Theorem cdj3i 12013

Description: Two ways to express "A and B are completely disjoint subspaces." (1) <=> (3) in Lemma 5 of [Holland] p. 1520.

Hypotheses

Ref	Expression
cdj3.1	|- A e. SH
cdj3.2	|- B e. SH
cdj3.3	|- S = {<.x, y>. | (x e. (A +H B) /\ y = U.{z e. A | E.w e. B x = (z +h w)})}
cdj3.4	|- T = {<.x, y>. | (x e. (A +H B) /\ y = U.{w e. B | E.z e. A x = (z +h w)})}
cdj3.5	|- (ph <-> E.v e. RR (0 < v /\ A.u e. (A +H B)(normh` (S` u)) <_ (v x. (normh` u))))
cdj3.6	|- (ps <-> E.v e. RR (0 < v /\ A.u e. (A +H B)(normh` (T` u)) <_ (v x. (normh` u))))

Assertion

Ref	Expression
cdj3i	|- (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 /\ ph /\ ps))

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

Proof of Theorem cdj3i

Step	Hyp	Ref	Expression
1		cdj3.1	. . . 4 |- A e. SH
2		cdj3.2	. . . 4 |- B e. SH
3	1, 2	cdj3lem1 12006	. . 3 |- (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		cdj3.3	. . . . 5 |- S = {<.x, y>. | (x e. (A +H B) /\ y = U.{z e. A | E.w e. B x = (z +h w)})}
5	1, 2, 4	cdj3lem2b 12009	. . . 4 |- (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))))
6		cdj3.5	. . . 4 |- (ph <-> E.v e. RR (0 < v /\ A.u e. (A +H B)(normh` (S` u)) <_ (v x. (normh` u))))
7	5, 6	sylibr 217	. . 3 |- (E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y)))) -> ph)
8		cdj3.4	. . . . 5 |- T = {<.x, y>. | (x e. (A +H B) /\ y = U.{w e. B | E.z e. A x = (z +h w)})}
9	1, 2, 8	cdj3lem3b 12012	. . . 4 |- (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` (T` u)) <_ (v x. (normh` u))))
10		cdj3.6	. . . 4 |- (ps <-> E.v e. RR (0 < v /\ A.u e. (A +H B)(normh` (T` u)) <_ (v x. (normh` u))))
11	9, 10	sylibr 217	. . 3 |- (E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y)))) -> ps)
12	3, 7, 11	3jca 1050	. 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 /\ ph /\ ps))
13		readdcl 6455	. . . . . . . . . 10 |- ((f e. RR /\ g e. RR) -> (f + g) e. RR)
14		breq2 3342	. . . . . . . . . . . . 13 |- (v = (f + g) -> (0 < v <-> 0 < (f + g)))
15		opreq1 4889	. . . . . . . . . . . . . . . 16 |- (v = (f + g) -> (v x. (normh` (t +h h))) = ((f + g) x. (normh` (t +h h))))
16	15	breq2d 3350	. . . . . . . . . . . . . . 15 |- (v = (f + g) -> (((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h))) <-> ((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h)))))
17	16	2ralbidv 2140	. . . . . . . . . . . . . 14 |- (v = (f + g) -> (A.t e. A A.h e. B ((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h))) <-> A.t e. A A.h e. B ((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h)))))
18		fveq2 4681	. . . . . . . . . . . . . . . . 17 |- (x = t -> (normh` x) = (normh` t))
19	18	opreq1d 4897	. . . . . . . . . . . . . . . 16 |- (x = t -> ((normh` x) + (normh` y)) = ((normh` t) + (normh` y)))
20		opreq1 4889	. . . . . . . . . . . . . . . . . 18 |- (x = t -> (x +h y) = (t +h y))
21	20	fveq2d 4685	. . . . . . . . . . . . . . . . 17 |- (x = t -> (normh` (x +h y)) = (normh` (t +h y)))
22	21	opreq2d 4898	. . . . . . . . . . . . . . . 16 |- (x = t -> (v x. (normh` (x +h y))) = (v x. (normh` (t +h y))))
23	19, 22	breq12d 3351	. . . . . . . . . . . . . . 15 |- (x = t -> (((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) <-> ((normh` t) + (normh` y)) <_ (v x. (normh` (t +h y)))))
24		fveq2 4681	. . . . . . . . . . . . . . . . 17 |- (y = h -> (normh` y) = (normh` h))
25	24	opreq2d 4898	. . . . . . . . . . . . . . . 16 |- (y = h -> ((normh` t) + (normh` y)) = ((normh` t) + (normh` h)))
