [,) a ) ) q )
138	54, 55, 56, 57, 137	syl22anc 1327	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> p ( `' D " ( 0 [,) a ) ) q )
139	48	simprd 479	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> A = ( `' D " ( 0 [,) a ) ) )
140	139	breqd 4664	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> ( p A q <-> p ( `' D " ( 0 [,) a ) ) q ) )
141	138, 140	mpbird 247	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ r e. X ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> p A q )
142		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) -> <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) )
143		df-br 4654	. . . . . . . . . . . . 13 |- ( p ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) q <-> <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) )
144	142, 143	sylibr 224	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) -> p ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) q )
145		vex 3203	. . . . . . . . . . . . 13 |- p e. _V
146		vex 3203	. . . . . . . . . . . . 13 |- q e. _V
147	145, 146	brco 5292	. . . . . . . . . . . 12 |- ( p ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) q <-> E. r ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) )
148	144, 147	sylib 208	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) -> E. r ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) )
149	26	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ ( X X. X ) )
150	149, 28	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ ran ( X X. X ) )
151	34	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( X X. X ) = X )
152	150, 151	sseqtrd 3641	. . . . . . . . . . . . . . . . . . . 20 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ X )
153	152	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ p ( `' D " ( 0 [,) ( a / 2 ) ) ) r ) -> ran ( `' D " ( 0 [,) ( a / 2 ) ) ) C_ X )
154		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 |- r e. _V
155	145, 154	brelrn 5356	. . . . . . . . . . . . . . . . . . . 20 |- ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r -> r e. ran ( `' D " ( 0 [,) ( a / 2 ) ) ) )
156	155	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ p ( `' D " ( 0 [,) ( a / 2 ) ) ) r ) -> r e. ran ( `' D " ( 0 [,) ( a / 2 ) ) ) )
157	153, 156	sseldd 3604	. . . . . . . . . . . . . . . . . 18 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ p ( `' D " ( 0 [,) ( a / 2 ) ) ) r ) -> r e. X )
158	157	adantrr 753	. . . . . . . . . . . . . . . . 17 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) -> r e. X )
159	158	ex 450	. . . . . . . . . . . . . . . 16 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) -> r e. X ) )
160	159	ancrd 577	. . . . . . . . . . . . . . 15 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) -> ( r e. X /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) ) )
161	160	eximdv 1846	. . . . . . . . . . . . . 14 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( E. r ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) -> E. r ( r e. X /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) ) )
162	161	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( E. r ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) -> E. r ( r e. X /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) ) )
163	162	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) -> ( E. r ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) -> E. r ( r e. X /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) ) )
164	163	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) -> ( E. r ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) -> E. r ( r e. X /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) ) )
165	148, 164	mpd 15	. . . . . . . . . 10 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) -> E. r ( r e. X /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) )
166		df-rex 2918	. . . . . . . . . 10 |- ( E. r e. X ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) <-> E. r ( r e. X /\ ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) ) )
167	165, 166	sylibr 224	. . . . . . . . 9 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) -> E. r e. X ( p ( `' D " ( 0 [,) ( a / 2 ) ) ) r /\ r ( `' D " ( 0 [,) ( a / 2 ) ) ) q ) )
168	141, 167	r19.29a 3078	. . . . . . . 8 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) -> p A q )
169		df-br 4654	. . . . . . . 8 |- ( p A q <-> <. p , q >. e. A )
170	168, 169	sylib 208	. . . . . . 7 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ p e. X /\ q e. X ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) -> <. p , q >. e. A )
171	43, 44, 45, 46, 170	syl31anc 1329	. . . . . 6 |- ( ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) /\ ( p e. X /\ q e. X ) ) -> <. p , q >. e. A )
172	42, 171	mpdan 702	. . . . 5 |- ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) /\ <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) ) -> <. p , q >. e. A )
173	172	ex 450	. . . 4 |- ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( <. p , q >. e. ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) -> <. p , q >. e. A ) )
174	20, 173	relssdv 5212	. . 3 |- ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) C_ A )
175		id 22	. . . . . 6 |- ( v = ( `' D " ( 0 [,) ( a / 2 ) ) ) -> v = ( `' D " ( 0 [,) ( a / 2 ) ) ) )
176	175, 175	coeq12d 5286	. . . . 5 |- ( v = ( `' D " ( 0 [,) ( a / 2 ) ) ) -> ( v o. v ) = ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) )
177	176	sseq1d 3632	. . . 4 |- ( v = ( `' D " ( 0 [,) ( a / 2 ) ) ) -> ( ( v o. v ) C_ A <-> ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) C_ A ) )
178	177	rspcev 3309	. . 3 |- ( ( ( `' D " ( 0 [,) ( a / 2 ) ) ) e. F /\ ( ( `' D " ( 0 [,) ( a / 2 ) ) ) o. ( `' D " ( 0 [,) ( a / 2 ) ) ) ) C_ A ) -> E. v e. F ( v o. v ) C_ A )
179	18, 174, 178	syl2anc 693	. 2 |- ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> E. v e. F ( v o. v ) C_ A )
180	10	metustel 22355	. . . 4 |- ( D e. (PsMet ` X ) -> ( A e. F <-> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) ) )
181	180	adantl 482	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( A e. F <-> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) ) )
182	181	biimpa 501	. 2 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) -> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) )
183	179, 182	r19.29a 3078	1 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ A e. F ) -> E. v e. F ( v o. v ) C_ A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   (/)c0 3915   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   Rel wrel 5119   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   / cdiv 10684   2c2 11070   RR+crp 11832   +ecxad 11944   [,)cico 12177  PsMetcpsmet 19730

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-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

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-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-psmet 19738

This theorem is referenced by:  metust  22363