Theorem metxp 9111

Description: The direct product of two metric spaces. Definition 14-1.5 of [Gleason] p. 225.

Hypotheses

Ref	Expression
metxp.1	|- X = dom dom B
metxp.3	|- Y = dom dom C
metxp.5	|- B e. Met
metxp.6	|- C e. Met
metxp.7	|- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}

Assertion

Ref	Expression
metxp	|- D e. Met

Distinct variable groups:   x,y,z,B   x,C,y,z   x,X,y,z   x,Y,y,z

Proof of Theorem metxp

Step	Hyp	Ref	Expression
1		metxp.1	. . . 4 |- X = dom dom B
2		metxp.5	. . . . . 6 |- B e. Met
3		dmexg 4206	. . . . . 6 |- (B e. Met -> dom B e. _V)
4	2, 3	ax-mp 7	. . . . 5 |- dom B e. _V
5	4	dmex 4208	. . . 4 |- dom dom B e. _V
6	1, 5	eqeltri 1967	. . 3 |- X e. _V
7		metxp.3	. . . 4 |- Y = dom dom C
8		metxp.6	. . . . . 6 |- C e. Met
9		dmexg 4206	. . . . . 6 |- (C e. Met -> dom C e. _V)
10	8, 9	ax-mp 7	. . . . 5 |- dom C e. _V
11	10	dmex 4208	. . . 4 |- dom dom C e. _V
12	7, 11	eqeltri 1967	. . 3 |- Y e. _V
13	6, 12	xpex 4096	. 2 |- (X X. Y) e. _V
14		ffnoprv 4943	. . 3 |- (D:((X X. Y) X. (X X. Y))-->RR <-> (D Fn ((X X. Y) X. (X X. Y)) /\ A.w e. (X X. Y)A.v e. (X X. Y)(wDv) e. RR))
15		ltso 6681	. . . . 5 |- < Or RR
16	15	supex 5667	. . . 4 |- sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ) e. _V
17		metxp.7	. . . 4 |- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}
18	16, 17	fnoprab2 5064	. . 3 |- D Fn ((X X. Y) X. (X X. Y))
19	1, 7, 2, 8, 17	metxpcl 9109	. . . 4 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (wDv) e. RR)
20	19	rgen2a 2160	. . 3 |- A.w e. (X X. Y)A.v e. (X X. Y)(wDv) e. RR
21	14, 18, 20	mpbir2an 800	. 2 |- D:((X X. Y) X. (X X. Y))-->RR
22		eqid 1884	. . . . 5 |- (1st` w) = (1st` w)
23		eqid 1884	. . . . 5 |- (2nd` w) = (2nd` w)
24		eqid 1884	. . . . 5 |- (1st` v) = (1st` v)
25		eqid 1884	. . . . 5 |- (2nd` v) = (2nd` v)
26	1, 7, 2, 8, 17, 22, 23, 24, 25	metxpdval 9106	. . . 4 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (wDv) = if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))))
27	26	eqeq1d 1892	. . 3 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> ((wDv) = 0 <-> if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = 0))
28		iftrue 2989	. . . . . . . 8 |- (((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)) -> if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = ((1st` w)B(1st` v)))
29	28	eqeq1d 1892	. . . . . . 7 |- (((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)) -> (if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = 0 <-> ((1st` w)B(1st` v)) = 0))
30	29	adantl 424	. . . . . 6 |- (((w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v))) -> (if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = 0 <-> ((1st` w)B(1st` v)) = 0))
31		breq2 3342	. . . . . . . . . 10 |- (((1st` w)B(1st` v)) = 0 -> (((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)) <-> ((2nd` w)C(2nd` v)) < 0))
32	31	biimpac 462	. . . . . . . . 9 |- ((((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)) /\ ((1st` w)B(1st` v)) = 0) -> ((2nd` w)C(2nd` v)) < 0)
33	32	adantll 428	. . . . . . . 8 |- ((((w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v))) /\ ((1st` w)B(1st` v)) = 0) -> ((2nd` w)C(2nd` v)) < 0)