26		opreq2 4890	. . . . . . . . . . . . . . . . . 18 |- (y = h -> (t +h y) = (t +h h))
27	26	fveq2d 4685	. . . . . . . . . . . . . . . . 17 |- (y = h -> (normh` (t +h y)) = (normh` (t +h h)))
28	27	opreq2d 4898	. . . . . . . . . . . . . . . 16 |- (y = h -> (v x. (normh` (t +h y))) = (v x. (normh` (t +h h))))
29	25, 28	breq12d 3351	. . . . . . . . . . . . . . 15 |- (y = h -> (((normh` t) + (normh` y)) <_ (v x. (normh` (t +h y))) <-> ((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h)))))
30	23, 29	cbvral2v 2283	. . . . . . . . . . . . . 14 |- (A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) <-> A.t e. A A.h e. B ((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h))))
31	17, 30	syl5bb 591	. . . . . . . . . . . . 13 |- (v = (f + g) -> (A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) <-> A.t e. A A.h e. B ((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h)))))
32	14, 31	anbi12d 690	. . . . . . . . . . . 12 |- (v = (f + g) -> ((0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y)))) <-> (0 < (f + g) /\ A.t e. A A.h e. B ((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h))))))
33	32	rcla4ev 2381	. . . . . . . . . . 11 |- (((f + g) e. RR /\ (0 < (f + g) /\ A.t e. A A.h e. B ((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h))))) -> E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y)))))
34	33	ex 402	. . . . . . . . . 10 |- ((f + g) e. RR -> ((0 < (f + g) /\ A.t e. A A.h e. B ((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h)))) -> E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))))))
35	13, 34	syl 12	. . . . . . . . 9 |- ((f e. RR /\ g e. RR) -> ((0 < (f + g) /\ A.t e. A A.h e. B ((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h)))) -> E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))))))
36	35	adantl 424	. . . . . . . 8 |- (((A i^i B) = 0H /\ (f e. RR /\ g e. RR)) -> ((0 < (f + g) /\ A.t e. A A.h e. B ((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h)))) -> E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))))))
37		addgt0 6831	. . . . . . . . . 10 |- (((f e. RR /\ g e. RR) /\ (0 < f /\ 0 < g)) -> 0 < (f + g))
38	37	ex 402	. . . . . . . . 9 |- ((f e. RR /\ g e. RR) -> ((0 < f /\ 0 < g) -> 0 < (f + g)))
39	38	adantl 424	. . . . . . . 8 |- (((A i^i B) = 0H /\ (f e. RR /\ g e. RR)) -> ((0 < f /\ 0 < g) -> 0 < (f + g)))
40	1, 2	shsvai 10966	. . . . . . . . . . . 12 |- ((t e. A /\ h e. B) -> (t +h h) e. (A +H B))
41		fveq2 4681	. . . . . . . . . . . . . . . 16 |- (u = (t +h h) -> (S` u) = (S` (t +h h)))
42	41	fveq2d 4685	. . . . . . . . . . . . . . 15 |- (u = (t +h h) -> (normh` (S` u)) = (normh` (S` (t +h h))))
43		fveq2 4681	. . . . . . . . . . . . . . . 16 |- (u = (t +h h) -> (normh` u) = (normh` (t +h h)))
44	43	opreq2d 4898	. . . . . . . . . . . . . . 15 |- (u = (t +h h) -> (f x. (normh` u)) = (f x. (normh` (t +h h))))
45	42, 44	breq12d 3351	. . . . . . . . . . . . . 14 |- (u = (t +h h) -> ((normh` (S` u)) <_ (f x. (normh` u)) <-> (normh` (S` (t +h h))) <_ (f x. (normh` (t +h h)))))
46	45	rcla4v 2376	. . . . . . . . . . . . 13 |- ((t +h h) e. (A +H B) -> (A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u)) -> (normh` (S` (t +h h))) <_ (f x. (normh` (t +h h)))))
47		fveq2 4681	. . . . . . . . . . . . . . . 16 |- (u = (t +h h) -> (T` u) = (T` (t +h h)))
48	47	fveq2d 4685	. . . . . . . . . . . . . . 15 |- (u = (t +h h) -> (normh` (T` u)) = (normh` (T` (t +h h))))
