Theorem cmbrt 5494

Description: Binary relation expressing A commutes with B. Definition of commutes in [Kalmbach] p. 20.

Assertion

Ref	Expression
cmbrt	|- ((A e. CH /\ B e. CH) -> (A Com B <-> A = ((A i^i B) vH (A i^i (_|_` B)))))

Proof of Theorem cmbrt

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 5 |- (x = A -> (x e. CH <-> A e. CH))
2	1	anbi1d 469	. . . 4 |- (x = A -> ((x e. CH /\ y e. CH) <-> (A e. CH /\ y e. CH)))
3		id 9	. . . . 5 |- (x = A -> x = A)
4		ineq1 1638	. . . . . 6 |- (x = A -> (x i^i y) = (A i^i y))
5		ineq1 1638	. . . . . 6 |- (x = A -> (x i^i (_|_` y)) = (A i^i (_|_` y)))
6	4, 5	opreq12d 3014	. . . . 5 |- (x = A -> ((x i^i y) vH (x i^i (_|_` y))) = ((A i^i y) vH (A i^i (_|_` y))))
7	3, 6	cleq12d 1115	. . . 4 |- (x = A -> (x = ((x i^i y) vH (x i^i (_|_` y))) <-> A = ((A i^i y) vH (A i^i (_|_` y)))))
8	2, 7	anbi12d 476	. . 3 |- (x = A -> (((x e. CH /\ y e. CH) /\ x = ((x i^i y) vH (x i^i (_|_` y)))) <-> ((A e. CH /\ y e. CH) /\ A = ((A i^i y) vH (A i^i (_|_` y))))))
9		eleq1 1149	. . . . 5 |- (y = B -> (y e. CH <-> B e. CH))
10	9	anbi2d 468	. . . 4 |- (y = B -> ((A e. CH /\ y e. CH) <-> (A e. CH /\ B e. CH)))
11		ineq2 1639	. . . . . 6 |- (y = B -> (A i^i y) = (A i^i B))
12		fveq2 2832	. . . . . . 7 |- (y = B -> (_|_` y) = (_|_` B))
13	12	ineq2d 1645	. . . . . 6 |- (y = B -> (A i^i (_|_` y)) = (A i^i (_|_` B)))
14	11, 13	opreq12d 3014	. . . . 5 |- (y = B -> ((A i^i y) vH (A i^i (_|_` y))) = ((A i^i B) vH (A i^i (_|_` B))))
15	14	cleq2d 1112	. . . 4 |- (y = B -> (A = ((A i^i y) vH (A i^i (_|_` y))) <-> A = ((A i^i B) vH (A i^i (_|_` B)))))
16	10, 15	anbi12d 476	. . 3 |- (y = B -> (((A e. CH /\ y e. CH) /\ A = ((A i^i y) vH (A i^i (_|_` y)))) <-> ((A e. CH /\ B e. CH) /\ A = ((A i^i B) vH (A i^i (_|_` B))))))
17		df-cm 5493	. . 3 |- Com = {<.x, y>. | ((x e. CH /\ y e. CH) /\ x = ((x i^i y) vH (x i^i (_|_` y))))}
18	8, 16, 17	brabg 2116	. 2 |- ((A e. CH /\ B e. CH) -> (A Com B <-> ((A e. CH /\ B e. CH) /\ A = ((A i^i B) vH (A i^i (_|_` B))))))
19		ibar 487	. 2 |- ((A e. CH /\ B e. CH) -> (A = ((A i^i B) vH (A i^i (_|_` B))) <-> ((A e. CH /\ B e. CH) /\ A = ((A i^i B) vH (A i^i (_|_` B))))))
20	18, 19	bitr4d 409	1 |- ((A e. CH /\ B e. CH) -> (A Com B <-> A = ((A i^i B) vH (A i^i (_|_` B)))))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092   i^i cin 1486   class class class wbr 2054  ` cfv 2422  (class class class)co 3001  CHcch 4968  _|_cort 4969   vH chj 4972   Com ccm 4975

This theorem is referenced by:  cmbr 5499

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-cm 5493

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