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Theorem cmbr 23069
 Description: Binary relation expressing commutes with . Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cmbr

Proof of Theorem cmbr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2490 . . . . 5
21anbi1d 686 . . . 4
3 id 20 . . . . 5
4 ineq1 3522 . . . . . 6
5 ineq1 3522 . . . . . 6
64, 5oveq12d 6085 . . . . 5
73, 6eqeq12d 2444 . . . 4
82, 7anbi12d 692 . . 3
9 eleq1 2490 . . . . 5
109anbi2d 685 . . . 4
11 ineq2 3523 . . . . . 6
12 fveq2 5714 . . . . . . 7
1312ineq2d 3529 . . . . . 6
1411, 13oveq12d 6085 . . . . 5
1514eqeq2d 2441 . . . 4
1610, 15anbi12d 692 . . 3
17 df-cm 23068 . . 3
188, 16, 17brabg 4461 . 2
1918bianabs 851 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725   cin 3306   class class class wbr 4199  cfv 5440  (class class class)co 6067  cch 22415  cort 22416   chj 22419   ccm 22422 This theorem is referenced by:  cmbri  23075  cm2j  23105 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 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390 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 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-rex 2698  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-iota 5404  df-fv 5448  df-ov 6070  df-cm 23068
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