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Theorem idcnop 23476
 Description: The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
idcnop

Proof of Theorem idcnop
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oi 5705 . . 3
2 f1of 5666 . . 3
31, 2ax-mp 8 . 2
4 id 20 . . . 4
5 fvresi 5916 . . . . . . . . 9
6 fvresi 5916 . . . . . . . . 9
75, 6oveqan12rd 6093 . . . . . . . 8
87fveq2d 5724 . . . . . . 7
98breq1d 4214 . . . . . 6
109biimprd 215 . . . . 5
1110ralrimiva 2781 . . . 4
12 breq2 4208 . . . . . . 7
1312imbi1d 309 . . . . . 6
1413ralbidv 2717 . . . . 5
1514rspcev 3044 . . . 4
164, 11, 15syl2anr 465 . . 3
1716rgen2 2794 . 2
18 elcnop 23352 . 2
193, 17, 18mpbir2an 887 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wcel 1725  wral 2697  wrex 2698   class class class wbr 4204   cid 4485   cres 4872  wf 5442  wf1o 5445  cfv 5446  (class class class)co 6073   clt 9112  crp 10604  chil 22414  cno 22418   cmv 22420  ccop 22441 This theorem is referenced by:  nmcopex  23524  nmcoplb  23525 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-hilex 22494 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-cnop 23335
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