Theorem ubthlem9 9880

Description: Lemma for ubthi 9889. Evaluate the operator value at x in terms of the operator value at Q - p.

Hypotheses

Ref	Expression
ubthlem7.1	|- X = (BaseSet` U)
ubthlem7.7	|- U e. NrmCVec
ubthlem7.n	|- L = (norm` U)
ubthlem7.g	|- G = (+v` U)
ubthlem7.m	|- M = (-v` U)
ubthlem7.r	|- R = (.s` U)
ubthlem7.z	|- Z = (0v` U)
ubthlem7.q	|- Q = (pG(((r / 2) x. (1 / (L` x)))Rx))
ubthlem9.5	|- B = (U BLnOp W)
ubthlem9.6	|- T:NN-->B
ubthlem9.8	|- W e. NrmCVec
ubthlem9.s	|- S = (.s` W)

Assertion

Ref	Expression
ubthlem9	|- ((n e. NN /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> ((T` n)` x) = (((2 / r) x. (L` x))S((T` n)` (QMp))))

Distinct variable groups:   Q,n   T,n   n,p

Proof of Theorem ubthlem9

Step	Hyp	Ref	Expression
1		ubthlem7.1	. . . . 5 |- X = (BaseSet` U)
2		ubthlem7.7	. . . . 5 |- U e. NrmCVec
3		ubthlem7.n	. . . . 5 |- L = (norm` U)
4		ubthlem7.g	. . . . 5 |- G = (+v` U)
5		ubthlem7.m	. . . . 5 |- M = (-v` U)
6		ubthlem7.r	. . . . 5 |- R = (.s` U)
7		ubthlem7.z	. . . . 5 |- Z = (0v` U)
8		ubthlem7.q	. . . . 5 |- Q = (pG(((r / 2) x. (1 / (L` x)))Rx))
9	1, 2, 3, 4, 5, 6, 7, 8	ubthlem8 9879	. . . 4 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> x = (((2 / r) x. (L` x))R(QMp)))
10	9	fveq2d 4685	. . 3 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> ((T` n)` x) = ((T` n)` (((2 / r) x. (L` x))R(QMp))))
11	10	adantl 424	. 2 |- ((n e. NN /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> ((T` n)` x) = ((T` n)` (((2 / r) x. (L` x))R(QMp))))
12		ubthlem9.8	. . . 4 |- W e. NrmCVec
13		ubthlem9.s	. . . . . 6 |- S = (.s` W)
14		eqid 1884	. . . . . 6 |- (U LnOp W) = (U LnOp W)
15	1, 6, 13, 14	lnomul 9760	. . . . 5 |- (((U e. NrmCVec /\ W e. NrmCVec /\ (T` n) e. (U LnOp W)) /\ (((2 / r) x. (L` x)) e. CC /\ (QMp) e. X)) -> ((T` n)` (((2 / r) x. (L` x))R(QMp))) = (((2 / r) x. (L` x))S((T` n)` (QMp))))
16	2, 15	mp3anl1 1185	. . . 4 |- (((W e. NrmCVec /\ (T` n) e. (U LnOp W)) /\ (((2 / r) x. (L` x)) e. CC /\ (QMp) e. X)) -> ((T` n)` (((2 / r) x. (L` x))R(QMp))) = (((2 / r) x. (L` x))S((T` n)` (QMp))))
17	12, 16	mpanl1 770	. . 3 |- (((T` n) e. (U LnOp W) /\ (((2 / r) x. (L` x)) e. CC /\ (QMp) e. X)) -> ((T` n)` (((2 / r) x. (L` x))R(QMp))) = (((2 / r) x. (L` x))S((T` n)` (QMp))))
18		ubthlem9.6	. . . . 5 |- T:NN-->B
19	18	ffvelrni 4788	. . . 4 |- (n e. NN -> (T` n) e. B)
20		ubthlem9.5	. . . . . 6 |- B = (U BLnOp W)
21	14, 20	bloln 9784	. . . . 5 |- ((U e. NrmCVec /\ W e. NrmCVec /\ (T` n) e. B) -> (T` n) e. (U LnOp W))
22	2, 12, 21	mp3an12 1181	. . . 4 |- ((T` n) e. B -> (T` n) e. (U LnOp W))
23	19, 22	syl 12	. . 3 |- (n e. NN -> (T` n) e. (U LnOp W))
