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Theorem cmbr 26628
 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 2529 . . . . 5
21anbi1d 704 . . . 4
3 id 22 . . . . 5
4 ineq1 3689 . . . . . 6
5 ineq1 3689 . . . . . 6
64, 5oveq12d 6314 . . . . 5
73, 6eqeq12d 2479 . . . 4
82, 7anbi12d 710 . . 3
9 eleq1 2529 . . . . 5
109anbi2d 703 . . . 4
11 ineq2 3690 . . . . . 6
12 fveq2 5872 . . . . . . 7
1312ineq2d 3696 . . . . . 6
1411, 13oveq12d 6314 . . . . 5
1514eqeq2d 2471 . . . 4
1610, 15anbi12d 710 . . 3
17 df-cm 26627 . . 3
188, 16, 17brabg 4775 . 2
1918bianabs 880 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395   wcel 1819   cin 3470   class class class wbr 4456  cfv 5594  (class class class)co 6296  cch 25972  cort 25973   chj 25976   ccm 25979 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-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-cm 26627 This theorem is referenced by:  cmbri  26634  cm2j  26664
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