Theorem cdj3lem2b 10364

Description: Lemma for cdj3 10368. The first-component function S is bounded if the subspaces are completely disjoint.

Hypotheses

Ref	Expression
cdj3lem2.1	|- A e. SH
cdj3lem2.2	|- B e. SH
cdj3lem2.3	|- S = {<.x, y>. | (x e. (A +H B) /\ y = U.{z e. A | E.w e. B x = (z +h w)})}

Assertion

Ref	Expression
cdj3lem2b	|- (E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y)))) -> E.v e. RR (0 < v /\ A.u e. (A +H B)(normh` (S` u)) <_ (v x. (normh` u))))

Distinct variable groups:   x,y,z,w,v,u,A   x,B,y,z,w,v,u   v,S,u

Proof of Theorem cdj3lem2b

Step	Hyp	Ref	Expression
1		cdj3lem2.1	. . 3 |- A e. SH
2		cdj3lem2.2	. . 3 |- B e. SH
3	1, 2	cdj3lem1 10361	. 2 |- (E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y)))) -> (A i^i B) = 0H)
4	1, 2	cdjreu 10359	. . . . . . . . . 10 |- ((u e. (A +H B) /\ (A i^i B) = 0H) -> E!t e. A E.h e. B u = (t +h h))
5		reurex 1928	. . . . . . . . . 10 |- (E!t e. A E.h e. B u = (t +h h) -> E.t e. A E.h e. B u = (t +h h))
6	4, 5	syl 10	. . . . . . . . 9 |- ((u e. (A +H B) /\ (A i^i B) = 0H) -> E.t e. A E.h e. B u = (t +h h))
7	6	adantrr 395	. . . . . . . 8 |- ((u e. (A +H B) /\ ((A i^i B) = 0H /\ v e. RR)) -> E.t e. A E.h e. B u = (t +h h))
8		fveq2 3724	. . . . . . . . . . . . . . . 16 |- (x = t -> (normh` x) = (normh` t))
9	8	opreq1d 3975	. . . . . . . . . . . . . . 15 |- (x = t -> ((normh` x) + (normh` y)) = ((normh` t) + (normh` y)))
10		opreq1 3968	. . . . . . . . . . . . . . . . 17 |- (x = t -> (x +h y) = (t +h y))
11	10	fveq2d 3728	. . . . . . . . . . . . . . . 16 |- (x = t -> (normh` (x +h y)) = (normh` (t +h y)))
12	11	opreq2d 3976	. . . . . . . . . . . . . . 15 |- (x = t -> (v x. (normh` (x +h y))) = (v x. (normh` (t +h y))))
13	9, 12	breq12d 2631	. . . . . . . . . . . . . 14 |- (x = t -> (((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) <-> ((normh` t) + (normh` y)) <_ (v x. (normh` (t +h y)))))
14		fveq2 3724	. . . . . . . . . . . . . . . 16 |- (y = h -> (normh` y) = (normh` h))
15	14	opreq2d 3976	. . . . . . . . . . . . . . 15 |- (y = h -> ((normh` t) + (normh` y)) = ((normh` t) + (normh` h)))
16		opreq2 3969	. . . . . . . . . . . . . . . . 17 |- (y = h -> (t +h y) = (t +h h))
17	16	fveq2d 3728	. . . . . . . . . . . . . . . 16 |- (y = h -> (normh` (t +h y)) = (normh` (t +h h)))
18	17	opreq2d 3976	. . . . . . . . . . . . . . 15 |- (y = h -> (v x. (normh` (t +h y))) = (v x. (normh` (t +h h))))
19	15, 18	breq12d 2631	. . . . . . . . . . . . . 14 |- (y = h -> (((normh` t) + (normh` y)) <_ (v x. (normh` (t +h y))) <-> ((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h)))))
20	13, 19	rcla42v 1880	. . . . . . . . . . . . 13 |- ((t e. A /\ h e. B) -> (A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) -> ((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h)))))
21		fveq2 3724	. . . . . . . . . . . . . . . . 17 |- (u = (t +h h) -> (S` u) = (S` (t +h h)))
22	21	fveq2d 3728	. . . . . . . . . . . . . . . 16 |- (u = (t +h h) -> (normh` (S` u)) = (normh` (S` (t +h h))))
23		fveq2 3724	. . . . . . . . . . . . . . . . 17 |- (u = (t +h h) -> (normh` u) = (normh` (t +h h)))
24	23	opreq2d 3976	. . . . . . . . . . . . . . . 16 |- (u = (t +h h) -> (v x. (normh` u)) = (v x. (normh` (t +h h))))
25	22, 24	breq12d 2631	. . . . . . . . . . . . . . 15 |- (u = (t +h h) -> ((normh` (S` u)) <_ (v x. (normh` u)) <-> (normh` (S` (t +h h))) <_ (v x. (normh` (t +h h)))))
26		cdj3lem2.3	. . . . . . . . . . . . . . . . . . . 20 |- S = {<.x, y>. | (x e. (A +H B) /\ y = U.{z e. A | E.w e. B x = (z +h w)})}
27	1, 2, 26	cdj3lem2 10362	. . . . . . . . . . . . . . . . . . 19 |- ((t e. A /\ h e. B /\ (A i^i B) = 0H) -> (S` (t +h h)) = t)
28	27	3expa 833	. . . . . . . . . . . . . . . . . 18 |- (((t e. A /\ h e. B) /\ (A i^i B) = 0H) -> (S` (t +h h)) = t)