34	7	metge0 9096	. . . . . . . . . . . . 13 |- ((C e. Met /\ (2nd` w) e. Y /\ (2nd` v) e. Y) -> 0 <_ ((2nd` w)C(2nd` v)))
35	8, 34	mp3an1 1178	. . . . . . . . . . . 12 |- (((2nd` w) e. Y /\ (2nd` v) e. Y) -> 0 <_ ((2nd` w)C(2nd` v)))
36		lenlt 6679	. . . . . . . . . . . . 13 |- ((0 e. RR /\ ((2nd` w)C(2nd` v)) e. RR) -> (0 <_ ((2nd` w)C(2nd` v)) <-> -. ((2nd` w)C(2nd` v)) < 0))
37		0re 6603	. . . . . . . . . . . . 13 |- 0 e. RR
38	7	metcl 9088	. . . . . . . . . . . . . 14 |- ((C e. Met /\ (2nd` w) e. Y /\ (2nd` v) e. Y) -> ((2nd` w)C(2nd` v)) e. RR)
39	8, 38	mp3an1 1178	. . . . . . . . . . . . 13 |- (((2nd` w) e. Y /\ (2nd` v) e. Y) -> ((2nd` w)C(2nd` v)) e. RR)
40	36, 37, 39	sylancr 526	. . . . . . . . . . . 12 |- (((2nd` w) e. Y /\ (2nd` v) e. Y) -> (0 <_ ((2nd` w)C(2nd` v)) <-> -. ((2nd` w)C(2nd` v)) < 0))
41	35, 40	mpbid 212	. . . . . . . . . . 11 |- (((2nd` w) e. Y /\ (2nd` v) e. Y) -> -. ((2nd` w)C(2nd` v)) < 0)
42		elxp7 5042	. . . . . . . . . . . . 13 |- (w e. (X X. Y) <-> (w e. (_V X. _V) /\ ((1st` w) e. X /\ (2nd` w) e. Y)))
43	42	simprbi 353	. . . . . . . . . . . 12 |- (w e. (X X. Y) -> ((1st` w) e. X /\ (2nd` w) e. Y))
44	43	simprd 352	. . . . . . . . . . 11 |- (w e. (X X. Y) -> (2nd` w) e. Y)
45		elxp7 5042	. . . . . . . . . . . . 13 |- (v e. (X X. Y) <-> (v e. (_V X. _V) /\ ((1st` v) e. X /\ (2nd` v) e. Y)))
46	45	simprbi 353	. . . . . . . . . . . 12 |- (v e. (X X. Y) -> ((1st` v) e. X /\ (2nd` v) e. Y))
47	46	simprd 352	. . . . . . . . . . 11 |- (v e. (X X. Y) -> (2nd` v) e. Y)
48	41, 44, 47	syl2an 503	. . . . . . . . . 10 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> -. ((2nd` w)C(2nd` v)) < 0)
49	48	pm2.21d 94	. . . . . . . . 9 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (((2nd` w)C(2nd` v)) < 0 -> w = v))
50	49	ad2antrr 440	. . . . . . . 8 |- ((((w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v))) /\ ((1st` w)B(1st` v)) = 0) -> (((2nd` w)C(2nd` v)) < 0 -> w = v))
51	33, 50	mpd 29	. . . . . . 7 |- ((((w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v))) /\ ((1st` w)B(1st` v)) = 0) -> w = v)
52	51	ex 402	. . . . . 6 |- (((w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v))) -> (((1st` w)B(1st` v)) = 0 -> w = v))
53	30, 52	sylbid 220	. . . . 5 |- (((w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v))) -> (if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = 0 -> w = v))
54		iffalse 2991	. . . . . . . 8 |- (-. ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)) -> if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = ((2nd` w)C(2nd` v)))
55	54	eqeq1d 1892	. . . . . . 7 |- (-. ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)) -> (if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = 0 <-> ((2nd` w)C(2nd` v)) = 0))
56	55	adantl 424	. . . . . 6 |- (((w e. (X X. Y) /\ v e. (X X. Y)) /\ -. ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v))) -> (if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = 0 <-> ((2nd` w)C(2nd` v)) = 0))
