Theorem elunop2 11367

Description: An operator is unitary iff it is linear, onto, and idempotent in the norm. Similar to theorem in [AkhiezerGlazman] p. 73, and its converse.

Assertion

Ref	Expression
elunop2	|- (T e. UniOp <-> (T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)))

Distinct variable group:   x,T

Proof of Theorem elunop2

Step	Hyp	Ref	Expression
1		unoplin 11273	. . 3 |- (T e. UniOp -> T e. LinOp)
2		elunop 11228	. . . 4 |- (T e. UniOp <-> (T:~H-onto->~H /\ A.x e. ~H A.y e. ~H ((T` x) .ih (T` y)) = (x .ih y)))
3	2	pm3.26bi 347	. . 3 |- (T e. UniOp -> T:~H-onto->~H)
4		unopnorm 11270	. . . 4 |- ((T e. UniOp /\ x e. ~H) -> (normh` (T` x)) = (normh` x))
5	4	r19.21aiva 2010	. . 3 |- (T e. UniOp -> A.x e. ~H (normh` (T` x)) = (normh` x))
6	1, 3, 5	3jca 929	. 2 |- (T e. UniOp -> (T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)))
7		eleq1 1794	. . 3 |- (T = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> (T e. UniOp <-> if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) e. UniOp))
8		eleq1 1794	. . . . . . 7 |- (T = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> (T e. LinOp <-> if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) e. LinOp))
9		foeq1 4424	. . . . . . 7 |- (T = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> (T:~H-onto->~H <-> if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)):~H-onto->~H))
10		fveq1 4491	. . . . . . . . . . 11 |- (T = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> (T` y) = (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y))
11	10	fveq2d 4496	. . . . . . . . . 10 |- (T = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> (normh` (T` y)) = (normh` (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y)))
12	11	eqeq1d 1729	. . . . . . . . 9 |- (T = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> ((normh` (T` y)) = (normh` y) <-> (normh` (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y)) = (normh` y)))
13	12	ralbidv 1957	. . . . . . . 8 |- (T = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> (A.y e. ~H (normh` (T` y)) = (normh` y) <-> A.y e. ~H (normh` (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y)) = (normh` y)))
14		fveq2 4492	. . . . . . . . . . 11 |- (x = y -> (T` x) = (T` y))
15	14	fveq2d 4496	. . . . . . . . . 10 |- (x = y -> (normh` (T` x)) = (normh` (T` y)))
16		fveq2 4492	. . . . . . . . . 10 |- (x = y -> (normh` x) = (normh` y))
17	15, 16	eqeq12d 1736	. . . . . . . . 9 |- (x = y -> ((normh` (T` x)) = (normh` x) <-> (normh` (T` y)) = (normh` y)))
18	17	cbvralv 2113	. . . . . . . 8 |- (A.x e. ~H (normh` (T` x)) = (normh` x) <-> A.y e. ~H (normh` (T` y)) = (normh` y))
19	13, 18	syl5bb 588	. . . . . . 7 |- (T = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> (A.x e. ~H (normh` (T` x)) = (normh` x) <-> A.y e. ~H (normh` (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y)) = (normh` y)))
20	8, 9, 19	3anbi123d 1015	. . . . . 6 |- (T = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> ((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)) <-> (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) e. LinOp /\ if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)):~H-onto->~H /\ A.y e. ~H (normh` (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y)) = (normh` y))))
21		eleq1 1794	. . . . . . 7 |- ((_I |` ~H) = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> ((_I |` ~H) e. LinOp <-> if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) e. LinOp))
22		foeq1 4424	. . . . . . 7 |- ((_I |` ~H) = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> ((_I |` ~H):~H-onto->~H <-> if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)):~H-onto->~H))
23		fveq1 4491	. . . . . . . . . 10 |- ((_I |` ~H) = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> ((_I |` ~H)` y) = (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y))
24	23	fveq2d 4496	. . . . . . . . 9 |- ((_I |` ~H) = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> (normh` ((_I |` ~H)` y)) = (normh` (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y)))
25	24	eqeq1d 1729	. . . . . . . 8 |- ((_I |` ~H) = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> ((normh` ((_I |` ~H)` y)) = (normh` y) <-> (normh` (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y)) = (normh` y)))
26	25	ralbidv 1957	. . . . . . 7 |- ((_I |` ~H) = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> (A.y e. ~H (normh` ((_I |` ~H)` y)) = (normh` y) <-> A.y e. ~H (normh` (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y)) = (normh` y)))
27	21, 22, 26	3anbi123d 1015	. . . . . 6 |- ((_I |` ~H) = if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) -> (((_I |` ~H) e. LinOp /\ (_I |` ~H):~H-onto->~H /\ A.y e. ~H (normh` ((_I |` ~H)` y)) = (normh` y)) <-> (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) e. LinOp /\ if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)):~H-onto->~H /\ A.y e. ~H (normh` (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y)) = (normh` y))))
28		idlnop 11346	. . . . . . 7 |- (_I |` ~H) e. LinOp
29		f1oi 4482	. . . . . . . 8 |- (_I |` ~H):~H-1-1-onto->~H
30		f1ofo 4454	. . . . . . . 8 |- ((_I |` ~H):~H-1-1-onto->~H -> (_I |` ~H):~H-onto->~H)
31	29, 30	ax-mp 7	. . . . . . 7 |- (_I |` ~H):~H-onto->~H
32		fvresi 4630	. . . . . . . . 9 |- (y e. ~H -> ((_I |` ~H)` y) = y)
33	32	fveq2d 4496	. . . . . . . 8 |- (y e. ~H -> (normh` ((_I |` ~H)` y)) = (normh` y))
34	33	rgen 1993	. . . . . . 7 |- A.y e. ~H (normh` ((_I |` ~H)` y)) = (normh` y)
35	28, 31, 34	3pm3.2i 927	. . . . . 6 |- ((_I |` ~H) e. LinOp /\ (_I |` ~H):~H-onto->~H /\ A.y e. ~H (normh` ((_I |` ~H)` y)) = (normh` y))
36	20, 27, 35	elimhyp 2845	. . . . 5 |- (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) e. LinOp /\ if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)):~H-onto->~H /\ A.y e. ~H (normh` (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y)) = (normh` y))
37	36	3simp1i 881	. . . 4 |- if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) e. LinOp
38	36	3simp2i 882	. . . 4 |- if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)):~H-onto->~H
39	36	3simp3i 883	. . . 4 |- A.y e. ~H (normh` (if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H))` y)) = (normh` y)
40	37, 38, 39	lnopunii 11366	. . 3 |- if((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)), T, (_I |` ~H)) e. UniOp
41	7, 40	dedth 2835	. 2 |- ((T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)) -> T e. UniOp)
42	6, 41	impbii 173	1 |- (T e. UniOp <-> (T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)))

