Theorem cmbri 28449

Description: Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pjoml2.1	|- A e. CH
pjoml2.2	|- B e. CH

Assertion

Ref	Expression
cmbri	|- ( A C_H B <-> A = ( ( A i^i B ) vH ( A i^i ( _|_ ` B ) ) ) )

Proof of Theorem cmbri

Step	Hyp	Ref	Expression
1		pjoml2.1	. 2 |- A e. CH
2		pjoml2.2	. 2 |- B e. CH
3		cmbr 28443	. 2 |- ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> A = ( ( A i^i B ) vH ( A i^i ( _|_ ` B ) ) ) ) )
4	1, 2, 3	mp2an 708	1 |- ( A C_H B <-> A = ( ( A i^i B ) vH ( A i^i ( _|_ ` B ) ) ) )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   i^i cin 3573   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CHcch 27786   _|_cort 27787   vH chj 27790   C_H ccm 27793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653  df-cm 28442

This theorem is referenced by:  cmcmlem  28450  cmcm2i  28452  cmbr2i  28455  cmbr3i  28459  pjclem1  29054  pjci  29059