57	2, 1, 22, 24	metxplem1 9103	. . . . . . . . 9 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` w)B(1st` v)) e. RR)
58	8, 7, 23, 25	metxplem2 9104	. . . . . . . . 9 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` w)C(2nd` v)) e. RR)
59		lenlt 6679	. . . . . . . . 9 |- ((((1st` w)B(1st` v)) e. RR /\ ((2nd` w)C(2nd` v)) e. RR) -> (((1st` w)B(1st` v)) <_ ((2nd` w)C(2nd` v)) <-> -. ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v))))
60	57, 58, 59	syl11anc 524	. . . . . . . 8 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (((1st` w)B(1st` v)) <_ ((2nd` w)C(2nd` v)) <-> -. ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v))))
61		breq2 3342	. . . . . . . . . . . 12 |- (((2nd` w)C(2nd` v)) = 0 -> (((1st` w)B(1st` v)) <_ ((2nd` w)C(2nd` v)) <-> ((1st` w)B(1st` v)) <_ 0))
62	1	metge0 9096	. . . . . . . . . . . . . . . 16 |- ((B e. Met /\ (1st` w) e. X /\ (1st` v) e. X) -> 0 <_ ((1st` w)B(1st` v)))
63	2, 62	mp3an1 1178	. . . . . . . . . . . . . . 15 |- (((1st` w) e. X /\ (1st` v) e. X) -> 0 <_ ((1st` w)B(1st` v)))
64	63	biantrud 795	. . . . . . . . . . . . . 14 |- (((1st` w) e. X /\ (1st` v) e. X) -> (((1st` w)B(1st` v)) <_ 0 <-> (((1st` w)B(1st` v)) <_ 0 /\ 0 <_ ((1st` w)B(1st` v)))))
65		letri3 6687	. . . . . . . . . . . . . . 15 |- ((((1st` w)B(1st` v)) e. RR /\ 0 e. RR) -> (((1st` w)B(1st` v)) = 0 <-> (((1st` w)B(1st` v)) <_ 0 /\ 0 <_ ((1st` w)B(1st` v)))))
66	1	metcl 9088	. . . . . . . . . . . . . . . 16 |- ((B e. Met /\ (1st` w) e. X /\ (1st` v) e. X) -> ((1st` w)B(1st` v)) e. RR)
67	2, 66	mp3an1 1178	. . . . . . . . . . . . . . 15 |- (((1st` w) e. X /\ (1st` v) e. X) -> ((1st` w)B(1st` v)) e. RR)
68	65, 67, 37	sylancl 525	. . . . . . . . . . . . . 14 |- (((1st` w) e. X /\ (1st` v) e. X) -> (((1st` w)B(1st` v)) = 0 <-> (((1st` w)B(1st` v)) <_ 0 /\ 0 <_ ((1st` w)B(1st` v)))))
69	64, 68	bitr4d 590	. . . . . . . . . . . . 13 |- (((1st` w) e. X /\ (1st` v) e. X) -> (((1st` w)B(1st` v)) <_ 0 <-> ((1st` w)B(1st` v)) = 0))
70	43	simplld 348	. . . . . . . . . . . . 13 |- (w e. (X X. Y) -> (1st` w) e. X)
71	46	simplld 348	. . . . . . . . . . . . 13 |- (v e. (X X. Y) -> (1st` v) e. X)
72	69, 70, 71	syl2an 503	. . . . . . . . . . . 12 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (((1st` w)B(1st` v)) <_ 0 <-> ((1st` w)B(1st` v)) = 0))
73	61, 72	sylan9bbr 600	. . . . . . . . . . 11 |- (((w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` w)C(2nd` v)) = 0) -> (((1st` w)B(1st` v)) <_ ((2nd` w)C(2nd` v)) <-> ((1st` w)B(1st` v)) = 0))
74	7	meteq0 9089	. . . . . . . . . . . . . . . . . 18 |- ((C e. Met /\ (2nd` w) e. Y /\ (2nd` v) e. Y) -> (((2nd` w)C(2nd` v)) = 0 <-> (2nd` w) = (2nd` v)))
75	8, 74	mp3an1 1178	. . . . . . . . . . . . . . . . 17 |- (((2nd` w) e. Y /\ (2nd` v) e. Y) -> (((2nd` w)C(2nd` v)) = 0 <-> (2nd` w) = (2nd` v)))
76	1	meteq0 9089	. . . . . . . . . . . . . . . . . 18 |- ((B e. Met /\ (1st` w) e. X /\ (1st` v) e. X) -> (((1st` w)B(1st` v)) = 0 <-> (1st` w) = (1st` v)))
77	2, 76	mp3an1 1178	. . . . . . . . . . . . . . . . 17 |- (((1st` w) e. X /\ (1st` v) e. X) -> (((1st` w)B(1st` v)) = 0 <-> (1st` w) = (1st` v)))
