Theorem metxplem4 9110

Description: Lemma for metxp 9111. Triangle inequality. Warning: The HTML proof page is 0.6 megabyte in size.

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
metxplem4	|- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (wDv) <_ ((uDw) + (uDv)))

Distinct variable groups:   x,y,z,B   x,C,y,z   v,u,w,D   x,u,y,z,X,v,w   u,Y,v,w,x,y,z

Proof of Theorem metxplem4

Step	Hyp	Ref	Expression
1		metxp.6	. . . 4 |- C e. Met
2		metxp.3	. . . 4 |- Y = dom dom C
3		eqid 1884	. . . 4 |- (2nd` w) = (2nd` w)
4		eqid 1884	. . . 4 |- (2nd` v) = (2nd` v)
5	1, 2, 3, 4	metxplem2 9104	. . 3 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` w)C(2nd` v)) e. RR)
6	5	3adant1 894	. 2 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` w)C(2nd` v)) e. RR)
7		metxp.5	. . . 4 |- B e. Met
8		metxp.1	. . . 4 |- X = dom dom B
9		eqid 1884	. . . 4 |- (1st` w) = (1st` w)
10		eqid 1884	. . . 4 |- (1st` v) = (1st` v)
11	7, 8, 9, 10	metxplem1 9103	. . 3 |- ((w e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` w)B(1st` v)) e. RR)
12	11	3adant1 894	. 2 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` w)B(1st` v)) e. RR)
13		metxp.7	. . . . 5 |- 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, < ))}
14	8, 2, 7, 1, 13, 9, 3, 10, 4	metxptval 9107	. . . 4 |- (((w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` w)C(2nd` v)) <_ ((1st` w)B(1st` v))) -> (wDv) = ((1st` w)B(1st` v)))
15	14	3adantl1 1032	. . 3 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` w)C(2nd` v)) <_ ((1st` w)B(1st` v))) -> (wDv) = ((1st` w)B(1st` v)))
16		eqid 1884	. . . . . . 7 |- (2nd` u) = (2nd` u)
17	1, 2, 16, 3	metxplem2 9104	. . . . . 6 |- ((u e. (X X. Y) /\ w e. (X X. Y)) -> ((2nd` u)C(2nd` w)) e. RR)
18	17	3adant3 896	. . . . 5 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` u)C(2nd` w)) e. RR)
19		eqid 1884	. . . . . . 7 |- (1st` u) = (1st` u)
20	7, 8, 19, 9	metxplem1 9103	. . . . . 6 |- ((u e. (X X. Y) /\ w e. (X X. Y)) -> ((1st` u)B(1st` w)) e. RR)
21	20	3adant3 896	. . . . 5 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` u)B(1st` w)) e. RR)
22	1, 2, 16, 4	metxplem2 9104	. . . . . . . . 9 |- ((u e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` u)C(2nd` v)) e. RR)
23	22	3adant2 895	. . . . . . . 8 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` u)C(2nd` v)) e. RR)
24	7, 8, 19, 10	metxplem1 9103	. . . . . . . . 9 |- ((u e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` u)B(1st` v)) e. RR)
25	24	3adant2 895	. . . . . . . 8 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` u)B(1st` v)) e. RR)
26		leid 6701	. . . . . . . . . . . . 13 |- (((1st` u)B(1st` w)) e. RR -> ((1st` u)B(1st` w)) <_ ((1st` u)B(1st` w)))
27	20, 26	syl 12	. . . . . . . . . . . 12 |- ((u e. (X X. Y) /\ w e. (X X. Y)) -> ((1st` u)B(1st` w)) <_ ((1st` u)B(1st` w)))
28	27	3adant3 896	. . . . . . . . . . 11 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` u)B(1st` w)) <_ ((1st` u)B(1st` w)))
29		leid 6701	. . . . . . . . . . . . . 14 |- (((1st` u)B(1st` v)) e. RR -> ((1st` u)B(1st` v)) <_ ((1st` u)B(1st` v)))
