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Theorem leopg 27167
 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 6304 . . . 4
21eleq1d 2526 . . 3
31fveq1d 5874 . . . . . 6
43oveq1d 6311 . . . . 5
54breq2d 4468 . . . 4
65ralbidv 2896 . . 3
72, 6anbi12d 710 . 2
8 oveq1 6303 . . . 4
98eleq1d 2526 . . 3
108fveq1d 5874 . . . . . 6
1110oveq1d 6311 . . . . 5
1211breq2d 4468 . . . 4
1312ralbidv 2896 . . 3
149, 13anbi12d 710 . 2
15 df-leop 26897 . 2
167, 14, 15brabg 4775 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395   wcel 1819  wral 2807   class class class wbr 4456  cfv 5594  (class class class)co 6296  cc0 9509   cle 9646  chil 25962   csp 25965   chod 25983  cho 25993   cleo 26001 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-iota 5557  df-fv 5602  df-ov 6299  df-leop 26897 This theorem is referenced by:  leop  27168  leoprf2  27172
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