78	75, 77	bi2anan9r 695	. . . . . . . . . . . . . . . 16 |- ((((1st` w) e. X /\ (1st` v) e. X) /\ ((2nd` w) e. Y /\ (2nd` v) e. Y)) -> ((((2nd` w)C(2nd` v)) = 0 /\ ((1st` w)B(1st` v)) = 0) <-> ((2nd` w) = (2nd` v) /\ (1st` w) = (1st` v))))
79	78	an4s 566	. . . . . . . . . . . . . . 15 |- ((((1st` w) e. X /\ (2nd` w) e. Y) /\ ((1st` v) e. X /\ (2nd` v) e. Y)) -> ((((2nd` w)C(2nd` v)) = 0 /\ ((1st` w)B(1st` v)) = 0) <-> ((2nd` w) = (2nd` v) /\ (1st` w) = (1st` v))))
80	79, 43, 46	syl2an 503	. . . . . . . . . . . . . 14 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> ((((2nd` w)C(2nd` v)) = 0 /\ ((1st` w)B(1st` v)) = 0) <-> ((2nd` w) = (2nd` v) /\ (1st` w) = (1st` v))))
81		xpopth 5046	. . . . . . . . . . . . . . 15 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (((1st` w) = (1st` v) /\ (2nd` w) = (2nd` v)) <-> w = v))
82		ancom 482	. . . . . . . . . . . . . . 15 |- (((2nd` w) = (2nd` v) /\ (1st` w) = (1st` v)) <-> ((1st` w) = (1st` v) /\ (2nd` w) = (2nd` v)))
83	81, 82	syl5bb 591	. . . . . . . . . . . . . 14 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (((2nd` w) = (2nd` v) /\ (1st` w) = (1st` v)) <-> w = v))
84	80, 83	bitrd 587	. . . . . . . . . . . . 13 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> ((((2nd` w)C(2nd` v)) = 0 /\ ((1st` w)B(1st` v)) = 0) <-> w = v))
85	84	biimpd 170	. . . . . . . . . . . 12 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> ((((2nd` w)C(2nd` v)) = 0 /\ ((1st` w)B(1st` v)) = 0) -> w = v))
86	85	expdimp 406	. . . . . . . . . . 11 |- (((w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` w)C(2nd` v)) = 0) -> (((1st` w)B(1st` v)) = 0 -> w = v))
87	73, 86	sylbid 220	. . . . . . . . . 10 |- (((w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` w)C(2nd` v)) = 0) -> (((1st` w)B(1st` v)) <_ ((2nd` w)C(2nd` v)) -> w = v))
88	87	ex 402	. . . . . . . . 9 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (((2nd` w)C(2nd` v)) = 0 -> (((1st` w)B(1st` v)) <_ ((2nd` w)C(2nd` v)) -> w = v)))
89	88	com23 36	. . . . . . . 8 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (((1st` w)B(1st` v)) <_ ((2nd` w)C(2nd` v)) -> (((2nd` w)C(2nd` v)) = 0 -> w = v)))
90	60, 89	sylbird 222	. . . . . . 7 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (-. ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)) -> (((2nd` w)C(2nd` v)) = 0 -> w = v)))
91	90	imp 377	. . . . . 6 |- (((w e. (X X. Y) /\ v e. (X X. Y)) /\ -. ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v))) -> (((2nd` w)C(2nd` v)) = 0 -> w = v))
92	56, 91	sylbid 220	. . . . 5 |- (((w e. (X X. Y) /\ v e. (X X. Y)) /\ -. ((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v))) -> (if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = 0 -> w = v))
93	53, 92	pm2.61dan 535	. . . 4 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = 0 -> w = v))
94		ifeq12 2992	. . . . . . . . . 10 |- ((0 = ((1st` w)B(1st` v)) /\ 0 = ((2nd` w)C(2nd` v))) -> if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), 0, 0) = if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))))
95	1	met0 9092	. . . . . . . . . . . 12 |- ((B e. Met /\ (1st` w) e. X) -> ((1st` w)B(1st` w)) = 0)