30	24, 29	syl 12	. . . . . . . . . . . . 13 |- ((u e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` u)B(1st` v)) <_ ((1st` u)B(1st` v)))
31	30	3adant2 895	. . . . . . . . . . . 12 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` u)B(1st` v)) <_ ((1st` u)B(1st` v)))
32		elxp7 5042	. . . . . . . . . . . . . . . 16 |- (u e. (X X. Y) <-> (u e. (_V X. _V) /\ ((1st` u) e. X /\ (2nd` u) e. Y)))
33	32	simprbi 353	. . . . . . . . . . . . . . 15 |- (u e. (X X. Y) -> ((1st` u) e. X /\ (2nd` u) e. Y))
34	33	simplld 348	. . . . . . . . . . . . . 14 |- (u e. (X X. Y) -> (1st` u) e. X)
35		elxp7 5042	. . . . . . . . . . . . . . . 16 |- (w e. (X X. Y) <-> (w e. (_V X. _V) /\ ((1st` w) e. X /\ (2nd` w) e. Y)))
36	35	simprbi 353	. . . . . . . . . . . . . . 15 |- (w e. (X X. Y) -> ((1st` w) e. X /\ (2nd` w) e. Y))
37	36	simplld 348	. . . . . . . . . . . . . 14 |- (w e. (X X. Y) -> (1st` w) e. X)
38		elxp7 5042	. . . . . . . . . . . . . . . 16 |- (v e. (X X. Y) <-> (v e. (_V X. _V) /\ ((1st` v) e. X /\ (2nd` v) e. Y)))
39	38	simprbi 353	. . . . . . . . . . . . . . 15 |- (v e. (X X. Y) -> ((1st` v) e. X /\ (2nd` v) e. Y))
40	39	simplld 348	. . . . . . . . . . . . . 14 |- (v e. (X X. Y) -> (1st` v) e. X)
41	34, 37, 40	3anim123i 1053	. . . . . . . . . . . . 13 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` u) e. X /\ (1st` w) e. X /\ (1st` v) e. X))
42	7, 8, 21, 25, 41	metxplem3 9105	. . . . . . . . . . . 12 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((1st` u)B(1st` w)) <_ ((1st` u)B(1st` w)) -> (((1st` u)B(1st` v)) <_ ((1st` u)B(1st` v)) -> ((1st` w)B(1st` v)) <_ (((1st` u)B(1st` w)) + ((1st` u)B(1st` v))))))
43	31, 42	mpid 58	. . . . . . . . . . 11 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((1st` u)B(1st` w)) <_ ((1st` u)B(1st` w)) -> ((1st` w)B(1st` v)) <_ (((1st` u)B(1st` w)) + ((1st` u)B(1st` v)))))
44	28, 43	mpd 29	. . . . . . . . . 10 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` w)B(1st` v)) <_ (((1st` u)B(1st` w)) + ((1st` u)B(1st` v))))
45	44	adantr 425	. . . . . . . . 9 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> ((1st` w)B(1st` v)) <_ (((1st` u)B(1st` w)) + ((1st` u)B(1st` v))))
46	8, 2, 7, 1, 13, 19, 16, 10, 4	metxptval 9107	. . . . . . . . . . 11 |- (((u e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> (uDv) = ((1st` u)B(1st` v)))
47	46	3adantl2 1033	. . . . . . . . . 10 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> (uDv) = ((1st` u)B(1st` v)))
48	47	opreq2d 4898	. . . . . . . . 9 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> (((1st` u)B(1st` w)) + (uDv)) = (((1st` u)B(1st` w)) + ((1st` u)B(1st` v))))
49	45, 48	breqtrrd 3363	. . . . . . . 8 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> ((1st` w)B(1st` v)) <_ (((1st` u)B(1st` w)) + (uDv)))
50	7, 8, 21, 23, 41	metxplem3 9105	. . . . . . . . . . 11 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((1st` u)B(1st` w)) <_ ((1st` u)B(1st` w)) -> (((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v)) -> ((1st` w)B(1st` v)) <_ (((1st` u)B(1st` w)) + ((2nd` u)C(2nd` v))))))
