Theorem ubthlem9 8533

Description: Lemma for ubthi 8540. 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 8532	. . . 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 3734	. . 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 390	. 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 1478	. . . . . 6 |- (U LnOp W) = (U LnOp W)
15	1, 6, 13, 14	lnomul 8417	. . . . 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 912	. . . 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 708	. . 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 3821	. . . 4 |- (n e. NN -> (T` n) e. B)
20		ubthlem9.5	. . . . . 6 |- B = (U BLnOp W)
21	14, 20	bloln 8440	. . . . 5 |- ((U e. NrmCVec /\ W e. NrmCVec /\ (T` n) e. B) -> (T` n) e. (U LnOp W))
22	2, 12, 21	mp3an12 908	. . . 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 5285	. . . . . . 7 |- (((2 / r) e. CC /\ (L` x) e. CC) -> ((2 / r) x. (L` x)) e. CC)
25		gt0ne0t 5630	. . . . . . . 8 |- ((r e. RR /\ 0 < r) -> r =/= 0)
26		2cn 5982	. . . . . . . . . 10 |- 2 e. CC
27		divclt 5724	. . . . . . . . . 10 |- ((2 e. CC /\ r e. CC /\ r =/= 0) -> (2 / r) e. CC)
28	26, 27	mp3an1 905	. . . . . . . . 9 |- ((r e. CC /\ r =/= 0) -> (2 / r) e. CC)
29		recnt 5325	. . . . . . . . 9 |- (r e. RR -> r e. CC)
30	28, 29	sylan 450	. . . . . . . 8 |- ((r e. RR /\ r =/= 0) -> (2 / r) e. CC)
31	25, 30	syldan 469	. . . . . . 7 |- ((r e. RR /\ 0 < r) -> (2 / r) e. CC)
32	1, 3	nvcl 8283	. . . . . . . . 9 |- ((U e. NrmCVec /\ x e. X) -> (L` x) e. RR)
33	2, 32	mpan 697	. . . . . . . 8 |- (x e. X -> (L` x) e. RR)
34	33	recnd 5327	. . . . . . 7 |- (x e. X -> (L` x) e. CC)
35	24, 31, 34	syl2an 456	. . . . . 6 |- (((r e. RR /\ 0 < r) /\ x e. X) -> ((2 / r) x. (L` x)) e. CC)
36	35	adantrr 397	. . . . 5 |- (((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)) -> ((2 / r) x. (L` x)) e. CC)
37	36	adantl 390	. . . 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 8263	. . . . . 6 |- ((U e. NrmCVec /\ Q e. X /\ p e. X) -> (QMp) e. X)
39	2, 38	mp3an1 905	. . . . 5 |- ((Q e. X /\ p e. X) -> (QMp) e. X)
40	1, 2, 3, 4, 5, 6, 7, 8	ubthlem7 8531	. . . . . 6 |- ((p e. X /\ (r e. RR /\ (x e. X /\ x =/= Z))) -> Q e. X)
41	40	adantrlr 403	. . . . 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 473	. . . 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 456	. 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 1510	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 958   e. wcel 960   =/= wne 1588   class class class wbr 2624  -->wf 3184  ` cfv 3188  (class class class)co 3969  CCcc 5244  RRcr 5245  0cc0 5246  1c1 5247   x. cmul 5251   / cdiv 5306  NNcn 5308   < clt 5498  2c2 5963  NrmCVeccnv 8199  +vcpv 8200  Basecba 8201  .scns 8202  0vcn0v 8203  -vcnsb 8204  normcnm 8205   LnOp clno 8397   BLnOp cblo 8399

This theorem is referenced by:  ubthlem12 8536

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-2 5972  df-sqr 6671  df-re 6752  df-im 6753  df-cj 6754  df-abs 6755  df-grp 8034  df-gid 8035  df-ginv 8036  df-gdiv 8037  df-abl 8096  df-vc 8161  df-nv 8207  df-va 8210  df-ba 8211  df-sm 8212  df-0v 8213  df-vs 8214  df-nm 8215  df-lno 8401  df-blo 8403

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