96	2, 95	mpan 759	. . . . . . . . . . 11 |- ((1st` w) e. X -> ((1st` w)B(1st` w)) = 0)
97		fveq2 4681	. . . . . . . . . . . 12 |- (w = v -> (1st` w) = (1st` v))
98	97	opreq2d 4898	. . . . . . . . . . 11 |- (w = v -> ((1st` w)B(1st` w)) = ((1st` w)B(1st` v)))
99	96, 98	sylan9req 1950	. . . . . . . . . 10 |- (((1st` w) e. X /\ w = v) -> 0 = ((1st` w)B(1st` v)))
100	7	met0 9092	. . . . . . . . . . . 12 |- ((C e. Met /\ (2nd` w) e. Y) -> ((2nd` w)C(2nd` w)) = 0)
101	8, 100	mpan 759	. . . . . . . . . . 11 |- ((2nd` w) e. Y -> ((2nd` w)C(2nd` w)) = 0)
102		fveq2 4681	. . . . . . . . . . . 12 |- (w = v -> (2nd` w) = (2nd` v))
103	102	opreq2d 4898	. . . . . . . . . . 11 |- (w = v -> ((2nd` w)C(2nd` w)) = ((2nd` w)C(2nd` v)))
104	101, 103	sylan9req 1950	. . . . . . . . . 10 |- (((2nd` w) e. Y /\ w = v) -> 0 = ((2nd` w)C(2nd` v)))
105	94, 99, 104	syl2an 503	. . . . . . . . 9 |- ((((1st` w) e. X /\ w = v) /\ ((2nd` w) e. Y /\ w = v)) -> if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), 0, 0) = if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))))
106	105	anandirs 571	. . . . . . . 8 |- ((((1st` w) e. X /\ (2nd` w) e. Y) /\ w = v) -> if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), 0, 0) = if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))))
107	106, 43	sylan 497	. . . . . . 7 |- ((w e. (X X. Y) /\ w = v) -> if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), 0, 0) = if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))))
108		ifid 3003	. . . . . . 7 |- if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), 0, 0) = 0
109	107, 108	syl5reqr 1943	. . . . . 6 |- ((w e. (X X. Y) /\ w = v) -> if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = 0)
110	109	ex 402	. . . . 5 |- (w e. (X X. Y) -> (w = v -> if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = 0))
111	110	adantr 425	. . . 4 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (w = v -> if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = 0))
112	93, 111	impbid 574	. . 3 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> (if(((2nd` w)C(2nd` v)) < ((1st` w)B(1st` v)), ((1st` w)B(1st` v)), ((2nd` w)C(2nd` v))) = 0 <-> w = v))
113	27, 112	bitrd 587	. 2 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> ((wDv) = 0 <-> w = v))
114	1, 7, 2, 8, 17	metxplem4 9110	. . 3 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (wDv) <_ ((uDw) + (uDv)))
115	114	3coml 1075	. 2 |- ((w e. (X X. Y) /\ v e. (X X. Y) /\ u e. (X X. Y)) -> (wDv) <_ ((uDw) + (uDv)))
116	13, 21, 113, 115	ismeti 9079	1 |- D e. Met

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  ifcif 2982  {cpr 3045   class class class wbr 3338   X. cxp 3984  dom cdm 3986   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  supcsup 5663  RRcr 6385  0cc0 6386   + caddc 6389   <_ cle 6448   < clt 6653  Metcme 9066

This theorem is referenced by:  cn2met 9185  xplmi 9251  xplmi2 9252  xplm 9253  xpcn 9254  oprcn 9255  bopcn 9263  vacnlem6 9672  txmet 15925

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-2 7154  df-met 9070

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