29	28	fveq2d 3728	. . . . . . . . . . . . . . . . 17 |- (((t e. A /\ h e. B) /\ (A i^i B) = 0H) -> (normh` (S` (t +h h))) = (normh` t))
30	29	ad2ant2r 409	. . . . . . . . . . . . . . . 16 |- ((((t e. A /\ h e. B) /\ ((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h)))) /\ ((A i^i B) = 0H /\ v e. RR)) -> (normh` (S` (t +h h))) = (normh` t))
31	2	shel 9082	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (h e. B -> h e. H~)
32		normge0t 8992	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (h e. H~ -> 0 <_ (normh` h))
33	31, 32	syl 10	. . . . . . . . . . . . . . . . . . . . . . 23 |- (h e. B -> 0 <_ (normh` h))
34	33	adantl 388	. . . . . . . . . . . . . . . . . . . . . 22 |- ((t e. A /\ h e. B) -> 0 <_ (normh` h))
35		addge01t 5672	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((normh` t) e. RR /\ (normh` h) e. RR) -> (0 <_ (normh` h) <-> (normh` t) <_ ((normh` t) + (normh` h))))
36	1	shel 9082	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (t e. A -> t e. H~)
37		normclt 8991	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (t e. H~ -> (normh` t) e. RR)
38	36, 37	syl 10	. . . . . . . . . . . . . . . . . . . . . . 23 |- (t e. A -> (normh` t) e. RR)
39		normclt 8991	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (h e. H~ -> (normh` h) e. RR)
40	31, 39	syl 10	. . . . . . . . . . . . . . . . . . . . . . 23 |- (h e. B -> (normh` h) e. RR)
41	35, 38, 40	syl2an 454	. . . . . . . . . . . . . . . . . . . . . 22 |- ((t e. A /\ h e. B) -> (0 <_ (normh` h) <-> (normh` t) <_ ((normh` t) + (normh` h))))
42	34, 41	mpbid 195	. . . . . . . . . . . . . . . . . . . . 21 |- ((t e. A /\ h e. B) -> (normh` t) <_ ((normh` t) + (normh` h)))
43	42	adantr 389	. . . . . . . . . . . . . . . . . . . 20 |- (((t e. A /\ h e. B) /\ v e. RR) -> (normh` t) <_ ((normh` t) + (normh` h)))
44		letrt 5525	. . . . . . . . . . . . . . . . . . . . 21 |- (((normh` t) e. RR /\ ((normh` t) + (normh` h)) e. RR /\ (v x. (normh` (t +h h))) e. RR) -> (((normh` t) <_ ((normh` t) + (normh` h)) /\ ((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h)))) -> (normh` t) <_ (v x. (normh` (t +h h)))))
45	38	ad2antrr 404	. . . . . . . . . . . . . . . . . . . . 21 |- (((t e. A /\ h e. B) /\ v e. RR) -> (normh` t) e. RR)
46		axaddrcl 5272	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((normh` t) e. RR /\ (normh` h) e. RR) -> ((normh` t) + (normh` h)) e. RR)
47	46, 38, 40	syl2an 454	. . . . . . . . . . . . . . . . . . . . . 22 |- ((t e. A /\ h e. B) -> ((normh` t) + (normh` h)) e. RR)
48	47	adantr 389	. . . . . . . . . . . . . . . . . . . . 21 |- (((t e. A /\ h e. B) /\ v e. RR) -> ((normh` t) + (normh` h)) e. RR)
49		axmulrcl 5274	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((v e. RR /\ (normh` (t +h h)) e. RR) -> (v x. (normh` (t +h h))) e. RR)
50		hvaddclt 8882	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((t e. H~ /\ h e. H~) -> (t +h h) e. H~)
51	50, 36, 31	syl2an 454	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((t e. A /\ h e. B) -> (t +h h) e. H~)
52		normclt 8991	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((t +h h) e. H~ -> (normh` (t +h h)) e. RR)
53	51, 52	syl 10	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((t e. A /\ h e. B) -> (normh` (t +h h)) e. RR)
54	49, 53	sylan2 451	. . . . . . . . . . . . . . . . . . . . . 22 |- ((v e. RR /\ (t e. A /\ h e. B)) -> (v x. (normh` (t +h h))) e. RR)
55	54	ancoms 436	. . . . . . . . . . . . . . . . . . . . 21 |- (((t e. A /\ h e. B) /\ v e. RR) -> (v x. (normh` (t +h h))) e. RR)
56	44, 45, 48, 55	syl3anc 858	. . . . . . . . . . . . . . . . . . . 20 |- (((t e. A /\ h e. B) /\ v e. RR) -> (((normh` t) <_ ((normh` t) + (normh` h)) /\ ((normh` t) + (