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Theorem bloval 25400
 Description: The class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
bloval.3
bloval.4
bloval.5
Assertion
Ref Expression
bloval
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem bloval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bloval.5 . 2
2 oveq1 6291 . . . 4
3 oveq1 6291 . . . . . 6
43fveq1d 5868 . . . . 5
54breq1d 4457 . . . 4
62, 5rabeqbidv 3108 . . 3
7 oveq2 6292 . . . . 5
8 bloval.4 . . . . 5
97, 8syl6eqr 2526 . . . 4
10 oveq2 6292 . . . . . . 7
11 bloval.3 . . . . . . 7
1210, 11syl6eqr 2526 . . . . . 6
1312fveq1d 5868 . . . . 5
1413breq1d 4457 . . . 4
159, 14rabeqbidv 3108 . . 3
16 df-blo 25365 . . 3
17 ovex 6309 . . . . 5
188, 17eqeltri 2551 . . . 4
1918rabex 4598 . . 3
206, 15, 16, 19ovmpt2 6422 . 2
211, 20syl5eq 2520 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1379   wcel 1767  crab 2818  cvv 3113   class class class wbr 4447  cfv 5588  (class class class)co 6284   cpnf 9625   clt 9628  cnv 25181   clno 25359  cnmoo 25360   cblo 25361 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-blo 25365 This theorem is referenced by:  isblo  25401  hhbloi  26525
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