51	28, 50	mpd 29	. . . . . . . . . 10 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v)) -> ((1st` w)B(1st` v)) <_ (((1st` u)B(1st` w)) + ((2nd` u)C(2nd` v)))))
52	51	imp 377	. . . . . . . . 9 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> ((1st` w)B(1st` v)) <_ (((1st` u)B(1st` w)) + ((2nd` u)C(2nd` v))))
53	8, 2, 7, 1, 13, 19, 16, 10, 4	metxpfval 9108	. . . . . . . . . . 11 |- (((u e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> (uDv) = ((2nd` u)C(2nd` v)))
54	53	3adantl2 1033	. . . . . . . . . 10 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> (uDv) = ((2nd` u)C(2nd` v)))
55	54	opreq2d 4898	. . . . . . . . 9 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> (((1st` u)B(1st` w)) + (uDv)) = (((1st` u)B(1st` w)) + ((2nd` u)C(2nd` v))))
56	52, 55	breqtrrd 3363	. . . . . . . 8 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> ((1st` w)B(1st` v)) <_ (((1st` u)B(1st` w)) + (uDv)))
57	23, 25, 49, 56	lecasei 6804	. . . . . . 7 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` w)B(1st` v)) <_ (((1st` u)B(1st` w)) + (uDv)))
58	57	adantr 425	. . . . . 6 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) -> ((1st` w)B(1st` v)) <_ (((1st` u)B(1st` w)) + (uDv)))
59	8, 2, 7, 1, 13, 19, 16, 9, 3	metxptval 9107	. . . . . . . 8 |- (((u e. (X X. Y) /\ w e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) -> (uDw) = ((1st` u)B(1st` w)))
60	59	3adantl3 1034	. . . . . . 7 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) -> (uDw) = ((1st` u)B(1st` w)))
61	60	opreq1d 4897	. . . . . 6 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) -> ((uDw) + (uDv)) = (((1st` u)B(1st` w)) + (uDv)))
62	58, 61	breqtrrd 3363	. . . . 5 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) -> ((1st` w)B(1st` v)) <_ ((uDw) + (uDv)))
63	23	adantr 425	. . . . . . 7 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) -> ((2nd` u)C(2nd` v)) e. RR)
64	25	adantr 425	. . . . . . 7 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) -> ((1st` u)B(1st` v)) e. RR)
65	7, 8, 18, 25, 41	metxplem3 9105	. . . . . . . . . . 11 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w)) -> (((1st` u)B(1st` v)) <_ ((1st` u)B(1st` v)) -> ((1st` w)B(1st` v)) <_ (((2nd` u)C(2nd` w)) + ((1st` u)B(1st` v))))))
66	31, 65	mpid 58	. . . . . . . . . 10 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w)) -> ((1st` w)B(1st` v)) <_ (((2nd` u)C(2nd` w)) + ((1st` u)B(1st` v)))))
67	66	imp 377	. . . . . . . . 9 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) -> ((1st` w)B(1st` v)) <_ (((2nd` u)C(2nd` w)) + ((1st` u)B(1st` v))))
68	67	adantr 425	. . . . . . . 8 |- ((((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> ((1st` w)B(1st` v)) <_ (((2nd` u)C(2nd` w)) + ((1st` u)B(1st` v))))
69	47	opreq2d 4898	. . . . . . . . 9 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> (((2nd` u)C(2nd` w)) + (uDv)) = (((2nd` u)C(2nd` w)) + ((1st` u)B(1st` v))))
70	69	adantlr 429	. . . . . . . 8 |- ((((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> (((2nd` u)C(2nd` w)) + (uDv)) = (((2nd` u)C(2nd` w)) + ((1st` u)B(1st` v))))
71	68, 70	breqtrrd 3363	. . . . . . 7 |- ((((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> ((1st` w)B(1st` v)) <_ (((2nd` u)C(2nd` w)) + (uDv)))
