Theorem cdj3lem2b 12009

Description: Lemma for cdj3i 12013. 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 12006	. 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	cdjreui 12004	. . . . . . . . . 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 2440	. . . . . . . . . 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 12	. . . . . . . . 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 431	. . . . . . . 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 4681	. . . . . . . . . . . . . . . 16 |- (x = t -> (normh` x) = (normh` t))
9	8	opreq1d 4897	. . . . . . . . . . . . . . 15 |- (x = t -> ((normh` x) + (normh` y)) = ((normh` t) + (normh` y)))
10		opreq1 4889	. . . . . . . . . . . . . . . . 17 |- (x = t -> (x +h y) = (t +h y))
11	10	fveq2d 4685	. . . . . . . . . . . . . . . 16 |- (x = t -> (normh` (x +h y)) = (normh` (t +h y)))
12	11	opreq2d 4898	. . . . . . . . . . . . . . 15 |- (x = t -> (v x. (normh` (x +h y))) = (v x. (normh` (t +h y))))
13	9, 12	breq12d 3351	. . . . . . . . . . . . . 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 4681	. . . . . . . . . . . . . . . 16 |- (y = h -> (normh` y) = (normh` h))
15	14	opreq2d 4898	. . . . . . . . . . . . . . 15 |- (y = h -> ((normh` t) + (normh` y)) = ((normh` t) + (normh` h)))
16		opreq2 4890	. . . . . . . . . . . . . . . . 17 |- (y = h -> (t +h y) = (t +h h))
17	16	fveq2d 4685	. . . . . . . . . . . . . . . 16 |- (y = h -> (normh` (t +h y)) = (normh` (t +h h)))
18	17	opreq2d 4898	. . . . . . . . . . . . . . 15 |- (y = h -> (v x. (normh` (t +h y))) = (v x. (normh` (t +h h))))
19	15, 18	breq12d 3351	. . . . . . . . . . . . . 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 2384	. . . . . . . . . . . . 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 4681	. . . . . . . . . . . . . . . . 17 |- (u = (t +h h) -> (S` u) = (S` (t +h h)))
22	21	fveq2d 4685	. . . . . . . . . . . . . . . 16 |- (u = (t +h h) -> (normh` (S` u)) = (normh` (S` (t +h h))))
23		fveq2 4681	. . . . . . . . . . . . . . . . 17 |- (u = (t +h h) -> (normh` u) = (normh` (t +h h)))
24	23	opreq2d 4898	. . . . . . . . . . . . . . . 16 |- (u = (t +h h) -> (v x. (normh` u)) = (v x. (normh` (t +h h))))
25	22, 24	breq12d 3351	. . . . . . . . . . . . . . 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 12007	. . . . . . . . . . . . . . . . . . 19 |- ((t e. A /\ h e. B /\ (A i^i B) = 0H) -> (S` (t +h h)) = t)
28	27	3expa 1067	. . . . . . . . . . . . . . . . . 18 |- (((t e. A /\ h e. B) /\ (A i^i B) = 0H) -> (S` (t +h h)) = t)
29	28	fveq2d 4685	. . . . . . . . . . . . . . . . 17 |- (((t e. A /\ h e. B) /\ (A i^i B) = 0H) -> (normh` (S` (t +h h))) = (normh` t))
30	29	ad2ant2r 445	. . . . . . . . . . . . . . . 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	sheli 10715	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (h e. B -> h e. ~H)
32		normge0 10625	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (h e. ~H -> 0 <_ (normh` h))
33	31, 32	syl 12	. . . . . . . . . . . . . . . . . . . . . . 23 |- (h e. B -> 0 <_ (normh` h))
34	33	adantl 424	. . . . . . . . . . . . . . . . . . . . . 22 |- ((t e. A /\ h e. B) -> 0 <_ (normh` h))
35		addge01 6861	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((normh` t) e. RR /\ (normh` h) e. RR) -> (0 <_ (normh` h) <-> (normh` t) <_ ((normh` t) + (normh` h))))