49	43	opreq2d 4898	. . . . . . . . . . . . . . 15 |- (u = (t +h h) -> (g x. (normh` u)) = (g x. (normh` (t +h h))))
50	48, 49	breq12d 3351	. . . . . . . . . . . . . 14 |- (u = (t +h h) -> ((normh` (T` u)) <_ (g x. (normh` u)) <-> (normh` (T` (t +h h))) <_ (g x. (normh` (t +h h)))))
51	50	rcla4v 2376	. . . . . . . . . . . . 13 |- ((t +h h) e. (A +H B) -> (A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u)) -> (normh` (T` (t +h h))) <_ (g x. (normh` (t +h h)))))
52	46, 51	anim12d 617	. . . . . . . . . . . 12 |- ((t +h h) e. (A +H B) -> ((A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u)) /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u))) -> ((normh` (S` (t +h h))) <_ (f x. (normh` (t +h h))) /\ (normh` (T` (t +h h))) <_ (g x. (normh` (t +h h))))))
53	40, 52	syl 12	. . . . . . . . . . 11 |- ((t e. A /\ h e. B) -> ((A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u)) /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u))) -> ((normh` (S` (t +h h))) <_ (f x. (normh` (t +h h))) /\ (normh` (T` (t +h h))) <_ (g x. (normh` (t +h h))))))
54	53	adantl 424	. . . . . . . . . 10 |- ((((A i^i B) = 0H /\ (f e. RR /\ g e. RR)) /\ (t e. A /\ h e. B)) -> ((A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u)) /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u))) -> ((normh` (S` (t +h h))) <_ (f x. (normh` (t +h h))) /\ (normh` (T` (t +h h))) <_ (g x. (normh` (t +h h))))))
55	1	sheli 10715	. . . . . . . . . . . . . . . 16 |- (t e. A -> t e. ~H)
56		normcl 10624	. . . . . . . . . . . . . . . 16 |- (t e. ~H -> (normh` t) e. RR)
57	55, 56	syl 12	. . . . . . . . . . . . . . 15 |- (t e. A -> (normh` t) e. RR)
58	2	sheli 10715	. . . . . . . . . . . . . . . 16 |- (h e. B -> h e. ~H)
59		normcl 10624	. . . . . . . . . . . . . . . 16 |- (h e. ~H -> (normh` h) e. RR)
60	58, 59	syl 12	. . . . . . . . . . . . . . 15 |- (h e. B -> (normh` h) e. RR)
61	57, 60	anim12i 360	. . . . . . . . . . . . . 14 |- ((t e. A /\ h e. B) -> ((normh` t) e. RR /\ (normh` h) e. RR))
62	61	adantl 424	. . . . . . . . . . . . 13 |- (((f e. RR /\ g e. RR) /\ (t e. A /\ h e. B)) -> ((normh` t) e. RR /\ (normh` h) e. RR))
63		remulcl 6457	. . . . . . . . . . . . . . 15 |- ((f e. RR /\ (normh` (t +h h)) e. RR) -> (f x. (normh` (t +h h))) e. RR)
64		hvaddcl 10514	. . . . . . . . . . . . . . . . 17 |- ((t e. ~H /\ h e. ~H) -> (t +h h) e. ~H)
65	64, 55, 58	syl2an 503	. . . . . . . . . . . . . . . 16 |- ((t e. A /\ h e. B) -> (t +h h) e. ~H)
66		normcl 10624	. . . . . . . . . . . . . . . 16 |- ((t +h h) e. ~H -> (normh` (t +h h)) e. RR)
67	65, 66	syl 12	. . . . . . . . . . . . . . 15 |- ((t e. A /\ h e. B) -> (normh` (t +h h)) e. RR)
68	63, 67	sylan2 500	. . . . . . . . . . . . . 14 |- ((f e. RR /\ (t e. A /\ h e. B)) -> (f x. (normh` (t +h h))) e. RR)
69	68	adantlr 429	. . . . . . . . . . . . 13 |- (((f e. RR /\ g e. RR) /\ (t e. A /\ h e. B)) -> (f x. (normh` (t +h h))) e. RR)
70		remulcl 6457	. . . . . . . . . . . . . . 15 |- ((g e. RR /\ (normh` (t +h h)) e. RR) -> (g x. (normh` (t +h h))) e. RR)
71	70, 67	sylan2 500	. . . . . . . . . . . . . 14 |- ((g e. RR /\ (t e. A /\ h e. B)) -> (g x. (normh` (t +h h))) e. RR)
72	71	adantll 428	. . . . . . . . . . . . 13 |- (((f e. RR /\ g e. RR) /\ (t e. A /\ h e. B)) -> (g x. (normh` (t +h h))) e. RR)
73		le2add 6828	. . . . . . . . . . . . 13 |- ((((normh` t) e. RR /\ (normh` h) e. RR) /\ ((f x. (normh` (t +h h))) e. RR /\ (g x. (normh` (t +h h))) e. RR)) -> (((normh` t) <_ (f x. (normh` (t +h h))) /\ (normh` h) <_ (g x. (normh` (t +h h)))) -> ((normh` t) + (normh` h)) <_ ((f x. (normh` (t +h h))) + (g x. (normh` (t +h h))))))