Syntax hints:   <-> wb 162   /\ w3a 855   = wceq 1136   e. wcel 1138  A.wral 1939  ifcif 2806  _Icid 3397   |` cres 3799  -onto->wfo 3807  -1-1-onto->wf1o 3808  ` cfv 3809  (class class class)co 4695  ~Hchil 10212   .ih csp 10217  normhcno 10218  LinOpclo 10240  UniOpcuo 10242

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-9 1145  ax-10 1146  ax-11 1147  ax-12 1148  ax-13 1149  ax-14 1150  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426  ax-ext 1702  ax-rep 3243  ax-sep 3253  ax-nul 3260  ax-pow 3296  ax-pr 3339  ax-un 3601  ax-inf2 5540  ax-hilex 10293  ax-hfvadd 10294  ax-hvcom 10295  ax-hvass 10296  ax-hv0cl 10297  ax-hvaddid 10298  ax-hfvmul 10299  ax-hvmulid 10300  ax-hvdistr2 10303  ax-hvmul0 10304  ax-hfi 10371  ax-his1 10374  ax-his2 10375  ax-his3 10376  ax-his4 10377

This theorem depends on definitions:  df-bi 163  df-or 240  df-an 241  df-3or 856  df-3an 857  df-ex 1165  df-sb 1374  df-eu 1613  df-mo 1614  df-clab 1709  df-cleq 1714  df-clel 1717  df-ne 1856  df-nel 1857  df-ral 1943  df-rex 1944  df-reu 1945  df-rab 1946  df-v 2127  df-sbc 2287  df-csb 2374  df-dif 2430  df-un 2433  df-in 2436  df-ss 2438  df-pss 2440  df-nul 2702  df-if 2807  df-pw 2859  df-sn 2873  df-pr 2874  df-tp 2876  df-op 2877  df-uni 3000  df-int 3037  df-iun 3079  df-br 3159  df-opab 3214  df-tr 3230  df-eprel 3398  df-id 3401  df-po 3406  df-so 3419  df-fr 3440  df-we 3459  df-ord 3475  df-on 3476  df-lim 3477  df-suc 3478  df-om 3761  df-xp 3811  df-rel 3812  df-cnv 3813  df-co 3814  df-dm 3815  df-rn 3816  df-res 3817  df-ima 3818  df-fun 3819  df-fn 3820  df-f 3821  df-f1 3822  df-fo 3823  df-f1o 3824  df-fv 3825  df-opr 4697  df-oprab 4698  df-mpt 4817  df-1st 4831  df-2nd 4832  df-iota 4900  df-rdg 4951  df-1o 4988  df-oadd 4990  df-omul 4991  df-er 5129  df-ec 5131  df-qs 5134  df-en 5238  df-dom 5239  df-sdom 5240  df-undef 5367  df-riota 5371  df-sup 5474  df-ni 5948  df-pli 5949  df-mi 5950  df-lti 5951  df-plpq 5983  df-mpq 5984  df-enq 5985  df-nq 5986  df-plq 5987  df-mq 5988  df-rq 5989  df-ltq 5990  df-1q 5991  df-np 6034  df-1p 6035  df-plp 6036  df-mp 6037  df-ltp 6038  df-plpr 6112  df-mpr 6113  df-enr 6114  df-nr 6115  df-plr 6116  df-mr 6117  df-ltr 6118  df-0r 6119  df-1r 6120  df-m1r 6121  df-c 6188  df-0 6189  df-1 6190  df-i 6191  df-r 6192  df-plus 6193  df-mul 6194  df-lt 6195  df-sub 6307  df-neg 6309  df-pnf 6450  df-mnf 6451  df-xr 6452  df-ltxr 6453  df-le 6454  df-div 6688  df-n 6903  df-2 6949  df-n0 7104  df-z 7140  df-seq1 7516  df-exp 7607  df-sqr 7715  df-re 7796  df-im 7797  df-cj 7798  df-hnorm 10261  df-hvsub 10264  df-lnop 11196  df-unop 11198

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