36	1	sheli 10715	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (t e. A -> t e. ~H)
37		normcl 10624	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (t e. ~H -> (normh` t) e. RR)
38	36, 37	syl 12	. . . . . . . . . . . . . . . . . . . . . . 23 |- (t e. A -> (normh` t) e. RR)
39		normcl 10624	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (h e. ~H -> (normh` h) e. RR)
40	31, 39	syl 12	. . . . . . . . . . . . . . . . . . . . . . 23 |- (h e. B -> (normh` h) e. RR)
41	35, 38, 40	syl2an 503	. . . . . . . . . . . . . . . . . . . . . 22 |- ((t e. A /\ h e. B) -> (0 <_ (normh` h) <-> (normh` t) <_ ((normh` t) + (normh` h))))
42	34, 41	mpbid 212	. . . . . . . . . . . . . . . . . . . . 21 |- ((t e. A /\ h e. B) -> (normh` t) <_ ((normh` t) + (normh` h)))
43	42	adantr 425	. . . . . . . . . . . . . . . . . . . 20 |- (((t e. A /\ h e. B) /\ v e. RR) -> (normh` t) <_ ((normh` t) + (normh` h)))
44	38	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . 21 |- (((t e. A /\ h e. B) /\ v e. RR) -> (normh` t) e. RR)
45		readdcl 6455	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((normh` t) e. RR /\ (normh` h) e. RR) -> ((normh` t) + (normh` h)) e. RR)
46	45, 38, 40	syl2an 503	. . . . . . . . . . . . . . . . . . . . . 22 |- ((t e. A /\ h e. B) -> ((normh` t) + (normh` h)) e. RR)
47	46	adantr 425	. . . . . . . . . . . . . . . . . . . . 21 |- (((t e. A /\ h e. B) /\ v e. RR) -> ((normh` t) + (normh` h)) e. RR)
48		remulcl 6457	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((v e. RR /\ (normh` (t +h h)) e. RR) -> (v x. (normh` (t +h h))) e. RR)
49		hvaddcl 10514	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((t e. ~H /\ h e. ~H) -> (t +h h) e. ~H)
50	49, 36, 31	syl2an 503	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((t e. A /\ h e. B) -> (t +h h) e. ~H)
51		normcl 10624	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((t +h h) e. ~H -> (normh` (t +h h)) e. RR)
52	50, 51	syl 12	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((t e. A /\ h e. B) -> (normh` (t +h h)) e. RR)
53	48, 52	sylan2 500	. . . . . . . . . . . . . . . . . . . . . 22 |- ((v e. RR /\ (t e. A /\ h e. B)) -> (v x. (normh` (t +h h))) e. RR)
54	53	ancoms 484	. . . . . . . . . . . . . . . . . . . . 21 |- (((t e. A /\ h e. B) /\ v e. RR) -> (v x. (normh` (t +h h))) e. RR)
55		letr 6695	. . . . . . . . . . . . . . . . . . . . 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)))))
56	44, 47, 54, 55	syl111anc 1100	. . . . . . . . . . . . . . . . . . . 20 |- (((t e. A /\ h e. B) /\ v 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)))))
57	43, 56	mpand 765	. . . . . . . . . . . . . . . . . . 19 |- (((t e. A /\ h e. B) /\ v e. RR) -> (((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h))) -> (normh` t) <_ (v x. (normh` (t +h h)))))
58	57	imp 377	. . . . . . . . . . . . . . . . . 18 |- ((((t e. A /\ h e. B) /\ v e. RR) /\ ((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h)))) -> (normh` t) <_ (v x. (normh` (t +h h))))
59	58	an1rs 547	. . . . . . . . . . . . . . . . 17 |- ((((t e. A /\ h e. B) /\ ((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h)))) /\ v e. RR) -> (normh` t) <_ (v x. (normh` (t +h h))))
60	59	adantrl 430	. . . . . . . . . . . . . . . 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` t) <_ (v x. (normh` (t +h h))))