74	62, 69, 72, 73	syl12anc 1098	. . . . . . . . . . . 12 |- (((f e. RR /\ g e. RR) /\ (t e. A /\ h e. B)) -> (((normh` t) <_ (f x. (normh` (t +h h))) /\ (normh` h) <_ (g x. (normh` (t +h h)))) -> ((normh` t) + (normh` h)) <_ ((f x. (normh` (t +h h))) + (g x. (normh` (t +h h))))))
75	74	adantll 428	. . . . . . . . . . 11 |- ((((A i^i B) = 0H /\ (f e. RR /\ g e. RR)) /\ (t e. A /\ h e. B)) -> (((normh` t) <_ (f x. (normh` (t +h h))) /\ (normh` h) <_ (g x. (normh` (t +h h)))) -> ((normh` t) + (normh` h)) <_ ((f x. (normh` (t +h h))) + (g x. (normh` (t +h h))))))
76	1, 2, 4	cdj3lem2 12007	. . . . . . . . . . . . . . . . 17 |- ((t e. A /\ h e. B /\ (A i^i B) = 0H) -> (S` (t +h h)) = t)
77	76	fveq2d 4685	. . . . . . . . . . . . . . . 16 |- ((t e. A /\ h e. B /\ (A i^i B) = 0H) -> (normh` (S` (t +h h))) = (normh` t))
78	77	breq1d 3348	. . . . . . . . . . . . . . 15 |- ((t e. A /\ h e. B /\ (A i^i B) = 0H) -> ((normh` (S` (t +h h))) <_ (f x. (normh` (t +h h))) <-> (normh` t) <_ (f x. (normh` (t +h h)))))
79	1, 2, 8	cdj3lem3 12010	. . . . . . . . . . . . . . . . 17 |- ((t e. A /\ h e. B /\ (A i^i B) = 0H) -> (T` (t +h h)) = h)
80	79	fveq2d 4685	. . . . . . . . . . . . . . . 16 |- ((t e. A /\ h e. B /\ (A i^i B) = 0H) -> (normh` (T` (t +h h))) = (normh` h))
81	80	breq1d 3348	. . . . . . . . . . . . . . 15 |- ((t e. A /\ h e. B /\ (A i^i B) = 0H) -> ((normh` (T` (t +h h))) <_ (g x. (normh` (t +h h))) <-> (normh` h) <_ (g x. (normh` (t +h h)))))
82	78, 81	anbi12d 690	. . . . . . . . . . . . . 14 |- ((t e. A /\ h e. B /\ (A i^i B) = 0H) -> (((normh` (S` (t +h h))) <_ (f x. (normh` (t +h h))) /\ (normh` (T` (t +h h))) <_ (g x. (normh` (t +h h)))) <-> ((normh` t) <_ (f x. (normh` (t +h h))) /\ (normh` h) <_ (g x. (normh` (t +h h))))))
83	82	3expa 1067	. . . . . . . . . . . . 13 |- (((t e. A /\ h e. B) /\ (A i^i B) = 0H) -> (((normh` (S` (t +h h))) <_ (f x. (normh` (t +h h))) /\ (normh` (T` (t +h h))) <_ (g x. (normh` (t +h h)))) <-> ((normh` t) <_ (f x. (normh` (t +h h))) /\ (normh` h) <_ (g x. (normh` (t +h h))))))
84	83	ancoms 484	. . . . . . . . . . . 12 |- (((A i^i B) = 0H /\ (t e. A /\ h e. B)) -> (((normh` (S` (t +h h))) <_ (f x. (normh` (t +h h))) /\ (normh` (T` (t +h h))) <_ (g x. (normh` (t +h h)))) <-> ((normh` t) <_ (f x. (normh` (t +h h))) /\ (normh` h) <_ (g x. (normh` (t +h h))))))
85	84	adantlr 429	. . . . . . . . . . 11 |- ((((A i^i B) = 0H /\ (f e. RR /\ g e. RR)) /\ (t e. A /\ h e. B)) -> (((normh` (S` (t +h h))) <_ (f x. (normh` (t +h h))) /\ (normh` (T` (t +h h))) <_ (g x. (normh` (t +h h)))) <-> ((normh` t) <_ (f x. (normh` (t +h h))) /\ (normh` h) <_ (g x. (normh` (t +h h))))))
86		adddir 6472	. . . . . . . . . . . . . . 15 |- ((f e. CC /\ g e. CC /\ (normh` (t +h h)) e. CC) -> ((f + g) x. (normh` (t +h h))) = ((f x. (normh` (t +h h))) + (g x. (normh` (t +h h)))))
87		recn 6466	. . . . . . . . . . . . . . 15 |- (f e. RR -> f e. CC)
88		recn 6466	. . . . . . . . . . . . . . 15 |- (g e. RR -> g e. CC)
89	67	recnd 6468	. . . . . . . . . . . . . . 15 |- ((t e. A /\ h e. B) -> (normh` (t +h h)) e. CC)
90	86, 87, 88, 89	syl3an 1139	. . . . . . . . . . . . . 14 |- ((f e. RR /\ g e. RR /\ (t e. A /\ h e. B)) -> ((f + g) x. (normh` (t +h h))) = ((f x. (normh` (t +h h))) + (g x. (normh` (t +h h)))))