72	7, 8, 18, 23, 41	metxplem3 9105	. . . . . . . . 9 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w)) -> (((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v)) -> ((1st` w)B(1st` v)) <_ (((2nd` u)C(2nd` w)) + ((2nd` u)C(2nd` v))))))
73	72	imp31 389	. . . . . . . 8 |- ((((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> ((1st` w)B(1st` v)) <_ (((2nd` u)C(2nd` w)) + ((2nd` u)C(2nd` v))))
74	54	opreq2d 4898	. . . . . . . . 9 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> (((2nd` u)C(2nd` w)) + (uDv)) = (((2nd` u)C(2nd` w)) + ((2nd` u)C(2nd` v))))
75	74	adantlr 429	. . . . . . . 8 |- ((((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> (((2nd` u)C(2nd` w)) + (uDv)) = (((2nd` u)C(2nd` w)) + ((2nd` u)C(2nd` v))))
76	73, 75	breqtrrd 3363	. . . . . . 7 |- ((((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> ((1st` w)B(1st` v)) <_ (((2nd` u)C(2nd` w)) + (uDv)))
77	63, 64, 71, 76	lecasei 6804	. . . . . 6 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) -> ((1st` w)B(1st` v)) <_ (((2nd` u)C(2nd` w)) + (uDv)))
78	8, 2, 7, 1, 13, 19, 16, 9, 3	metxpfval 9108	. . . . . . . 8 |- (((u e. (X X. Y) /\ w e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) -> (uDw) = ((2nd` u)C(2nd` w)))
79	78	3adantl3 1034	. . . . . . 7 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) -> (uDw) = ((2nd` u)C(2nd` w)))
80	79	opreq1d 4897	. . . . . 6 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) -> ((uDw) + (uDv)) = (((2nd` u)C(2nd` w)) + (uDv)))
81	77, 80	breqtrrd 3363	. . . . 5 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) -> ((1st` w)B(1st` v)) <_ ((uDw) + (uDv)))
82	18, 21, 62, 81	lecasei 6804	. . . 4 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((1st` w)B(1st` v)) <_ ((uDw) + (uDv)))
83	82	adantr 425	. . 3 |- (((u e. (X X. Y) /\ 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)) <_ ((uDw) + (uDv)))
84	15, 83	eqbrtrd 3357	. 2 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` w)C(2nd` v)) <_ ((1st` w)B(1st` v))) -> (wDv) <_ ((uDw) + (uDv)))
85	8, 2, 7, 1, 13, 9, 3, 10, 4	metxpfval 9108	. . . 4 |- (((w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` w)B(1st` v)) <_ ((2nd` w)C(2nd` v))) -> (wDv) = ((2nd` w)C(2nd` v)))
86	85	3adantl1 1032	. . 3 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` w)B(1st` v)) <_ ((2nd` w)C(2nd` v))) -> (wDv) = ((2nd` w)C(2nd` v)))
87	23	adantr 425	. . . . . . 7 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) -> ((2nd` u)C(2nd` v)) e. RR)
88	25	adantr 425	. . . . . . 7 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) -> ((1st` u)B(1st` v)) e. RR)
89	33	simprd 352	. . . . . . . . . . 11 |- (u e. (X X. Y) -> (2nd` u) e. Y)
90	36	simprd 352	. . . . . . . . . . 11 |- (w e. (X X. Y) -> (2nd` w) e. Y)
91	39	simprd 352	. . . . . . . . . . 11 |- (v e. (X X. Y) -> (2nd` v) e. Y)
92	89, 90, 91	3anim123i 1053	. . . . . . . . . 10 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` u) e. Y /\ (2nd` w) e. Y /\ (2nd` v) e. Y))