24		mulcl 6456	. . . . . . 7 |- (((2 / r) e. CC /\ (L` x) e. CC) -> ((2 / r) x. (L` x)) e. CC)
25		gt0ne0 6800	. . . . . . . 8 |- ((r e. RR /\ 0 < r) -> r =/= 0)
26		2cn 7164	. . . . . . . . . 10 |- 2 e. CC
27		divcl 6901	. . . . . . . . . 10 |- ((2 e. CC /\ r e. CC /\ r =/= 0) -> (2 / r) e. CC)
28	26, 27	mp3an1 1178	. . . . . . . . 9 |- ((r e. CC /\ r =/= 0) -> (2 / r) e. CC)
29		recn 6466	. . . . . . . . 9 |- (r e. RR -> r e. CC)
30	28, 29	sylan 497	. . . . . . . 8 |- ((r e. RR /\ r =/= 0) -> (2 / r) e. CC)
31	25, 30	syldan 516	. . . . . . 7 |- ((r e. RR /\ 0 < r) -> (2 / r) e. CC)
32	1, 3	nvcl 9619	. . . . . . . . 9 |- ((U e. NrmCVec /\ x e. X) -> (L` x) e. RR)
33	2, 32	mpan 759	. . . . . . . 8 |- (x e. X -> (L` x) e. RR)
34	33	recnd 6468	. . . . . . 7 |- (x e. X -> (L` x) e. CC)
35	24, 31, 34	syl2an 503	. . . . . 6 |- (((r e. RR /\ 0 < r) /\ x e. X) -> ((2 / r) x. (L` x)) e. CC)
36	35	adantrr 431	. . . . 5 |- (((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)) -> ((2 / r) x. (L` x)) e. CC)
37	36	adantl 424	. . . 4 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> ((2 / r) x. (L` x)) e. CC)
38	1, 2, 3, 4, 5, 6, 7, 8	ubthlem7 9878	. . . . . 6 |- ((p e. X /\ (r e. RR /\ (x e. X /\ x =/= Z))) -> Q e. X)
39	38	adantrlr 437	. . . . 5 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> Q e. X)
40		simpl 346	. . . . 5 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> p e. X)
41	1, 5	nvmcl 9599	. . . . . 6 |- ((U e. NrmCVec /\ Q e. X /\ p e. X) -> (QMp) e. X)
42	2, 41	mp3an1 1178	. . . . 5 |- ((Q e. X /\ p e. X) -> (QMp) e. X)
43	39, 40, 42	syl11anc 524	. . . 4 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> (QMp) e. X)
44	37, 43	jca 310	. . 3 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> (((2 / r) x. (L` x)) e. CC /\ (QMp) e. X))
45	17, 23, 44	syl2an 503	. 2 |- ((n e. NN /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> ((T` n)` (((2 / r) x. (L` x))R(QMp))) = (((2 / r) x. (L` x))S((T` n)` (QMp))))
46	11, 45	eqtrd 1925	1 |- ((n e. NN /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> ((T` n)` x) = (((2 / r) x. (L` x))S((T` n)` (QMp))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   x. cmul 6391   / cdiv 6447  NNcn 6449   < clt 6653  2c2 7145  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  0vcn0v 9539  -vcnsb 9540  normcnm 9541   LnOp clno 9740   BLnOp cblo 9742

This theorem is referenced by:  ubthlem12 9883  ubthlem12OLD 9884

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-div 6892  df-2 7154  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-lno 9744  df-blo 9746

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