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Theorem leopg 22698
 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 5828 . . . 4
21eleq1d 2350 . . 3
31fveq1d 5488 . . . . . 6
43oveq1d 5835 . . . . 5
54breq2d 4036 . . . 4
65ralbidv 2564 . . 3
72, 6anbi12d 691 . 2
8 oveq1 5827 . . . 4
98eleq1d 2350 . . 3
108fveq1d 5488 . . . . . 6
1110oveq1d 5835 . . . . 5
1211breq2d 4036 . . . 4
1312ralbidv 2564 . . 3
149, 13anbi12d 691 . 2
15 df-leop 22428 . 2
167, 14, 15brabg 4283 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1623   wcel 1685  wral 2544   class class class wbr 4024  cfv 5221  (class class class)co 5820  cc0 8733   cle 8864  chil 21495   csp 21498   chod 21516  cho 21526   cleo 21534 This theorem is referenced by:  leop  22699  leoprf2  22703 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-ov 5823  df-leop 22428
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