91	90	3expa 1067	. . . . . . . . . . . . 13 |- (((f e. RR /\ g e. RR) /\ (t e. A /\ h e. B)) -> ((f + g) x. (normh` (t +h h))) = ((f x. (normh` (t +h h))) + (g x. (normh` (t +h h)))))
92	91	breq2d 3350	. . . . . . . . . . . 12 |- (((f e. RR /\ g e. RR) /\ (t e. A /\ h e. B)) -> (((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h))) <-> ((normh` t) + (normh` h)) <_ ((f x. (normh` (t +h h))) + (g x. (normh` (t +h h))))))
93	92	adantll 428	. . . . . . . . . . 11 |- ((((A i^i B) = 0H /\ (f e. RR /\ g e. RR)) /\ (t e. A /\ h e. B)) -> (((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h))) <-> ((normh` t) + (normh` h)) <_ ((f x. (normh` (t +h h))) + (g x. (normh` (t +h h))))))
94	75, 85, 93	3imtr4d 602	. . . . . . . . . 10 |- ((((A i^i B) = 0H /\ (f e. RR /\ g e. RR)) /\ (t e. A /\ h e. B)) -> (((normh` (S` (t +h h))) <_ (f x. (normh` (t +h h))) /\ (normh` (T` (t +h h))) <_ (g x. (normh` (t +h h)))) -> ((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h)))))
95	54, 94	syld 30	. . . . . . . . 9 |- ((((A i^i B) = 0H /\ (f e. RR /\ g e. RR)) /\ (t e. A /\ h e. B)) -> ((A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u)) /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u))) -> ((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h)))))
96	95	r19.21advva 2185	. . . . . . . 8 |- (((A i^i B) = 0H /\ (f e. RR /\ g e. RR)) -> ((A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u)) /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u))) -> A.t e. A A.h e. B ((normh` t) + (normh` h)) <_ ((f + g) x. (normh` (t +h h)))))
97	36, 39, 96	syl2and 508	. . . . . . 7 |- (((A i^i B) = 0H /\ (f e. RR /\ g e. RR)) -> (((0 < f /\ 0 < g) /\ (A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u)) /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u)))) -> E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))))))
98		an4 564	. . . . . . 7 |- (((0 < f /\ A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u))) /\ (0 < g /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u)))) <-> ((0 < f /\ 0 < g) /\ (A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u)) /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u)))))
99	97, 98	syl5ib 223	. . . . . 6 |- (((A i^i B) = 0H /\ (f e. RR /\ g e. RR)) -> (((0 < f /\ A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u))) /\ (0 < g /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u)))) -> E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))))))
100	99	ex 402	. . . . 5 |- ((A i^i B) = 0H -> ((f e. RR /\ g e. RR) -> (((0 < f /\ A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u))) /\ (0 < g /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u)))) -> E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y)))))))
101	100	r19.23advv 2218	. . . 4 |- ((A i^i B) = 0H -> (E.f e. RR E.g e. RR ((0 < f /\ A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u))) /\ (0 < g /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u)))) -> E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))))))
102		breq2 3342	. . . . . . . . 9 |- (v = f -> (0 < v <-> 0 < f))
103		opreq1 4889	. . . . . . . . . . 11 |- (v = f -> (v x. (normh` u)) = (f x. (normh` u)))
104	103	breq2d 3350	. . . . . . . . . 10 |- (v = f -> ((normh` (S` u)) <_ (v x. (normh` u)) <-> (normh` (S` u)) <_ (f x. (normh` u))))
105	104	ralbidv 2123	. . . . . . . . 9 |- (v = f -> (A.u e. (A +H B)(normh` (S` u)) <_ (v x. (normh` u)) <-> A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u))))
