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Theorem leopg 23625
 Description: Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
Assertion
Ref Expression
leopg
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem leopg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6089 . . . 4
21eleq1d 2502 . . 3
31fveq1d 5730 . . . . . 6
43oveq1d 6096 . . . . 5
54breq2d 4224 . . . 4
65ralbidv 2725 . . 3
72, 6anbi12d 692 . 2
8 oveq1 6088 . . . 4
98eleq1d 2502 . . 3
108fveq1d 5730 . . . . . 6
1110oveq1d 6096 . . . . 5
1211breq2d 4224 . . . 4
1312ralbidv 2725 . . 3
149, 13anbi12d 692 . 2
15 df-leop 23355 . 2
167, 14, 15brabg 4474 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2705   class class class wbr 4212  cfv 5454  (class class class)co 6081  cc0 8990   cle 9121  chil 22422   csp 22425   chod 22443  cho 22453   cleo 22461 This theorem is referenced by:  leop  23626  leoprf2  23630 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-iota 5418  df-fv 5462  df-ov 6084  df-leop 23355
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