61	30, 60	eqbrtrd 3357	. . . . . . . . . . . . . . 15 |- ((((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))) <_ (v x. (normh` (t +h h))))
62	25, 61	syl5cbir 228	. . . . . . . . . . . . . 14 |- ((((t e. A /\ h e. B) /\ ((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h)))) /\ ((A i^i B) = 0H /\ v e. RR)) -> (u = (t +h h) -> (normh` (S` u)) <_ (v x. (normh` u))))
63	62	exp31 407	. . . . . . . . . . . . 13 |- ((t e. A /\ h e. B) -> (((normh` t) + (normh` h)) <_ (v x. (normh` (t +h h))) -> (((A i^i B) = 0H /\ v e. RR) -> (u = (t +h h) -> (normh` (S` u)) <_ (v x. (normh` u))))))
64	20, 63	syld 30	. . . . . . . . . . . 12 |- ((t e. A /\ h e. B) -> (A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) -> (((A i^i B) = 0H /\ v e. RR) -> (u = (t +h h) -> (normh` (S` u)) <_ (v x. (normh` u))))))
65	64	com14 42	. . . . . . . . . . 11 |- (u = (t +h h) -> (A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) -> (((A i^i B) = 0H /\ v e. RR) -> ((t e. A /\ h e. B) -> (normh` (S` u)) <_ (v x. (normh` u))))))
66	65	com4t 44	. . . . . . . . . 10 |- (((A i^i B) = 0H /\ v e. RR) -> ((t e. A /\ h e. B) -> (u = (t +h h) -> (A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) -> (normh` (S` u)) <_ (v x. (normh` u))))))
67	66	r19.23advv 2218	. . . . . . . . 9 |- (((A i^i B) = 0H /\ v e. RR) -> (E.t e. A E.h e. B u = (t +h h) -> (A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) -> (normh` (S` u)) <_ (v x. (normh` u)))))
68	67	adantl 424	. . . . . . . 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) -> (A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) -> (normh` (S` u)) <_ (v x. (normh` u)))))
69	7, 68	mpd 29	. . . . . . 7 |- ((u e. (A +H B) /\ ((A i^i B) = 0H /\ v e. RR)) -> (A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) -> (normh` (S` u)) <_ (v x. (normh` u))))
70	69	ex 402	. . . . . 6 |- (u e. (A +H B) -> (((A i^i B) = 0H /\ v e. RR) -> (A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) -> (normh` (S` u)) <_ (v x. (normh` u)))))
71	70	com3l 38	. . . . 5 |- (((A i^i B) = 0H /\ v e. RR) -> (A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) -> (u e. (A +H B) -> (normh` (S` u)) <_ (v x. (normh` u)))))
72	71	r19.21adv 2181	. . . 4 |- (((A i^i B) = 0H /\ v e. RR) -> (A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y))) -> A.u e. (A +H B)(normh` (S` u)) <_ (v x. (normh` u))))
73	72	anim2d 620	. . 3 |- (((A i^i B) = 0H /\ v e. RR) -> ((0 < v /\ A.x e. A A.y e. B ((normh` x) + (normh` y)) <_ (v x. (normh` (x +h y)))) -> (0 < v /\ A.u e. (A +H B)(normh` (S` u)) <_ (v x. (normh` u)))))
74	73	reximdva 2203	. 2 |- ((A i^i B) = 0H -> (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)))))
75	3, 74	mpcom 60	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)))) -> E.v e. RR (0 < v /\ A.u e. (A +H B)(normh` (S` u)) <_ (v x. (normh` u))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107  {crab 2108   i^i cin 2592  U.cuni 3177   class class class wbr 3338  {copab 3395  ` cfv 3998  (class class class)co 4884  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 is referenced by:  cdj3lem3b 12012  cdj3i 12013

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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