106	102, 105	anbi12d 690	. . . . . . . 8 |- (v = f -> ((0 < v /\ A.u e. (A +H B)(normh` (S` u)) <_ (v x. (normh` u))) <-> (0 < f /\ A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u)))))
107	106	cbvrexv 2281	. . . . . . 7 |- (E.v e. RR (0 < v /\ A.u e. (A +H B)(normh` (S` u)) <_ (v x. (normh` u))) <-> E.f e. RR (0 < f /\ A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u))))
108	6, 107	bitri 190	. . . . . 6 |- (ph <-> E.f e. RR (0 < f /\ A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u))))
109		breq2 3342	. . . . . . . . 9 |- (v = g -> (0 < v <-> 0 < g))
110		opreq1 4889	. . . . . . . . . . 11 |- (v = g -> (v x. (normh` u)) = (g x. (normh` u)))
111	110	breq2d 3350	. . . . . . . . . 10 |- (v = g -> ((normh` (T` u)) <_ (v x. (normh` u)) <-> (normh` (T` u)) <_ (g x. (normh` u))))
112	111	ralbidv 2123	. . . . . . . . 9 |- (v = g -> (A.u e. (A +H B)(normh` (T` u)) <_ (v x. (normh` u)) <-> A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u))))
113	109, 112	anbi12d 690	. . . . . . . 8 |- (v = g -> ((0 < v /\ A.u e. (A +H B)(normh` (T` u)) <_ (v x. (normh` u))) <-> (0 < g /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u)))))
114	113	cbvrexv 2281	. . . . . . 7 |- (E.v e. RR (0 < v /\ A.u e. (A +H B)(normh` (T` u)) <_ (v x. (normh` u))) <-> E.g e. RR (0 < g /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u))))
115	10, 114	bitri 190	. . . . . 6 |- (ps <-> E.g e. RR (0 < g /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u))))
116	108, 115	anbi12i 540	. . . . 5 |- ((ph /\ ps) <-> (E.f e. RR (0 < f /\ A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u))) /\ E.g e. RR (0 < g /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u)))))
117		reeanv 2249	. . . . 5 |- (E.f e. RR E.g e. RR ((0 < f /\ A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u))) /\ (0 < g /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u)))) <-> (E.f e. RR (0 < f /\ A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u))) /\ E.g e. RR (0 < g /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u)))))
118	116, 117	bitr4i 193	. . . 4 |- ((ph /\ ps) <-> E.f e. RR E.g e. RR ((0 < f /\ A.u e. (A +H B)(normh` (S` u)) <_ (f x. (normh` u))) /\ (0 < g /\ A.u e. (A +H B)(normh` (T` u)) <_ (g x. (normh` u)))))
119	101, 118	syl5ib 223	. . 3 |- ((A i^i B) = 0H -> ((ph /\ ps) -> E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))))))
120	119	3impib 1065	. 2 |- (((A i^i B) = 0H /\ ph /\ ps) -> E.v e. RR (0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y)))))
121	12, 120	impbii 174	1 |- (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 /\ ph /\ ps))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  {crab 2108   i^i cin 2592  U.cuni 3177   class class class wbr 3338  {copab 3395  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   + caddc 6389   x. cmul 6391   <_ cle 6448   < clt 6653  ~Hchil 10420   +h cva 10421  normhcno 10426  SHcsh 10429   +H cph 10432  0Hc0h 10436

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  ax-hilex 10501  ax-hfvadd 10502  ax-hvcom 10503  ax-hvass 10504  ax-hv0cl 10505  ax-hvaddid 10506  ax-hfvmul 10507  ax-hvmulid 10508  ax-hvmulass 10509  ax-hvdistr1 10510  ax-hvdistr2 10511  ax-hvmul0 10512  ax-hfi 10579  ax-his1 10582  ax-his3 10584  ax-his4 10585

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-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-hnorm 10469  df-hvsub 10472  df-sh 10709  df-ch0 10758  df-shsum 10906

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