93	1, 2, 21, 25, 92	metxplem3 9105	. . . . . . . . 9 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w)) -> (((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v)) -> ((2nd` w)C(2nd` v)) <_ (((1st` u)B(1st` w)) + ((1st` u)B(1st` v))))))
94	93	imp31 389	. . . . . . . 8 |- ((((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> ((2nd` w)C(2nd` v)) <_ (((1st` u)B(1st` w)) + ((1st` u)B(1st` v))))
95	48	adantlr 429	. . . . . . . 8 |- ((((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> (((1st` u)B(1st` w)) + (uDv)) = (((1st` u)B(1st` w)) + ((1st` u)B(1st` v))))
96	94, 95	breqtrrd 3363	. . . . . . 7 |- ((((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> ((2nd` w)C(2nd` v)) <_ (((1st` u)B(1st` w)) + (uDv)))
97		leid 6701	. . . . . . . . . . . . 13 |- (((2nd` u)C(2nd` v)) e. RR -> ((2nd` u)C(2nd` v)) <_ ((2nd` u)C(2nd` v)))
98	22, 97	syl 12	. . . . . . . . . . . 12 |- ((u e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` u)C(2nd` v)) <_ ((2nd` u)C(2nd` v)))
99	98	3adant2 895	. . . . . . . . . . 11 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` u)C(2nd` v)) <_ ((2nd` u)C(2nd` v)))
100	1, 2, 21, 23, 92	metxplem3 9105	. . . . . . . . . . 11 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w)) -> (((2nd` u)C(2nd` v)) <_ ((2nd` u)C(2nd` v)) -> ((2nd` w)C(2nd` v)) <_ (((1st` u)B(1st` w)) + ((2nd` u)C(2nd` v))))))
101	99, 100	mpid 58	. . . . . . . . . 10 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w)) -> ((2nd` w)C(2nd` v)) <_ (((1st` u)B(1st` w)) + ((2nd` u)C(2nd` v)))))
102	101	imp 377	. . . . . . . . 9 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) -> ((2nd` w)C(2nd` v)) <_ (((1st` u)B(1st` w)) + ((2nd` u)C(2nd` v))))
103	102	adantr 425	. . . . . . . 8 |- ((((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> ((2nd` w)C(2nd` v)) <_ (((1st` u)B(1st` w)) + ((2nd` u)C(2nd` v))))
104	55	adantlr 429	. . . . . . . 8 |- ((((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> (((1st` u)B(1st` w)) + (uDv)) = (((1st` u)B(1st` w)) + ((2nd` u)C(2nd` v))))
105	103, 104	breqtrrd 3363	. . . . . . 7 |- ((((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> ((2nd` w)C(2nd` v)) <_ (((1st` u)B(1st` w)) + (uDv)))
106	87, 88, 96, 105	lecasei 6804	. . . . . 6 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) -> ((2nd` w)C(2nd` v)) <_ (((1st` u)B(1st` w)) + (uDv)))
107	106, 61	breqtrrd 3363	. . . . 5 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` w)) <_ ((1st` u)B(1st` w))) -> ((2nd` w)C(2nd` v)) <_ ((uDw) + (uDv)))
108		leid 6701	. . . . . . . . . . . . 13 |- (((2nd` u)C(2nd` w)) e. RR -> ((2nd` u)C(2nd` w)) <_ ((2nd` u)C(2nd` w)))
109	17, 108	syl 12	. . . . . . . . . . . 12 |- ((u e. (X X. Y) /\ w e. (X X. Y)) -> ((2nd` u)C(2nd` w)) <_ ((2nd` u)C(2nd` w)))
110	109	3adant3 896	. . . . . . . . . . 11 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` u)C(2nd` w)) <_ ((2nd` u)C(2nd` w)))
111	1, 2, 18, 25, 92	metxplem3 9105	. . . . . . . . . . 11 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((2nd` u)C(2nd` w)) <_ ((2nd` u)C(2nd` w)) -> (((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v)) -> ((2nd` w)C(2nd` v)) <_ (((2nd` u)C(2nd` w)) + ((1st` u)B(1st` v))))))
