Theorem ubthlem9 8537

Description: Lemma for ubthi 8544. Evaluate the operator value at x in terms of the operator value at Q - p.

Hypotheses

Ref	Expression
ubthlem7.1	|- X = (Base` 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 = (Base` 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 8536	. . . 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 3728	. . 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 388	. 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 1475	. . . . . 6 |- (U LnOp W) = (U LnOp W)
15	1, 6, 13, 14	lnomul 8421	. . . . 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 910	. . . 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 706	. . 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 3815	. . . 4 |- (n e. NN -> (T` n) e. B)
20		ubthlem9.5	. . . . . 6 |- B = (U BLnOp W)
21	14, 20	bloln 8444	. . . . 5 |- ((U e. NrmCVec /\ W e. NrmCVec /\ (T` n) e. B) -> (T` n) e. (U LnOp W))
22	2, 12, 21	mp3an12 906	. . . 4 |- ((T` n) e. B -> (T` n) e. (U LnOp W))
23	19, 22	syl 10	. . 3 |- (n e. NN -> (T` n) e. (U LnOp W))
24		axmulcl 5273	. . . . . . 7 |- (((2 / r) e. CC /\ (L` x) e. CC) -> ((2 / r) x. (L` x)) e. CC)
25		gt0ne0t 5618	. . . . . . . 8 |- ((r e. RR /\ 0 < r) -> r =/= 0)
26		2cn 5980	. . . . . . . . . 10 |- 2 e. CC
27		divclt 5712	. . . . . . . . . 10 |- ((2 e. CC /\ r e. CC /\ r =/= 0) -> (2 / r) e. CC)
28	26, 27	mp3an1 903	. . . . . . . . 9 |- ((r e. CC /\ r =/= 0) -> (2 / r) e. CC)
29		recnt 5313	. . . . . . . . 9 |- (r e. RR -> r e. CC)
30	28, 29	sylan 448	. . . . . . . 8 |- ((r e. RR /\ r =/= 0) -> (2 / r) e. CC)
31	25, 30	syldan 467	. . . . . . 7 |- ((r e. RR /\ 0 < r) -> (2 / r) e. CC)
32	1, 3	nvcl 8287	. . . . . . . . 9 |- ((U e. NrmCVec /\ x e. X) -> (L` x) e. RR)
33	2, 32	mpan 695	. . . . . . . 8 |- (x e. X -> (L` x) e. RR)
34	33	recnd 5315	. . . . . . 7 |- (x e. X -> (L` x) e. CC)
35	24, 31, 34	syl2an 454	. . . . . 6 |- (((r e. RR /\ 0 < r) /\ x e. X) -> ((2 / r) x. (L` x)) e. CC)
36	35	adantrr 395	. . . . 5 |- (((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)) -> ((2 / r) x. (L` x)) e. CC)
37	36	adantl 388	. . . 4 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> ((2 / r) x. (L` x)) e. CC)
38	1, 5	nvmcl 8267	. . . . . 6 |- ((U e. NrmCVec /\ Q e. X /\ p e. X) -> (QMp) e. X)
39	2, 38	mp3an1 903	. . . . 5 |- ((Q e. X /\ p e. X) -> (QMp) e. X)
40	1, 2, 3, 4, 5, 6, 7, 8	ubthlem7 8535	. . . . . 6 |- ((p e. X /\ (r e. RR /\ (x e. X /\ x =/= Z))) -> Q e. X)
41	40	adantrlr 401	. . . . 5 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> Q e. X)
42		pm3.26 319	. . . . 5 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> p e. X)
43	39, 41, 42	sylanc 471	. . . 4 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> (QMp) e. X)
44	37, 43	jca 288	. . 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 454	. 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 1507	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 223   = wceq 956   e. wcel 958   =/= wne 1585   class class class wbr 2619  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232  RRcr 5233  0cc0 5234  1c1 5235   x. cmul 5239   / cdiv 5294  NNcn 5296   < clt 5486  2c2 5961  NrmCVeccnv 8203  +vcpv 8204  Basecba 8205  .scns 8206  0vcn0v 8207  -vcnsb 8208  normcnm 8209   LnOp clno 8401   BLnOp cblo 8403

This theorem is referenced by:  ubthlem12 8540

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-sup 4574  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-2 5970  df-sqr 6670  df-re 6751  df-im 6752  df-cj 6753  df-abs 6754  df-grp 8037  df-gid 8038  df-ginv 8039  df-gdiv 8040  df-abl 8100  df-vc 8165  df-nv 8211  df-va 8214  df-ba 8215  df-sm 8216  df-0v 8217  df-vs 8218  df-nm 8219  df-lno 8405  df-blo 8407

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