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Theorem cnopc 23421
 Description: Basic continuity property of a continuous Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
cnopc
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem cnopc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elcnop 23365 . . . 4
21simprbi 452 . . 3
3 oveq2 6092 . . . . . . . 8
43fveq2d 5735 . . . . . . 7
54breq1d 4225 . . . . . 6
6 fveq2 5731 . . . . . . . . 9
76oveq2d 6100 . . . . . . . 8
87fveq2d 5735 . . . . . . 7
98breq1d 4225 . . . . . 6
105, 9imbi12d 313 . . . . 5
1110rexralbidv 2751 . . . 4
12 breq2 4219 . . . . . 6
1312imbi2d 309 . . . . 5
1413rexralbidv 2751 . . . 4
1511, 14rspc2v 3060 . . 3
162, 15syl5com 29 . 2
17163impib 1152 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707  wrex 2708   class class class wbr 4215  wf 5453  cfv 5457  (class class class)co 6084   clt 9125  crp 10617  chil 22427  cno 22431   cmv 22433  ccop 22454 This theorem is referenced by:  nmcopexi  23535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-hilex 22507 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-cnop 23348
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