112	110, 111	mpd 29	. . . . . . . . . 10 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v)) -> ((2nd` w)C(2nd` v)) <_ (((2nd` u)C(2nd` w)) + ((1st` u)B(1st` v)))))
113	112	imp 377	. . . . . . . . 9 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> ((2nd` w)C(2nd` v)) <_ (((2nd` u)C(2nd` w)) + ((1st` u)B(1st` v))))
114	113, 69	breqtrrd 3363	. . . . . . . 8 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((2nd` u)C(2nd` v)) <_ ((1st` u)B(1st` v))) -> ((2nd` w)C(2nd` v)) <_ (((2nd` u)C(2nd` w)) + (uDv)))
115	1, 2, 18, 23, 92	metxplem3 9105	. . . . . . . . . . . 12 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((2nd` u)C(2nd` w)) <_ ((2nd` u)C(2nd` w)) -> (((2nd` u)C(2nd` v)) <_ ((2nd` u)C(2nd` v)) -> ((2nd` w)C(2nd` v)) <_ (((2nd` u)C(2nd` w)) + ((2nd` u)C(2nd` v))))))
116	99, 115	mpid 58	. . . . . . . . . . 11 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (((2nd` u)C(2nd` w)) <_ ((2nd` u)C(2nd` w)) -> ((2nd` w)C(2nd` v)) <_ (((2nd` u)C(2nd` w)) + ((2nd` u)C(2nd` v)))))
117	110, 116	mpd 29	. . . . . . . . . 10 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` w)C(2nd` v)) <_ (((2nd` u)C(2nd` w)) + ((2nd` u)C(2nd` v))))
118	117	adantr 425	. . . . . . . . 9 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> ((2nd` w)C(2nd` v)) <_ (((2nd` u)C(2nd` w)) + ((2nd` u)C(2nd` v))))
119	118, 74	breqtrrd 3363	. . . . . . . 8 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` v)) <_ ((2nd` u)C(2nd` v))) -> ((2nd` w)C(2nd` v)) <_ (((2nd` u)C(2nd` w)) + (uDv)))
120	23, 25, 114, 119	lecasei 6804	. . . . . . 7 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` w)C(2nd` v)) <_ (((2nd` u)C(2nd` w)) + (uDv)))
121	120	adantr 425	. . . . . 6 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) -> ((2nd` w)C(2nd` v)) <_ (((2nd` u)C(2nd` w)) + (uDv)))
122	121, 80	breqtrrd 3363	. . . . 5 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` u)B(1st` w)) <_ ((2nd` u)C(2nd` w))) -> ((2nd` w)C(2nd` v)) <_ ((uDw) + (uDv)))
123	18, 21, 107, 122	lecasei 6804	. . . 4 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> ((2nd` w)C(2nd` v)) <_ ((uDw) + (uDv)))
124	123	adantr 425	. . 3 |- (((u e. (X X. Y) /\ 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)) <_ ((uDw) + (uDv)))
125	86, 124	eqbrtrd 3357	. 2 |- (((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) /\ ((1st` w)B(1st` v)) <_ ((2nd` w)C(2nd` v))) -> (wDv) <_ ((uDw) + (uDv)))
126	6, 12, 84, 125	lecasei 6804	1 |- ((u e. (X X. Y) /\ w e. (X X. Y) /\ v e. (X X. Y)) -> (wDv) <_ ((uDw) + (uDv)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  {cpr 3045   class class class wbr 3338   X. cxp 3984  dom cdm 3986  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  supcsup 5663  RRcr 6385   + caddc 6389   <_ cle 6448   < clt 6653  Metcme 9066

This theorem is referenced by:  metxp 9111

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-1st 5020  df-2nd 5021  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-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-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-met 9070

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